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2.1: Types of Data Representation

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Two common types of graphic displays are bar charts and histograms. Both bar charts and histograms use vertical or horizontal bars to represent the number of data points in each category or interval. The main difference graphically is that in a  bar chart  there are spaces between the bars and in a  histogram  there are not spaces between the bars. Why does this subtle difference exist and what does it imply about graphic displays in general?

Displaying Data

It is often easier for people to interpret relative sizes of data when that data is displayed graphically. Note that a  categorical variable  is a variable that can take on one of a limited number of values and a  quantitative variable  is a variable that takes on numerical values that represent a measurable quantity. Examples of categorical variables are tv stations, the state someone lives in, and eye color while examples of quantitative variables are the height of students or the population of a city. There are a few common ways of displaying data graphically that you should be familiar with. 

A  pie chart  shows the relative proportions of data in different categories.  Pie charts  are excellent ways of displaying categorical data with easily separable groups. The following pie chart shows six categories labeled A−F.  The size of each pie slice is determined by the central angle. Since there are 360 o  in a circle, the size of the central angle θ A  of category A can be found by:

CK-12 Foundation -  https://www.flickr.com/photos/slgc/16173880801  - CCSA

A  bar chart  displays frequencies of categories of data. The bar chart below has 5 categories, and shows the TV channel preferences for 53 adults. The horizontal axis could have also been labeled News, Sports, Local News, Comedy, Action Movies. The reason why the bars are separated by spaces is to emphasize the fact that they are categories and not continuous numbers. For example, just because you split your time between channel 8 and channel 44 does not mean on average you watch channel 26. Categories can be numbers so you need to be very careful.

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A  histogram  displays frequencies of quantitative data that has been sorted into intervals. The following is a histogram that shows the heights of a class of 53 students. Notice the largest category is 56-60 inches with 18 people.

A  boxplot  (also known as a  box and whiskers plot ) is another way to display quantitative data. It displays the five 5 number summary (minimum, Q1,  median , Q3, maximum). The box can either be vertically or horizontally displayed depending on the labeling of the axis. The box does not need to be perfectly symmetrical because it represents data that might not be perfectly symmetrical.

Earlier, you were asked about the difference between histograms and bar charts. The reason for the space in bar charts but no space in histograms is bar charts graph categorical variables while histograms graph quantitative variables. It would be extremely improper to forget the space with bar charts because you would run the risk of implying a spectrum from one side of the chart to the other. Note that in the bar chart where TV stations where shown, the station numbers were not listed horizontally in order by size. This was to emphasize the fact that the stations were categories.

Create a boxplot of the following numbers in your calculator.

8.5, 10.9, 9.1, 7.5, 7.2, 6, 2.3, 5.5

Enter the data into L1 by going into the Stat menu.

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Then turn the statplot on and choose boxplot.

Use Zoomstat to automatically center the window on the boxplot.

Create a pie chart to represent the preferences of 43 hungry students.

• Other – 5
• Burritos – 7
• Burgers – 9
• Pizza – 22

Create a bar chart representing the preference for sports of a group of 23 people.

• Football – 12
• Baseball – 10
• Basketball – 8
• Hockey – 3

Create a histogram for the income distribution of 200 million people.

• Below $50,000 is 100 million people
• Between $50,000 and $100,000 is 50 million people
• Between $100,000 and $150,000 is 40 million people
• Above $150,000 is 10 million people

1. What types of graphs show categorical data?

2. What types of graphs show quantitative data?

A math class of 30 students had the following grades:

3. Create a bar chart for this data.

4. Create a pie chart for this data.

5. Which graph do you think makes a better visual representation of the data?

A set of 20 exam scores is 67, 94, 88, 76, 85, 93, 55, 87, 80, 81, 80, 61, 90, 84, 75, 93, 75, 68, 100, 98

6. Create a histogram for this data. Use your best judgment to decide what the intervals should be.

7. Find the  five number summary  for this data.

8. Use the  five number summary  to create a boxplot for this data.

9. Describe the data shown in the boxplot below.

10. Describe the data shown in the histogram below.

A math class of 30 students has the following eye colors:

11. Create a bar chart for this data.

12. Create a pie chart for this data.

13. Which graph do you think makes a better visual representation of the data?

14. Suppose you have data that shows the breakdown of registered republicans by state. What types of graphs could you use to display this data?

15. From which types of graphs could you obtain information about the spread of the data? Note that spread is a measure of how spread out all of the data is.

Review (Answers)

To see the Review answers, open this  PDF file  and look for section 15.4. 

Additional Resources

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Practice: Types of Data Representation

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2: Graphical Representations of Data

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In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.

• 2.1: Introduction In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. Our emphasis will be on histograms and box plots.
• 2.2: Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs A stem-and-leaf plot is a way to plot data and look at the distribution, where all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends.  A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories.
• 2.3: Histograms, Frequency Polygons, and Time Series Graphs A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat.
• 2.4: Using Excel to Create Graphs Using technology to create graphs will make the graphs faster to create, more precise, and give the ability to use larger amounts of data. This section focuses on using Excel to create graphs.
• 2.5: Graphs that Deceive It's common to see graphs displayed in a misleading manner in social media and other instances. This could be done purposefully to make a point, or it could be accidental. Either way, it's important to recognize these instances to ensure you are not misled.
• 2.E: Graphical Representations of Data (Exercises) These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

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Representing data

Here you will learn about representing data, including how to create and interpret the different tables, charts, diagrams and graphs you can use to represent data.

Students first learn how to represent and interpret data in the first grade and expand their knowledge as they progress through elementary school, middle school and high school. Being data literate is essential for success in the real world.

What is representing data?

Representing data allows you to display and interpret collected data. Data literacy is essential to understanding the world around us.

There are different types of data that can be represented in different formats.

Let’s take a look at some of the different ways to represent data.

A histogram is a graphical representation used to display quantitative continuous data (numeric data). The graphical display uses bars that are different heights and each bar groups numbers into ranges. The horizontal axis represents the numerical range, and the vertical axis represents the frequency, which is the number of times the data falls in the particular numerical range.

For example, the frequency table shows the salaries of 157 employees at a small company. Create a histogram from the data.

Step by step guide : Histograms

Stem and leaf plot

A stem and leaf plot is a method of organizing numerical data based on the place value of the numbers.

Each number is split into two parts.

The first digit(s) form the stem,

The last digit forms the leaf.

For example, the data below represents the age of all the employees at Millstown Elementary School. Create a stem and leaf plot from the data

Step by step guide: Stem and Leaf Plot

Frequency distribution

A frequency distribution is a way of representing data from a frequency distribution table. Frequency distributions can be represented by frequency graphs such as pie graphs, bar graphs, line plots, vertical line graphs, and/or frequency polygons where the frequency is displayed on the vertical axis (y- axis ).

There are two types of data that can be represented using a frequency graph.

Categorical data – data that are words rather than numbers, for example, colors, makes of cars, types of music.

Numerical data – data that is in the form of numbers. There are two types of numerical data.

Here are some examples of frequency graphs:

Step by step guide: Frequency distribution

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Cumulative frequency

A cumulative frequency graph, also called an ogive, shows the frequencies of each category accumulated together. This allows you to analyze the distribution of the data in more detail than if you used a frequency polygon and calculate statistics.

Here is an example of a cumulative frequency graph along with the data set.

Similar to a frequency graph, the horizontal axis (x- axis ) represents the numerical interval and the vertical axis (y- axis ) represents the cumulative frequency.

A pie chart also known as a circle chart or pie graph is a visual representation of data that is made by a circle divided into sectors (pie slices). Each sector represents a part of the whole (whole pie). Pie charts are used to represent categorical data.

Here is an example of a pie chart that displays students’ favorite subjects in percentages at a particular school. Notice how each sector represents a percent of the whole circle.

The sectors of the circle graphs can be represented as the number data points in the category or as percents.

Step by step guide: Pie chart

A box plot also known as a box and whisker plot is a graph that represents the five number summary of a set of data.

The five number summary includes the following:

• Lowest value or smallest value
• Lower quartile or first quartile (Q1)
• Median , middle number , middle value , or second quartile (M)
• Upper quartile or third quartile (Q3)
• Highest value or largest value

Here is an example of a box plot for the given data set:

7, \, 4, \, 5, \, 6, \, 3, \, 4, \, 7, \, 10, \, 11, \, 8, \, 9, \, 2, \, 3, \, 8, \, 11, \, 12, \, 10

Like with a stem and leaf plot, it is helpful to put the data points in order from least to greatest.

2, \, 3, \, 3, \, 4, \, 4, \, 5, \, 6, \, 7, \, 7, \, 8, \, 8, \, 9, \, 10, \, 10, \, 11, \, 11, \, 12

Quartiles are values that divide the data set into three quarters. From the box plot, you can see that the first quartile is the value where the 25\% of the data set falls under.

The median or the second quartile is the value where 50\% of the data falls under and the third quartile (Q3) is the value where 75\% of the data set falls under.

From the box plot, you can also determine the interquartile range (IQR) which is found by finding the difference between Q1 and Q3.

Step by step guide: Box plot

Step by step guide: Quartile

Step by step guide: Interquartile range

Common Core State Standards

How does this relate to 6 th and 7 th grade math?

• Grade 6 – Statistics and Probability (6.SP.B.4) Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
• Grade 6 – Statistics and Probability (6.SP.A.3) Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
• Grade 6 – Ratios and Proportional Relationships (6.RP.3.c) Find a percent of a quantity as a rate per 100 (for example, 30\% of a quantity means \cfrac{30}{100} times the quantity); solve problems involving finding the whole, given a part and the percent.
• Grade 7 – Statistics and Probability (7.SP.B.3) Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10~{cm} greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
• Grade 7 – Statistics and Probability (7.SP.B.4) Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

How to represent data

For a more detailed step-by-step approach on how to represent data, go to the links highlighted in the “What is representing data” section above or follow the examples below.

Representing data examples

Example 1: stem and leaf diagram.

The data below represents the heights of trees at a tree farm in feet.

20ft, \, 15ft, \, 17ft, \, 29ft, \, 22ft, \, 13ft, \, 30ft, \, 25ft, \, 18ft, \, 27ft, \, 31ft

• Order the numbers from smallest to largest.

13, \, 15, \, 17, \, 18, \, 20, \, 22, \, 25, \, 27, \, 29, \, 30, \, 31

2 Split the numbers into two parts; the last part must be one digit only.

The numbers in the data will be split into tens and ones, so 13 will be 1 and 3 (1 represents 10 or 1 ten and 3 is 3 ones ).

3 Put the values into the diagram and create a key.

Example 2: histogram

Create histogram for the given test scores.

\begin{aligned} &82, \, 78, \, 77, \, 89, \, 90, \, 99, \, 97, \, 65, \, 66, \, 74, \, 78, \, 80, \, 78, \\ &92, \, 70, \, 85, \, 75, \, 85, \, 88, \, 79, \, 69, \, 88, \, 99, \, 84, \, 83, \, 91 \end{aligned}

Decide what bin size to use and how many bins are needed.

Bin size is the same as interval size.

The lowest test score is 65 , and the highest test score is 100. Let’s use bins (intervals) of 5. There are 7 bins all together.

Group the data by the bin sizes to find the frequency.

Create bars based on the bin sizes and frequencies within the bins.

Label the \textbf{x} and \textbf{y} axes with units.

Example 3: pie chart

24 pupils were asked which subject was their favorite. Here is a pie chart to show the results. How many students said science was their favorite subject?

Identify the categories.

There are 5 categories, science 25\%, English 13\%, history 20\%, art 10\%, and other 32\%.

Calculate and analyze the data.

There are 24 students that were surveyed, and 25\% said that science was their favorite subject.

24 \times 0.25=6

6 students say that science is their favorite subject.

Example 4: box plot

Create a box plot for the data below.

15, \, 11, \, 24, \, 13, \, 22, \, 17, \, 20, \, 25, \, 19, \, 10, \, 24

Determine the median and quartiles.

Placing the data in order from least to greatest.

10, \, 11, \, 13, \, 15, \, 17, \, 19, \, 20, \, 22, \, 24, \, 24, \, 25

Draw a scale, and mark the five key values: minimum value, lower quartile (LQ), median, upper quartile (UQ), and maximum value.

Join the lower quartile and upper quartile to form the box, and draw horizontal lines to the minimum and maximum values.

Example 5: frequency distribution

At the local zoo, the zoologist was taking a count of the animals.

Create a dot plot representing this data.

Read the question and determine what type of graph you need to create.

This question asks you to create a dot plot to represent the data.

Use the data to create the specific frequency graph.

Example 6: frequency polygon

A vet weighs all the dogs she sees in a week. Here are the results.

Draw a frequency polygon to show the results.

This question asks you to create a frequency polygon to represent the data.

Use the midpoints of the groups representing mass, 5, \, 15, \, 25 and so on, to label the horizontal axis. The frequency is on the vertical axis.

Plot points with a sharp pencil and crosses to be accurate, and then connect the points to create the frequency polygon.

Teaching tips for representing data

• Utilize interactive programs like Excel to allow students to spend time exploring how changing the bin size for a set of data affects the distribution of the data, therefore affecting the conclusions that might be drawn.
• Use project based learning activities where students can collect their own raw data and create and interpret tables, diagrams, and graphs.
• Provide visual aids or display examples of data representation around the classroom for students to differentiate between graph types including stem and leaf plots, bar graphs, histograms, dot plots, and box plots.

Easy mistakes to make

• Knowing when to label the horizontal axis as discrete groups or having it be a continuous scale The horizontal axis of a bar chart is divided into discrete categorical variables with gaps between the bars. Whereas on a histogram, values are on a continuous scale, so there are no gaps between the bins.
• Forgetting to order the data set before creating graphs If you are given a data set to represent on a box plot, stem and leaf plot, histogram, line plots, and/or bar graphs, make sure the list of values is in order before you start finding the key values. Listing the data in order is always a good practice to use when graphing and analyzing data.
• Not being precise when labeling axes When drawing graphs, diagrams, and charts, use a sharp pencil and a ruler so that you can be as accurate as possible. For pie charts, use a protractor to measure the angles accurately.

Practice representing data questions

1. Use the stem and leaf plot to determine the mode.

The value 157~{cm} occurs twice. Therefore, the mode is 157.

2. Use the box plot to determine the median.

On a box plot, the line in the box represents the median.

3. Which histogram represents the data?

The table shows the number of deer Karen sees in her yard over the course of a month.

Look to make sure the axes are numbered correctly. The horizontal axis should be labeled 0 to 20 counting by 5’ s. The vertical axis should be numbered 0 to 10.

Each bar height should be equal to the frequency in each interval.

4. Which pie chart represents the data in this frequency table?

The total of the frequencies is 40. The frequency of A is 10, which is a quarter of 40. So, section A needs to be a quarter of the pie chart.

Similarly section C needs to be a quarter too. The frequency of B is 20, which is half of 40. Section B needs to be half of the pie chart.

5. The table below shows the number of flowers in a garden.

Which dot plot represents the data in the table?

There are 4   types of flowers: tulip, lily, rose, and marigold. Label the horizontal axes with the flower types. For each flower, place the number of dots vertically that matches the frequency.

6. Which of these is the correct frequency polygon for the frequency table below?

The points should be plotted using the midpoints of the groups: 10, \, 30, \, 50, \, 70 and 90. They should be plotted using the correct frequencies: 1, \, 9, \, 8, \, 3 and 2. The points need joining up, but NOT the last and the first points.

Representing data FAQs

Continuous data can be any value within a range of values or in an interval. Discrete data is a specific value within a range.

When you study algebra 1, you will learn how to create scatter plots. A straight line going through the points on a scatter plot is known as a line of best fit.

The next lessons are

• Frequency table
• Frequency graph
• Sampling methods

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17 Data Visualization Techniques All Professionals Should Know

• 17 Sep 2019

There’s a growing demand for business analytics and data expertise in the workforce. But you don’t need to be a professional analyst to benefit from data-related skills.

Becoming skilled at common data visualization techniques can help you reap the rewards of data-driven decision-making , including increased confidence and potential cost savings. Learning how to effectively visualize data could be the first step toward using data analytics and data science to your advantage to add value to your organization.

Several data visualization techniques can help you become more effective in your role. Here are 17 essential data visualization techniques all professionals should know, as well as tips to help you effectively present your data.

Access your free e-book today.

What Is Data Visualization?

Data visualization is the process of creating graphical representations of information. This process helps the presenter communicate data in a way that’s easy for the viewer to interpret and draw conclusions.

There are many different techniques and tools you can leverage to visualize data, so you want to know which ones to use and when. Here are some of the most important data visualization techniques all professionals should know.

Data Visualization Techniques

The type of data visualization technique you leverage will vary based on the type of data you’re working with, in addition to the story you’re telling with your data .

Here are some important data visualization techniques to know:

• Gantt Chart
• Box and Whisker Plot
• Waterfall Chart
• Scatter Plot
• Pictogram Chart
• Highlight Table
• Bullet Graph
• Choropleth Map
• Network Diagram
• Correlation Matrices

1. Pie Chart

Pie charts are one of the most common and basic data visualization techniques, used across a wide range of applications. Pie charts are ideal for illustrating proportions, or part-to-whole comparisons.

Because pie charts are relatively simple and easy to read, they’re best suited for audiences who might be unfamiliar with the information or are only interested in the key takeaways. For viewers who require a more thorough explanation of the data, pie charts fall short in their ability to display complex information.

2. Bar Chart

The classic bar chart , or bar graph, is another common and easy-to-use method of data visualization. In this type of visualization, one axis of the chart shows the categories being compared, and the other, a measured value. The length of the bar indicates how each group measures according to the value.

One drawback is that labeling and clarity can become problematic when there are too many categories included. Like pie charts, they can also be too simple for more complex data sets.

3. Histogram

Unlike bar charts, histograms illustrate the distribution of data over a continuous interval or defined period. These visualizations are helpful in identifying where values are concentrated, as well as where there are gaps or unusual values.

Histograms are especially useful for showing the frequency of a particular occurrence. For instance, if you’d like to show how many clicks your website received each day over the last week, you can use a histogram. From this visualization, you can quickly determine which days your website saw the greatest and fewest number of clicks.

4. Gantt Chart

Gantt charts are particularly common in project management, as they’re useful in illustrating a project timeline or progression of tasks. In this type of chart, tasks to be performed are listed on the vertical axis and time intervals on the horizontal axis. Horizontal bars in the body of the chart represent the duration of each activity.

Utilizing Gantt charts to display timelines can be incredibly helpful, and enable team members to keep track of every aspect of a project. Even if you’re not a project management professional, familiarizing yourself with Gantt charts can help you stay organized.

5. Heat Map

A heat map is a type of visualization used to show differences in data through variations in color. These charts use color to communicate values in a way that makes it easy for the viewer to quickly identify trends. Having a clear legend is necessary in order for a user to successfully read and interpret a heatmap.

There are many possible applications of heat maps. For example, if you want to analyze which time of day a retail store makes the most sales, you can use a heat map that shows the day of the week on the vertical axis and time of day on the horizontal axis. Then, by shading in the matrix with colors that correspond to the number of sales at each time of day, you can identify trends in the data that allow you to determine the exact times your store experiences the most sales.

6. A Box and Whisker Plot

A box and whisker plot , or box plot, provides a visual summary of data through its quartiles. First, a box is drawn from the first quartile to the third of the data set. A line within the box represents the median. “Whiskers,” or lines, are then drawn extending from the box to the minimum (lower extreme) and maximum (upper extreme). Outliers are represented by individual points that are in-line with the whiskers.

This type of chart is helpful in quickly identifying whether or not the data is symmetrical or skewed, as well as providing a visual summary of the data set that can be easily interpreted.

7. Waterfall Chart

A waterfall chart is a visual representation that illustrates how a value changes as it’s influenced by different factors, such as time. The main goal of this chart is to show the viewer how a value has grown or declined over a defined period. For example, waterfall charts are popular for showing spending or earnings over time.

8. Area Chart

An area chart , or area graph, is a variation on a basic line graph in which the area underneath the line is shaded to represent the total value of each data point. When several data series must be compared on the same graph, stacked area charts are used.

This method of data visualization is useful for showing changes in one or more quantities over time, as well as showing how each quantity combines to make up the whole. Stacked area charts are effective in showing part-to-whole comparisons.

9. Scatter Plot

Another technique commonly used to display data is a scatter plot . A scatter plot displays data for two variables as represented by points plotted against the horizontal and vertical axis. This type of data visualization is useful in illustrating the relationships that exist between variables and can be used to identify trends or correlations in data.

Scatter plots are most effective for fairly large data sets, since it’s often easier to identify trends when there are more data points present. Additionally, the closer the data points are grouped together, the stronger the correlation or trend tends to be.

10. Pictogram Chart

Pictogram charts , or pictograph charts, are particularly useful for presenting simple data in a more visual and engaging way. These charts use icons to visualize data, with each icon representing a different value or category. For example, data about time might be represented by icons of clocks or watches. Each icon can correspond to either a single unit or a set number of units (for example, each icon represents 100 units).

In addition to making the data more engaging, pictogram charts are helpful in situations where language or cultural differences might be a barrier to the audience’s understanding of the data.

11. Timeline

Timelines are the most effective way to visualize a sequence of events in chronological order. They’re typically linear, with key events outlined along the axis. Timelines are used to communicate time-related information and display historical data.

Timelines allow you to highlight the most important events that occurred, or need to occur in the future, and make it easy for the viewer to identify any patterns appearing within the selected time period. While timelines are often relatively simple linear visualizations, they can be made more visually appealing by adding images, colors, fonts, and decorative shapes.

12. Highlight Table

A highlight table is a more engaging alternative to traditional tables. By highlighting cells in the table with color, you can make it easier for viewers to quickly spot trends and patterns in the data. These visualizations are useful for comparing categorical data.

Depending on the data visualization tool you’re using, you may be able to add conditional formatting rules to the table that automatically color cells that meet specified conditions. For instance, when using a highlight table to visualize a company’s sales data, you may color cells red if the sales data is below the goal, or green if sales were above the goal. Unlike a heat map, the colors in a highlight table are discrete and represent a single meaning or value.

13. Bullet Graph

A bullet graph is a variation of a bar graph that can act as an alternative to dashboard gauges to represent performance data. The main use for a bullet graph is to inform the viewer of how a business is performing in comparison to benchmarks that are in place for key business metrics.

In a bullet graph, the darker horizontal bar in the middle of the chart represents the actual value, while the vertical line represents a comparative value, or target. If the horizontal bar passes the vertical line, the target for that metric has been surpassed. Additionally, the segmented colored sections behind the horizontal bar represent range scores, such as “poor,” “fair,” or “good.”

14. Choropleth Maps

A choropleth map uses color, shading, and other patterns to visualize numerical values across geographic regions. These visualizations use a progression of color (or shading) on a spectrum to distinguish high values from low.

Choropleth maps allow viewers to see how a variable changes from one region to the next. A potential downside to this type of visualization is that the exact numerical values aren’t easily accessible because the colors represent a range of values. Some data visualization tools, however, allow you to add interactivity to your map so the exact values are accessible.

15. Word Cloud

A word cloud , or tag cloud, is a visual representation of text data in which the size of the word is proportional to its frequency. The more often a specific word appears in a dataset, the larger it appears in the visualization. In addition to size, words often appear bolder or follow a specific color scheme depending on their frequency.

Word clouds are often used on websites and blogs to identify significant keywords and compare differences in textual data between two sources. They are also useful when analyzing qualitative datasets, such as the specific words consumers used to describe a product.

16. Network Diagram

Network diagrams are a type of data visualization that represent relationships between qualitative data points. These visualizations are composed of nodes and links, also called edges. Nodes are singular data points that are connected to other nodes through edges, which show the relationship between multiple nodes.

There are many use cases for network diagrams, including depicting social networks, highlighting the relationships between employees at an organization, or visualizing product sales across geographic regions.

17. Correlation Matrix

A correlation matrix is a table that shows correlation coefficients between variables. Each cell represents the relationship between two variables, and a color scale is used to communicate whether the variables are correlated and to what extent.

Correlation matrices are useful to summarize and find patterns in large data sets. In business, a correlation matrix might be used to analyze how different data points about a specific product might be related, such as price, advertising spend, launch date, etc.

Other Data Visualization Options

While the examples listed above are some of the most commonly used techniques, there are many other ways you can visualize data to become a more effective communicator. Some other data visualization options include:

• Bubble clouds
• Circle views
• Dendrograms
• Dot distribution maps
• Open-high-low-close charts
• Polar areas
• Radial trees
• Ring Charts
• Sankey diagram
• Span charts
• Streamgraphs
• Wedge stack graphs
• Violin plots

Tips For Creating Effective Visualizations

Creating effective data visualizations requires more than just knowing how to choose the best technique for your needs. There are several considerations you should take into account to maximize your effectiveness when it comes to presenting data.

Related : What to Keep in Mind When Creating Data Visualizations in Excel

One of the most important steps is to evaluate your audience. For example, if you’re presenting financial data to a team that works in an unrelated department, you’ll want to choose a fairly simple illustration. On the other hand, if you’re presenting financial data to a team of finance experts, it’s likely you can safely include more complex information.

Another helpful tip is to avoid unnecessary distractions. Although visual elements like animation can be a great way to add interest, they can also distract from the key points the illustration is trying to convey and hinder the viewer’s ability to quickly understand the information.

Finally, be mindful of the colors you utilize, as well as your overall design. While it’s important that your graphs or charts are visually appealing, there are more practical reasons you might choose one color palette over another. For instance, using low contrast colors can make it difficult for your audience to discern differences between data points. Using colors that are too bold, however, can make the illustration overwhelming or distracting for the viewer.

Related : Bad Data Visualization: 5 Examples of Misleading Data

Visuals to Interpret and Share Information

No matter your role or title within an organization, data visualization is a skill that’s important for all professionals. Being able to effectively present complex data through easy-to-understand visual representations is invaluable when it comes to communicating information with members both inside and outside your business.

There’s no shortage in how data visualization can be applied in the real world. Data is playing an increasingly important role in the marketplace today, and data literacy is the first step in understanding how analytics can be used in business.

Are you interested in improving your analytical skills? Learn more about Business Analytics , our eight-week online course that can help you use data to generate insights and tackle business decisions.

This post was updated on January 20, 2022. It was originally published on September 17, 2019.

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Data Presentation - Tables

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Tables are a useful way to organize information using rows and columns. Tables are a versatile organization tool and can be used to communicate information on their own, or they can be used to accompany another data representation type (like a graph). Tables support a variety of parameters and can be used to keep track of frequencies, variable associations, and more.

For example, given below are the weights of 20 students in grade 10: \[50, 45, 48, 39, 40, 48, 54, 50, 48, 48, \\ 50, 39, 41, 46, 44, 43, 54, 57, 60, 45.\]

To find the frequency of \(48\) in this data, count the number of times that \(48\) appears in the list. There are \(4\) students that have this weight.

The list above has information about the weight of \(20\) students, and since the data has been arranged haphazardly, it is difficult to classify the students properly.

To make the information more clear, tabulate the given data.

\[\begin{array} \\ \text{Weights in kg} & & & \text{Frequency} \\ 39 & & & 2 \\ 40 & & & 1 \\ 41 & & & 1 \\ 43 & & & 1 \\ 44 & & & 1 \\ 45 & & & 2 \\ 46 & & & 1 \\ 48 & & & 4 \\ 50 & & & 3 \\ 54 & & & 2 \\ 57 & & & 1 \\ 60 & & & 1 \end{array}\]

This table makes the data more easy to understand.

Making a Table

Making and using tables.

To make a table, first decide how many rows and columns are needed to clearly display the data. To do this, consider how many variables are included in the data set.

The following is an example of a table where there are two variables.

The following is an example of a table with three variables.

A table is good for organizing quantitative data in a way that it is easy to look things up. For example, a table would be good way to associate a person’s name, age, and favorite food. However, when trying to communicate relations, such as how a person’s favorite food changes over time, a graph would be a better choice.

Using the table below, determine the average age of the group?

Good practices for making tables Label what each row or column represents Include units in labels when data is numerical Format data consistently (use consistent units and formatting)

What is wrong with this table? Flavor of Ice Cream Number Sold (cones) Chocolate 104 Vanilla two-hundred Strawberry 143 Coconut thirty Mango 126 Show answer Answer: The data isn’t consistently formatted. The number of cones sold is written in numbers in both symbols and words. It would be easier to understand if all entries were numerical symbols.

What is wrong with this table? Jack blue Sarah yellow Billy green Ron red Christina blue Margret purple Show answer Answer: There are no labels on the columns. It is not clear what the table is displaying — does the table show what color shirt each person is wearing? Do it show what each person's favorite color is? It isn't clear because labels are missing.

Many word processing softwares include tools for making tables. You can easily make tables in Microsoft Word and Excel and in Google Docs and Sheets.

Here is an example table (left blank) with which you could record information about a person's age, weight, and height.

Tables are used to present information in all types of fields. Geologists might make a table to record data about types of rocks they find while doing field work, political researchers might create a table to record information about potential voters, and physicists might make a table to record observations about the speed of a ball rolled on various surfaces.

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Data representation

Computers use binary - the digits 0 and 1 - to store data. A binary digit, or bit, is the smallest unit of data in computing. It is represented by a 0 or a 1. Binary numbers are made up of binary digits (bits), eg the binary number 1001. The circuits in a computer's processor are made up of billions of transistors. A transistor is a tiny switch that is activated by the electronic signals it receives. The digits 1 and 0 used in binary reflect the on and off states of a transistor. Computer programs are sets of instructions. Each instruction is translated into machine code - simple binary codes that activate the CPU. Programmers write computer code and this is converted by a translator into binary instructions that the processor can execute. All software, music, documents, and any other information that is processed by a computer, is also stored using binary. [1]

To include strings, integers, characters and colours. This should include considering the space taken by data, for instance the relation between the hexadecimal representation of colours and the number of colours available.

This video is superb place to understand this topic

• 1 How a file is stored on a computer
• 2 How an image is stored in a computer
• 3 The way in which data is represented in the computer.
• 6 Standards
• 7 References

How a file is stored on a computer [ edit ]

How an image is stored in a computer [ edit ]

The way in which data is represented in the computer. [ edit ].

To include strings, integers, characters and colours. This should include considering the space taken by data, for instance the relation between the hexadecimal representation of colours and the number of colours available [3] .

This helpful material is used with gratitude from a computer science wiki under a Creative Commons Attribution 3.0 License [4]

Sound [ edit ]

• Let's look at an oscilloscope
• The BBC has an excellent article on how computers represent sound

See Also [ edit ]

Standards [ edit ].

• Outline the way in which data is represented in the computer.

References [ edit ]

• ↑ http://www.bbc.co.uk/education/guides/zwsbwmn/revision/1
• ↑ https://ujjwalkarn.me/2016/08/11/intuitive-explanation-convnets/
• ↑ IBO Computer Science Guide, First exams 2014
• ↑ https://compsci2014.wikispaces.com/2.1.10+Outline+the+way+in+which+data+is+represented+in+the+computer

A unit of abstract mathematical system subject to the laws of arithmetic.

A natural number, a negative of a natural number, or zero.

Give a brief account.

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• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
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What are the different ways of Data Representation?

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The process of collecting the data and analyzing that data in large quantity is known as statistics. It is a branch of mathematics trading with the collection, analysis, interpretation, and presentation of numeral facts and figures.

It is a numerical statement that helps us to collect and analyze the data in large quantity the statistics are based on two of its concepts:

• Statistical Data 
• Statistical Science

Statistics must be expressed numerically and should be collected systematically.

Data Representation

The word data refers to constituting people, things, events, ideas. It can be a title, an integer, or anycast.  After collecting data the investigator has to condense them in tabular form to study their salient features. Such an arrangement is known as the presentation of data.

It refers to the process of condensing the collected data in a tabular form or graphically. This arrangement of data is known as Data Representation.

The row can be placed in different orders like it can be presented in ascending orders, descending order, or can be presented in alphabetical order. 

Example: Let the marks obtained by 10 students of class V in a class test, out of 50 according to their roll numbers, be: 39, 44, 49, 40, 22, 10, 45, 38, 15, 50 The data in the given form is known as raw data. The above given data can be placed in the serial order as shown below: Roll No. Marks 1 39 2 44 3 49 4 40 5 22 6 10 7 45 8 38 9 14 10 50 Now, if you want to analyse the standard of achievement of the students. If you arrange them in ascending or descending order, it will give you a better picture. Ascending order: 10, 15, 22, 38, 39, 40, 44. 45, 49, 50 Descending order: 50, 49, 45, 44, 40, 39, 38, 22, 15, 10 When the row is placed in ascending or descending order is known as arrayed data.

Types of Graphical Data Representation

Bar chart helps us to represent the collected data visually. The collected data can be visualized horizontally or vertically in a bar chart like amounts and frequency. It can be grouped or single. It helps us in comparing different items. By looking at all the bars, it is easy to say which types in a group of data influence the other.

Now let us understand bar chart by taking this example  Let the marks obtained by 5 students of class V in a class test, out of 10 according to their names, be: 7,8,4,9,6 The data in the given form is known as raw data. The above given data can be placed in the bar chart as shown below: Name Marks Akshay 7 Maya 8 Dhanvi 4 Jaslen 9 Muskan 6

A histogram is the graphical representation of data. It is similar to the appearance of a bar graph but there is a lot of difference between histogram and bar graph because a bar graph helps to measure the frequency of categorical data. A categorical data means it is based on two or more categories like gender, months, etc. Whereas histogram is used for quantitative data.

For example:

The graph which uses lines and points to present the change in time is known as a line graph. Line graphs can be based on the number of animals left on earth, the increasing population of the world day by day, or the increasing or decreasing the number of bitcoins day by day, etc. The line graphs tell us about the changes occurring across the world over time. In a  line graph, we can tell about two or more types of changes occurring around the world.

For Example:

Pie chart is a type of graph that involves a structural graphic representation of numerical proportion. It can be replaced in most cases by other plots like a bar chart, box plot, dot plot, etc. As per the research, it is shown that it is difficult to compare the different sections of a given pie chart, or if it is to compare data across different pie charts.

Frequency Distribution Table

A frequency distribution table is a chart that helps us to summarise the value and the frequency of the chart. This frequency distribution table has two columns, The first column consist of the list of the various outcome in the data, While the second column list the frequency of each outcome of the data. By putting this kind of data into a table it helps us to make it easier to understand and analyze the data. 

For Example: To create a frequency distribution table, we would first need to list all the outcomes in the data. In this example, the results are 0 runs, 1 run, 2 runs, and 3 runs. We would list these numerals in numerical ranking in the foremost queue. Subsequently, we ought to calculate how many times per result happened. They scored 0 runs in the 1st, 4th, 7th, and 8th innings, 1 run in the 2nd, 5th, and the 9th innings, 2 runs in the 6th inning, and 3 runs in the 3rd inning. We set the frequency of each result in the double queue. You can notice that the table is a vastly more useful method to show this data.  Baseball Team Runs Per Inning Number of Runs Frequency           0       4           1        3            2        1            3        1

Sample Questions

Question 1: Considering the school fee submission of 10 students of class 10th is given below:

In order to draw the bar graph for the data above, we prepare the frequency table as given below. Fee submission No. of Students Paid   6 Not paid    4 Now we have to represent the data by using the bar graph. It can be drawn by following the steps given below: Step 1: firstly we have to draw the two axis of the graph X-axis and the Y-axis. The varieties of the data must be put on the X-axis (the horizontal line) and the frequencies of the data must be put on the Y-axis (the vertical line) of the graph. Step 2: After drawing both the axis now we have to give the numeric scale to the Y-axis (the vertical line) of the graph It should be started from zero and ends up with the highest value of the data. Step 3: After the decision of the range at the Y-axis now we have to give it a suitable difference of the numeric scale. Like it can be 0,1,2,3…….or 0,10,20,30 either we can give it a numeric scale like 0,20,40,60… Step 4: Now on the X-axis we have to label it appropriately. Step 5: Now we have to draw the bars according to the data but we have to keep in mind that all the bars should be of the same length and there should be the same distance between each graph

Question 2: Watch the subsequent pie chart that denotes the money spent by Megha at the funfair. The suggested colour indicates the quantity paid for each variety. The total value of the data is 15 and the amount paid on each variety is diagnosed as follows:

Chocolates – 3

Wafers – 3

Toys – 2

Rides – 7

To convert this into pie chart percentage, we apply the formula:  (Frequency/Total Frequency) × 100 Let us convert the above data into a percentage: Amount paid on rides: (7/15) × 100 = 47% Amount paid on toys: (2/15) × 100 = 13% Amount paid on wafers: (3/15) × 100 = 20% Amount paid on chocolates: (3/15) × 100 = 20 %

Question 3: The line graph given below shows how Devdas’s height changes as he grows.

Given below is a line graph showing the height changes in Devdas’s as he grows. Observe the graph and answer the questions below.

(i) What was the height of  Devdas’s at 8 years? Answer: 65 inches (ii) What was the height of  Devdas’s at 6 years? Answer:  50 inches (iii) What was the height of  Devdas’s at 2 years? Answer: 35 inches (iv) How much has  Devdas’s grown from 2 to 8 years? Answer: 30 inches (v) When was  Devdas’s 35 inches tall? Answer: 2 years.

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Data Representation in Computer: Number Systems, Characters, Audio, Image and Video

• Post author: Anuj Kumar
• Post published: 16 July 2021
• Post category: Computer Science
• Post comments: 0 Comments

Table of Contents

• 1 What is Data Representation in Computer?
• 2.1 Binary Number System
• 2.2 Octal Number System
• 2.3 Decimal Number System
• 2.4 Hexadecimal Number System
• 3.4 Unicode
• 4 Data Representation of Audio, Image and Video
• 5.1 What is number system with example?

What is Data Representation in Computer?

A computer uses a fixed number of bits to represent a piece of data which could be a number, a character, image, sound, video, etc. Data representation is the method used internally to represent data in a computer. Let us see how various types of data can be represented in computer memory.

Before discussing data representation of numbers, let us see what a number system is.

Number Systems

Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.

A number is a mathematical object used to count, label, and measure. A number system is a systematic way to represent numbers. The number system we use in our day-to-day life is the decimal number system that uses 10 symbols or digits.

The number 289 is pronounced as two hundred and eighty-nine and it consists of the symbols 2, 8, and 9. Similarly, there are other number systems. Each has its own symbols and method for constructing a number.

A number system has a unique base, which depends upon the number of symbols. The number of symbols used in a number system is called the base or radix of a number system.

Let us discuss some of the number systems. Computer architecture supports the following number of systems:

Binary Number System

Octal number system, decimal number system, hexadecimal number system.

A Binary number system has only two digits that are 0 and 1. Every number (value) represents 0 and 1 in this number system. The base of the binary number system is 2 because it has only two digits.

The octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of the octal number system is 8, because it has only 8 digits.

The decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of the hexadecimal number system is 16, because it has 16 alphanumeric values.

Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15 .

Data Representation of Characters

There are different methods to represent characters . Some of them are discussed below:

The code called ASCII (pronounced ‘􀀏’.S-key”), which stands for American Standard Code for Information Interchange, uses 7 bits to represent each character in computer memory. The ASCII representation has been adopted as a standard by the U.S. government and is widely accepted.

A unique integer number is assigned to each character. This number called ASCII code of that character is converted into binary for storing in memory. For example, the ASCII code of A is 65, its binary equivalent in 7-bit is 1000001.

Since there are exactly 128 unique combinations of 7 bits, this 7-bit code can represent only128 characters. Another version is ASCII-8, also called extended ASCII, which uses 8 bits for each character, can represent 256 different characters.

For example, the letter A is represented by 01000001, B by 01000010 and so on. ASCII code is enough to represent all of the standard keyboard characters.

It stands for Extended Binary Coded Decimal Interchange Code. This is similar to ASCII and is an 8-bit code used in computers manufactured by International Business Machines (IBM). It is capable of encoding 256 characters.

If ASCII-coded data is to be used in a computer that uses EBCDIC representation, it is necessary to transform ASCII code to EBCDIC code. Similarly, if EBCDIC coded data is to be used in an ASCII computer, EBCDIC code has to be transformed to ASCII.

ISCII stands for Indian Standard Code for Information Interchange or Indian Script Code for Information Interchange. It is an encoding scheme for representing various writing systems of India. ISCII uses 8-bits for data representation.

It was evolved by a standardization committee under the Department of Electronics during 1986-88 and adopted by the Bureau of Indian Standards (BIS). Nowadays ISCII has been replaced by Unicode.

Using 8-bit ASCII we can represent only 256 characters. This cannot represent all characters of written languages of the world and other symbols. Unicode is developed to resolve this problem. It aims to provide a standard character encoding scheme, which is universal and efficient.

It provides a unique number for every character, no matter what the language and platform be. Unicode originally used 16 bits which can represent up to 65,536 characters. It is maintained by a non-profit organization called the Unicode Consortium.

The Consortium first published version 1.0.0 in 1991 and continues to develop standards based on that original work. Nowadays Unicode uses more than 16 bits and hence it can represent more characters. Unicode can represent characters in almost all written languages of the world.

Data Representation of Audio, Image and Video

In most cases, we may have to represent and process data other than numbers and characters. This may include audio data, images, and videos. We can see that like numbers and characters, the audio, image, and video data also carry information.

We will see different file formats for storing sound, image, and video .

Multimedia data such as audio, image, and video are stored in different types of files. The variety of file formats is due to the fact that there are quite a few approaches to compressing the data and a number of different ways of packaging the data.

For example, an image is most popularly stored in Joint Picture Experts Group (JPEG ) file format. An image file consists of two parts – header information and image data. Information such as the name of the file, size, modified data, file format, etc. is stored in the header part.

The intensity value of all pixels is stored in the data part of the file. The data can be stored uncompressed or compressed to reduce the file size. Normally, the image data is stored in compressed form. Let us understand what compression is.

Take a simple example of a pure black image of size 400X400 pixels. We can repeat the information black, black, …, black in all 16,0000 (400X400) pixels. This is the uncompressed form, while in the compressed form black is stored only once and information to repeat it 1,60,000 times is also stored.

Numerous such techniques are used to achieve compression. Depending on the application, images are stored in various file formats such as bitmap file format (BMP), Tagged Image File Format (TIFF), Graphics Interchange Format (GIF), Portable (Public) Network Graphic (PNG).

What we said about the header file information and compression is also applicable for audio and video files. Digital audio data can be stored in different file formats like WAV, MP3, MIDI, AIFF, etc. An audio file describes a format, sometimes referred to as the ‘container format’, for storing digital audio data.

For example, WAV file format typically contains uncompressed sound and MP3 files typically contain compressed audio data. The synthesized music data is stored in MIDI(Musical Instrument Digital Interface) files.

Similarly, video is also stored in different files such as AVI (Audio Video Interleave) – a file format designed to store both audio and video data in a standard package that allows synchronous audio with video playback, MP3, JPEG-2, WMV, etc.

FAQs About Data Representation in Computer

What is number system with example.

Let us discuss some of the number systems. Computer architecture supports the following number of systems: 1. Binary Number System 2. Octal Number System 3. Decimal Number System 4. Hexadecimal Number System

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Table of contents.

• Introduction to Functional Computer
• Fundamentals of Architectural Design

Data Representation

• Instruction Set Architecture : Instructions and Formats
• Instruction Set Architecture : Design Models
• Instruction Set Architecture : Addressing Modes
• Performance Measurements and Issues
• Computer Architecture Assessment 1
• Fixed Point Arithmetic : Addition and Subtraction
• Fixed Point Arithmetic : Multiplication
• Fixed Point Arithmetic : Division
• Floating Point Arithmetic
• Arithmetic Logic Unit Design
• CPU's Data Path
• CPU's Control Unit
• Control Unit Design
• Concepts of Pipelining
• Computer Architecture Assessment 2
• Pipeline Hazards
• Memory Characteristics and Organization
• Cache Memory
• Virtual Memory
• I/O Communication and I/O Controller
• Input/Output Data Transfer
• Direct Memory Access controller and I/O Processor
• CPU Interrupts and Interrupt Handling
• Computer Architecture Assessment 3

Course Computer Architecture

Digital computers store and process information in binary form as digital logic has only two values "1" and "0" or in other words "True or False" or also said as "ON or OFF". This system is called radix 2. We human generally deal with radix 10 i.e. decimal. As a matter of convenience there are many other representations like Octal (Radix 8), Hexadecimal (Radix 16), Binary coded decimal (BCD), Decimal etc.

Every computer's CPU has a width measured in terms of bits such as 8 bit CPU, 16 bit CPU, 32 bit CPU etc. Similarly, each memory location can store a fixed number of bits and is called memory width. Given the size of the CPU and Memory, it is for the programmer to handle his data representation. Most of the readers may be knowing that 4 bits form a Nibble, 8 bits form a byte. The word length is defined by the Instruction Set Architecture of the CPU. The word length may be equal to the width of the CPU.

The memory simply stores information as a binary pattern of 1's and 0's. It is to be interpreted as what the content of a memory location means. If the CPU is in the Fetch cycle, it interprets the fetched memory content to be instruction and decodes based on Instruction format. In the Execute cycle, the information from memory is considered as data. As a common man using a computer, we think computers handle English or other alphabets, special characters or numbers. A programmer considers memory content to be data types of the programming language he uses. Now recall figure 1.2 and 1.3 of chapter 1 to reinforce your thought that conversion happens from computer user interface to internal representation and storage.

• Data Representation in Computers

Information handled by a computer is classified as instruction and data. A broad overview of the internal representation of the information is illustrated in figure 3.1. No matter whether it is data in a numeric or non-numeric form or integer, everything is internally represented in Binary. It is up to the programmer to handle the interpretation of the binary pattern and this interpretation is called Data Representation . These data representation schemes are all standardized by international organizations.

Choice of Data representation to be used in a computer is decided by

• The number types to be represented (integer, real, signed, unsigned, etc.)
• Range of values likely to be represented (maximum and minimum to be represented)
• The Precision of the numbers i.e. maximum accuracy of representation (floating point single precision, double precision etc)
• If non-numeric i.e. character, character representation standard to be chosen. ASCII, EBCDIC, UTF are examples of character representation standards.
• The hardware support in terms of word width, instruction.

Before we go into the details, let us take an example of interpretation. Say a byte in Memory has value "0011 0001". Although there exists a possibility of so many interpretations as in figure 3.2, the program has only one interpretation as decided by the programmer and declared in the program.

• Fixed point Number Representation

Fixed point numbers are also known as whole numbers or Integers. The number of bits used in representing the integer also implies the maximum number that can be represented in the system hardware. However for the efficiency of storage and operations, one may choose to represent the integer with one Byte, two Bytes, Four bytes or more. This space allocation is translated from the definition used by the programmer while defining a variable as integer short or long and the Instruction Set Architecture.

In addition to the bit length definition for integers, we also have a choice to represent them as below:

• Unsigned Integer : A positive number including zero can be represented in this format. All the allotted bits are utilised in defining the number. So if one is using 8 bits to represent the unsigned integer, the range of values that can be represented is 28 i.e. "0" to "255". If 16 bits are used for representing then the range is 216 i.e. "0 to 65535".
• Signed Integer : In this format negative numbers, zero, and positive numbers can be represented. A sign bit indicates the magnitude direction as positive or negative. There are three possible representations for signed integer and these are Sign Magnitude format, 1's Compliment format and 2's Complement format .

Signed Integer – Sign Magnitude format: Most Significant Bit (MSB) is reserved for indicating the direction of the magnitude (value). A "0" on MSB means a positive number and a "1" on MSB means a negative number. If n bits are used for representation, n-1 bits indicate the absolute value of the number. Examples for n=8:

Examples for n=8:

0010 1111 = + 47 Decimal (Positive number)

1010 1111 = - 47 Decimal (Negative Number)

0111 1110 = +126 (Positive number)

1111 1110 = -126 (Negative Number)

0000 0000 = + 0 (Postive Number)

1000 0000 = - 0 (Negative Number)

Although this method is easy to understand, Sign Magnitude representation has several shortcomings like

• Zero can be represented in two ways causing redundancy and confusion.
• The total range for magnitude representation is limited to 2n-1, although n bits were accounted.
• The separate sign bit makes the addition and subtraction more complicated. Also, comparing two numbers is not straightforward.

Signed Integer – 1’s Complement format: In this format too, MSB is reserved as the sign bit. But the difference is in representing the Magnitude part of the value for negative numbers (magnitude) is inversed and hence called 1’s Complement form. The positive numbers are represented as it is in binary. Let us see some examples to better our understanding.

1101 0000 = - 47 Decimal (Negative Number)

1000 0001 = -126 (Negative Number)

1111 1111 = - 0 (Negative Number)

• Converting a given binary number to its 2's complement form

Step 1 . -x = x' + 1 where x' is the one's complement of x.

Step 2 Extend the data width of the number, fill up with sign extension i.e. MSB bit is used to fill the bits.

Example: -47 decimal over 8bit representation

As you can see zero is not getting represented with redundancy. There is only one way of representing zero. The other problem of the complexity of the arithmetic operation is also eliminated in 2’s complement representation. Subtraction is done as Addition.

More exercises on number conversion are left to the self-interest of readers.

• Floating Point Number system

The maximum number at best represented as a whole number is 2 n . In the Scientific world, we do come across numbers like Mass of an Electron is 9.10939 x 10-31 Kg. Velocity of light is 2.99792458 x 108 m/s. Imagine to write the number in a piece of paper without exponent and converting into binary for computer representation. Sure you are tired!!. It makes no sense to write a number in non- readable form or non- processible form. Hence we write such large or small numbers using exponent and mantissa. This is said to be Floating Point representation or real number representation. he real number system could have infinite values between 0 and 1.

Representation in computer

Unlike the two's complement representation for integer numbers, Floating Point number uses Sign and Magnitude representation for both mantissa and exponent . In the number 9.10939 x 1031, in decimal form, +31 is Exponent, 9.10939 is known as Fraction . Mantissa, Significand and fraction are synonymously used terms. In the computer, the representation is binary and the binary point is not fixed. For example, a number, say, 23.345 can be written as 2.3345 x 101 or 0.23345 x 102 or 2334.5 x 10-2. The representation 2.3345 x 101 is said to be in normalised form.

Floating-point numbers usually use multiple words in memory as we need to allot a sign bit, few bits for exponent and many bits for mantissa. There are standards for such allocation which we will see sooner.

• IEEE 754 Floating Point Representation

We have two standards known as Single Precision and Double Precision from IEEE. These standards enable portability among different computers. Figure 3.3 picturizes Single precision while figure 3.4 picturizes double precision. Single Precision uses 32bit format while double precision is 64 bits word length. As the name suggests double precision can represent fractions with larger accuracy. In both the cases, MSB is sign bit for the mantissa part, followed by Exponent and Mantissa. The exponent part has its sign bit.

It is to be noted that in Single Precision, we can represent an exponent in the range -127 to +127. It is possible as a result of arithmetic operations the resulting exponent may not fit in. This situation is called overflow in the case of positive exponent and underflow in the case of negative exponent. The Double Precision format has 11 bits for exponent meaning a number as large as -1023 to 1023 can be represented. The programmer has to make a choice between Single Precision and Double Precision declaration using his knowledge about the data being handled.

The Floating Point operations on the regular CPU is very very slow. Generally, a special purpose CPU known as Co-processor is used. This Co-processor works in tandem with the main CPU. The programmer should be using the float declaration only if his data is in real number form. Float declaration is not to be used generously.

• Decimal Numbers Representation

Decimal numbers (radix 10) are represented and processed in the system with the support of additional hardware. We deal with numbers in decimal format in everyday life. Some machines implement decimal arithmetic too, like floating-point arithmetic hardware. In such a case, the CPU uses decimal numbers in BCD (binary coded decimal) form and does BCD arithmetic operation. BCD operates on radix 10. This hardware operates without conversion to pure binary. It uses a nibble to represent a number in packed BCD form. BCD operations require not only special hardware but also decimal instruction set.

• Exceptions and Error Detection

All of us know that when we do arithmetic operations, we get answers which have more digits than the operands (Ex: 8 x 2= 16). This happens in computer arithmetic operations too. When the result size exceeds the allotted size of the variable or the register, it becomes an error and exception. The exception conditions associated with numbers and number operations are Overflow, Underflow, Truncation, Rounding and Multiple Precision . These are detected by the associated hardware in arithmetic Unit. These exceptions apply to both Fixed Point and Floating Point operations. Each of these exceptional conditions has a flag bit assigned in the Processor Status Word (PSW). We may discuss more in detail in the later chapters.

• Character Representation

Another data type is non-numeric and is largely character sets. We use a human-understandable character set to communicate with computer i.e. for both input and output. Standard character sets like EBCDIC and ASCII are chosen to represent alphabets, numbers and special characters. Nowadays Unicode standard is also in use for non-English language like Chinese, Hindi, Spanish, etc. These codes are accessible and available on the internet. Interested readers may access and learn more.

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Data representation exercises

Exercises not as directly relevant to this year’s class are marked with ⚠️.

DATAREP-1. Sizes and alignments

QUESTION DATAREP-1A. True or false: For any non-array type X, the size of X ( sizeof(X) ) is greater than or equal to the alignment of type X ( alignof(X) ).

True. This also mostly true for arrays. The exception is zero-length arrays: sizeof(X[0]) == 0 , but alignof(X[0]) == alignof(X) .

QUESTION DATAREP-1B. True or false: For any type X, the size of struct Y { X a; char newc; } is greater than the size of X.

QUESTION DATAREP-1C. True or false: For any types A1 ... An (with n ≥ 1), the size of struct Y is greater than the size of struct X , given:

struct X { A1 a1; ... An an; };

struct Y { A1 a1; ... An an; char newc; };

False (example: A1 = int , A2 = char )

QUESTION DATAREP-1D. True or false: For any types A1 ... An (with n ≥ 1), the size of struct Y is greater than the size of union X , given:

union X { A1 a1; ... An an; };

struct Y { A1 a1; ... An an; };

False (if n = 1 )

QUESTION DATAREP-1E. Assume that structure struct Y { ... } contains K char members and M int members, with K ≤ M , and nothing else. Write an expression defining the maximum sizeof(struct Y) .

QUESTION DATAREP-1F. You are given a structure struct Z { T1 a; T2 b; T3 c; } that contains no padding. What does (sizeof(T1) + sizeof(T2) + sizeof(T3)) % alignof(struct Z) equal?

QUESTION DATAREP-1G. Arrange the following types in increasing order by size. Sample answer: “1 < 2 = 4 < 3” (choose this if #1 has smaller size than #2, which has equal size to #4, which has smaller size than #3).

• struct minipoint { uint8_t x; uint8_t y; uint8_t z; }
• unsigned short[1]

#6 < #1 < #4 < #2 < #3 < #5

DATAREP-2. Expressions

QUESTION DATAREP-2A. Here are eight expressions. Group the expressions into four pairs so that the two expressions in each pair have the same value, and each pair has a different value from every other pair. There is one unique answer that meets these constraints. m has the same type and value everywhere it appears (there’s one unique value for m that meets the problem’s constraints). Assume an x86-32 machine: a 32-bit architecture in which pointers are 32 bits long.

• sizeof(&m)
• 16 >> 2

1—5; 2—7; 3—8; 4—6 1—5 is easy. m + ~m + 1 == m + (-m) == 0 , and m & ~m == 0 , giving us 3—8. Now what about the others? m & -m (#3) is either 0 or a power of 2, so it cannot be -1 (#2). The remaining possiblities are m and 1 . If (m & -m) == m , then the remaining pair would be 1 and -1, which clearly doesn’t work. Thus m & -m matches with 1, and m == -1 .

DATAREP-3. Hello binary

This problem locates 8-bit numbers horizontally and vertically in the following 16x16 image. Black pixels represent 1 bits and white pixels represent 0 bits. For horizontal arrangements, the most significant bit is on the left as usual. For vertical arrangements, the most significant bit is on top.

Examples : The 8-bit number 15 (hexadecimal 0x0F, binary 0b00001111) is located horizontally at 3,4, which means X=3, Y=4.

• The pixel at 3,4 is white, which has bit value 0.
• 4,4 is white, also 0.
• 5,4 is white, also 0.
• 6,4 is white, also 0.
• 7,4 is black, which has bit value 1.
• 8,4, 9,4, and 10,4 are black, giving three more 1s.
• Reading them all off, this is 0b00001111, or 15.

15 is also located horizontally at 7,6.

The 8-bit number 0 is located vertically at 0,0. It is also located horizontally at 0,0 and 1,0.

The 8-bit number 134 (hexadecimal 0x86, binary 0b10000110) is located vertically at 8,4.

QUESTION DATAREP-3A. Where is 3 located vertically? (All questions refer to 8-bit numbers.)

QUESTION DATAREP-3B. Where is 12 located horizontally?

QUESTION DATAREP-3C. Where is 255 located vertically?

DATAREP-4. Hello memory

Shintaro Tsuji wants to represent the image of Question DATAREP-3 in computer memory. He stores it in an array of 16-bit unsigned integers:

Row Y of the image is stored in integer cute[Y] .

QUESTION DATAREP-4A. What is sizeof(cute) , 2, 16, 32, or 64?

QUESTION DATAREP-4B. printf("%d\n", cute[0]); prints 16384 . Is Shintaro’s machine big-endian or little-endian?

Little-endian

DATAREP-5. Hello program

Now that Shintaro has represented the image in memory as an array of unsigned short objects, he can manipulate the image using C. For example, here’s a function.

Running swap produces the following image:

Shintaro has written several other functions. Here are some images (A is the original):

For each function, what image does that function create?

QUESTION DATAREP-5A.

H. The code flips all bits in the input.

QUESTION DATAREP-5B.

QUESTION DATAREP-5C.

The following programs generated the other images. Can you match them with their images?

f3 —I; f4 —B; f5 —C; f6 —F; f7 —G; f8 —A; f9 —E

DATAREP-6. Memory regions

Consider the following program:

This program allocates objects a through g on the heap and then stores those pointers in some stack and global variables. We recommend you draw a picture of the state setup creates.

QUESTION DATAREP-6A. Assume that (uintptr_t) a == 0x8300000 , and that malloc returns increasing addresses. Match each address to the most likely expression with that address value. The expressions are evaluated within the context of main . You will not reuse an expression.

1—D; 2—C; 3—E; 4—B; 5—A Since p has automatic storage duration, it is located on the stack, giving us 5—A. The global variable has static storage duration, and so does its component global.y ; so the pointer &global.y has an address that is below all heap-allocated pointers. This gives us 2—C. The remaining expressions go like this: global.y == e ; p.y == &e[1] , so *p.y == e[1] == (int) &d[100000] , and (int *) *p.y == &d[100000] ; p.x == g , so p.x[0] == g[0] == *g == c , and *p.x[0] == *c == (int) a . Address #4 has value 0x8300000, which by assumption is a ’s address; so 4—B. Address #3 is much larger than the other heap addresses, so 3—E. This leaves 1—D.

DATAREP-7. ⚠️ Garbage collection ⚠️

⚠️ Here is the top-level function for the conservative garbage collector we wrote in class.

This garbage collector is not correct because it doesn’t capture all memory roots.

Consider the program from the previous section, and assume that an object is reachable if do_stuff can access an address within the object via variable references and memory dereferences without casts or pointer arithmetic . Then:

QUESTION DATAREP-7A. Which reachable objects will m61_collect() free? Circle all that apply.

b , f . The collector searches the stack for roots. This yields just the values in struct ptrs p (the only pointer-containing variable with automatic storage duration at the time m61_collect is called). The objects directly pointed to by p are g and e . The collector then recursively marks objects pointed to by these objects. From g , it finds c . From e , it finds nothing. Then it checks one more time. From c , it finds the value of a ! Now, a is actually not a pointer here—the type of *c is int —so by the definition above, a is not actually reachable. But the collector doesn’t know this. Putting it together, the collector marks a , c , e , and g . It won’t free these objects; it will free the others ( b , d , and f ). But b and f are reachable from global .

QUESTION DATAREP-7B. Which unreachable objects will m61_collect() not free? Circle all that apply.

QUESTION DATAREP-7C. Conservative garbage collection in C is often slower than precise garbage collection in languages such as Java. Why? Circle all that apply.

• C is generally slower than other languages.
• Conservative garbage collectors must search all reachable memory for pointers. Precise garbage collectors can ignore values that do not contain pointers, such as large character buffers.
• C programs generally use the heap more than programs in other languages.
• None of the above.

DATAREP-8. Memory errors

The following function constructs and returns a lower-triangular matrix of size N . The elements are random 2-dimensional points in the unit square. The matrix is represented as an array of pointers to arrays.

This code is running on an x86- 32 machine ( size_t is 32 bits , not 64). You may assume that the machine has enough free physical memory and the process has enough available virtual address space to satisfy any memory allocation request.

QUESTION DATAREP-8A. Give a value of N so that, while make_random_lt_matrix(N) is running, no new fails, but a memory error (such as a null pointer dereference or an out-of-bounds dereference) happens on Line A. The memory error should happen specifically when i == 1 .

(This problem is probably easier when you write your answer in hexadecimal.)

We are asked to produce a value of N so that no memory error happens on Line A when i == 0 , but a memory error does happen when i == 1 . So reason that through. What memory errors could happen on Line A if malloc() returns non- nullptr ? There’s only one memory operation, namely the dereference m[i] . Perhaps this dereference is out of bounds. If no memory error happens when i == 0 , then a m[0] dereference must not cause a memory error. So the m object must contain at least 4 bytes. But a memory error does happen on Line A when i == 1 . So the m object must contain less than 8 bytes. How many bytes were allocated for m ? sizeof(point2_vector) * N == sizeof(point2 *) * N == 4 * N . So we have: (4 * N) ≥ 4 (4 * N) < 8 It seems like the only possible answer is N == 1 . But no, this doesn’t cause a memory error, because the loop body would never be executed with i == 1 ! The key insight is that the multiplications above use 32-bit unsigned computer arithmetic. Let’s write N as X + 1 . Then these inequalities become: 4 ≤ 4 * (X + 1) = 4 * X + 4 < 8 0 ≤ (4 * X) < 4 (Multiplication distributes over addition in computer arithmetic.) What values of X satisfy this inequality? It might be easier to see if we remember that multiplication by powers of two is equivalent to shifting: 0 ≤ (X << 2) < 4 The key insight is that this shift eliminates the top two bits of X . There are exactly four values for X that work: 0 , 0x40000000 , 0x80000000 , and 0xC0000000 . For any of these, 4 * X == 0 in 32-bit computer arithmetic, because 4× X = 0 (mod 2 32 ) in normal arithmetic. Plugging X back in to N , we see that N ∈ {0x40000001, 0x80000001, 0xC0000001} . These are the only values that work. Partial credit was awarded for values that acknowledged the possibility of overflow.

QUESTION DATAREP-8B. Give a value of N so that no new fails, and no memory error happens on Line A, but a memory error does happen on Line B.

If no memory error happens on Line A, then N < 2 30 (otherwise overflow would happen as seen above). But a memory error does happen on Line B. Line B dereferences m[i][j] , for 0 ≤ j ≤ i ; so how big is m[i] ? It was allocated on Line A with size `sizeof(point2) (i + 1) == 2 * sizeof(double) * (i + 1) == 16 * (i + 1) . If i + 1 ≥ 2<sup>32</sup> / 16 = 2<sup>28</sup>, this multiplication will overflow. Since i < N , we can finally reason that any N greater than or equal to 2<sup>28</sup> = 0x10000000 and less than 2<sup>30</sup> = 0x40000000` will cause the required memory error.

DATAREP-9. Data representation

Assume a 64-bit x86-64 architecture unless explicitly told otherwise.

Write your assumptions if a problem seems unclear, and write down your reasoning for partial credit.

QUESTION DATAREP-9A. Arrange the following values in increasing numeric order. Assume that x is an int with value 8192.

A possible answer might be “a < b < c = d < e < f < g < h.”

h < a = d = f < b < c < e < g

For each of the remaining questions, write one or more arguments that, when passed to the provided function, will cause it to return the integer 61 (which is 0x3d hexadecimal ). Write the expected number of arguments of the expected types.

QUESTION DATAREP-9B.

QUESTION DATAREP-9C.

"61"

QUESTION DATAREP-9D. Your answer should be different from the previous answer.

" 0x3d" , " 61 " , etc.

QUESTION DATAREP-9E. For this problem, you will also need to define a global variable. Give its type and value.

This code was compiled from: int f4 ( int a, int b) { extern unsigned char y; return (a & 5 ) * 2 + b - y; } A valid solution is a =0, b =61, unsigned char y =0.

DATAREP-10. Sizes and alignments

QUESTION DATAREP-10A. Use the following members to create a struct of size 16, using each member exactly once, and putting char a first; or say “impossible” if this is impossible.

• char a; (we’ve written this for you)
• unsigned char b;

QUESTION DATAREP-10B. Repeat Part A, but create a struct with size 12.

abdc, acbd, acdb, adbc, adcb, …

QUESTION DATAREP-10C. Repeat Part A, but create a struct with size 8.

QUESTION DATAREP-10D. Consider the following structs:

Give definitions for T, U, and V so that there is one byte of padding in struct x after x2 , and two bytes of padding in struct y after y1 .

Example: T = short[2] , U = char , V = int

DATAREP-11. Dynamic memory allocation

QUESTION DATAREP-11A. True or false?

• free(nullptr) is an error.
• malloc(0) can never return nullptr .

False, False

QUESTION DATAREP-11B. Give values for sz and nmemb so that calloc(sz, nmemb) will always return nullptr (on a 32-bit x86 machine), but malloc(sz * nmemb) might or might not return null.

(size_t) -1, (size_t) -1 —anything that causes an overflow

Consider the following 8 statements. ( p and q have type char* .)

• q = nullptr;
• p = (char*) malloc(12);
• q = (char*) malloc(8);

QUESTION DATAREP-11C. Put the statements in an order that would execute without error or evoking undefined behavior. Memory leaks count as errors. Use each statement exactly once. Sample answer: “abcdefgh.”

cdefghab (and others). Expect “OK”

QUESTION DATAREP-11D. Put the statements in an order that would cause one double-free error, and no other error or undefined behavior (except possibly one memory leak). Use each statement exactly once.

efghbcad (and others). Expect “double-free + memory leak”

QUESTION DATAREP-11E. Put the statements in an order that would cause one memory leak (one allocated piece of memory is not freed), and no other error or undefined behavior. Use each statement exactly once.

efghadbc (and others). Expect “memory leak”

QUESTION DATAREP-11F. Put the statements in an order that would cause one boundary write error, and no other error or undefined behavior. Use each statement exactly once.

eafhcgbd (and others). Expect “out of bounds write”

DATAREP-12. Pointers and debugging allocators

You are debugging some students’ m61 code from Problem Set 1. The codes use the following metadata:

Their linked-list manipulations in m61_malloc are similar.

But their linked-list manipulations in m61_free differ.

Alice’s code: void m61_free ( void * ptr, ...) { ... meta * m = (meta * ) ptr - 1 ; if (m -> next != nullptr ) { m -> next -> prev = m -> prev; } if (m -> prev == nullptr ) { mhead = nullptr ; } else { m -> prev -> next = m -> next; } ... }

Bob’s code: void m61_free ( void * ptr, ...) { ... meta * m = (meta * ) ptr - 1 ; if (m -> next) { m -> next -> prev = m -> prev; } if (m -> prev) { m -> prev -> next = m -> next; } ... }

Chris’s code: void m61_free ( void * ptr, ...) { ... meta * m = (meta * ) ptr - 1 ; m -> next -> prev = m -> prev; m -> prev -> next = m -> next; ... }

Donna’s code: void m61_free ( void * ptr, ...) { ... meta * m = (meta * ) ptr - 1 ; if (m -> next) { m -> next -> prev = m -> prev; } if (m -> prev) { m -> prev -> next = m -> next; } else { mhead = m -> next; } ... }

You may assume that all code not shown is correct.

QUESTION DATAREP-12A. Whose code will segmentation fault on this input? List all students that apply.

QUESTION DATAREP-12B. Whose code might report something like “ invalid free of pointer [ptr1], not allocated ” on this input? (Because a list traversal starting from mhead fails to find ptr1 .) List all students that apply. Don’t include students whose code would segfault before the report.

QUESTION DATAREP-12C. Whose code would improperly report something like “ LEAK CHECK: allocated object [ptr1] with size 1 ” on this input? (Because the mhead list appears not empty, although it should be.) List all students that apply. Don’t include students whose code would segfault before the report.

QUESTION DATAREP-12D. Whose linked-list code is correct for all inputs? List all that apply.

DATAREP-13. Arena allocation

Chimamanda Ngozi Adichie is a writing a program that needs to allocate and free a lot of nodes, where a node is defined as follows:

She uses an arena allocator variant. Here’s her code.

QUESTION DATAREP-13A. True or false?

• This allocator never has external fragmentation.
• This allocator never has internal fragmentation.

QUESTION DATAREP-13B. Chimamanda’s frenemy Paul Auster notices that if many nodes are allocated right in a row, every 1024th allocation seems much more expensive than the others. The reason is that every 1024th allocation initializes a new group, which in turn adds 1024 nodes to the free list. Chimamanda decides instead to allow a single element of the free list to represent many contiguous free nodes . The average allocation might get a tiny bit slower, but no allocation will be much slower than average. Here’s the start of her idea:

Complete this function by writing code to replace // ??? .

if (n -> key == 1 ) { a -> frees = n -> right; } else { a -> frees = n + 1 ; a -> frees -> key = n -> key - 1 ; a -> frees -> right = n -> right; } Another solution: if (n -> right) { a -> frees = n -> right; } else if (n -> key == 1 ) { a -> frees = NULL ; } else { a -> frees = n + 1 ; a -> frees -> key = n -> key - 1 ; }

QUESTION DATAREP-13C. Write a node_free function that works with the node_alloc function from the previous question.

void node_free (arena * a, node * n) { n -> right = a -> frees; n -> key = 1 ; a -> frees = n; } Or, if you use the solution above: void node_free (arena * a, node * n) { n -> right = a -> frees; a -> frees = n; }

QUESTION DATAREP-13D. Complete the following new function.

arena_group * node_find_group (arena * a, node * n) { for (arena_group * g = a -> groups; g; g = g -> next_group) { if ((uintptr_t) & g -> nodes[ 0 ] <= (uintptr_t) n && (uintptr_t) n <= (uintptr_t) & g -> nodes[ 1023 ]) { return g; } } return nullptr ; }

QUESTION DATAREP-13E. Chimamanda doesn’t like that the node_find_group function from part D takes O ( G ) time, where G is the number of allocated arena_groups. She remembers a library function that might help, posix_memalign :

The function posix_memalign() allocates size bytes and places the address of the allocated memory in *memptr . The address of the allocated memory will be a multiple of alignment , which must be a power of two and a multiple of sizeof(void*) . ...

“Cool,” she says, “I can use this to speed up node_find_group !” She now allocates a new group with the following code:

Given this allocation strategy, write a version of node_find_group that takes O (1) time.

arena_group * node_find_group (arena * a, node * n) { uintptr_t n_addr = (uintptr_t) n; return (arena_group * ) (n_addr - n_addr % 32768 ); }

DATAREP-14. Data representation

Sort the following expressions in ascending order by value, using the operators <, =, >. For example, if we gave you:

• int B = 0x6;

you would write C < A = B .

• unsigned char a = 0x191;
• char b = 0x293;
• unsigned long c = 0xFFFFFFFF;
• int d = 0xFFFFFFFF;
• int e = d + 3;
• size_t g = sizeof(*s) (given short *s )
• long h = 256;
• i = 0b100000000000000000000000000000000000 (binary)
• unsigned long j = 0xACE - 0x101;

b < d < e = g < a < h < j < c < f < i

DATAREP-15. Memory

For the following questions, select the part(s) of memory from the list below that best describes where you will find the object.

• between the heap and the stack
• in a read-only data segment
• in a text segment starting at address 0x08048000
• in a read/write data segment
• in a register

Assume the following code, compiled without optimization.

QUESTION DATAREP-15A. The value 0xdeadbeef, when we are returning from main.

7, in a register

QUESTION DATAREP-15B. The variable maxitems

4, in a read-only data segment

QUESTION DATAREP-15C. The structure s

6, in a read/write data segment

QUESTION DATAREP-15D. The structure at sp[9]

QUESTION DATAREP-15E. The variable u

2, stack, or 7, in a register

QUESTION DATAREP-15F. main

5, in a text segment starting at address 0x08048000

QUESTION DATAREP-15G. printf

3, between the heap and the stack

QUESTION DATAREP-15H. argc

QUESTION DATAREP-15I. The number the user enters

QUESTION DATAREP-15J. The variable L

DATAREP-16. Memory and pointers

⚠️ This question may benefit from Unit 4, kernel programming. ⚠️

If multiple processes are sharing data via mmap , they may have the file mapped at different virtual addresses. In this case, pointers to the same object will have different values in the different processes. One way to store pointers in mmapped memory so that multiple processes can access them consistently is using relative pointers. Rather than storing a regular pointer, you store the offset from the beginning of the mmapped region and add that to the address of the mapping to obtain a real pointer. An alternative representation is called self-relative pointers. In this case, you store the difference in address between the current location (i.e., the location containing the pointer) and the location to which you want to point. Neither representation addresses pointers between the mmapped region and the rest of the address space; you may assume such pointers do not exist.

QUESTION DATAREP-16A. State one advantage that relative pointers have over self-relative pointers.

The key thing to understand is that both of these approaches use relative pointers and both can be used to solve the problem of sharing a mapped region among processes that might have the region mapped at different addresses. Possible advantages: Within a region, you can safely use memcpy as moving pointers around inside the region does not change their value. If you copy a self relative pointer to a new location, its value has to change. That is, imagine that you have a self-relative pointer at offset 4 from the region and it points to the object at offset 64 from the region. The value of the self relative pointer is 60. If I copy that pointer to the offset 100 from the region, I have to change it to be -36. If you save the region as a uintptr_t or a char * , then you can simply add the offset to the region; self-relative-pointers will always be adding/subtracting from the address of the location storing the pointer, which may have a type other than char * , so you'd need to cast it before performing the addition/subtraction. You can use a larger region: if we assume that we have only N bits to store the pointer, then in the base+offset model, offset could be an unsigned value, which will be larger than the maximum offset possible with a signed pointer, which you need for the self-relative case. That is, although the number of values that can be represented by signed and unsigned numbers differs by one, the implementation must allow for a pointer from the beginning of the region to reference an item at the very last location of the region -- thus, your region size is limited by the largest positive number you can represent.

QUESTION DATAREP-16B. State one advantage that self-relative pointers have over relative pointers.

You don't have to know the address at which the region is mapped to use them. That is, given a location containing a self-relative pointer, you can find the target of that pointer.

For the following questions, assume the following setup:

QUESTION DATAREP-16C. Propose a type for TYPE1 and give 1 sentence why you chose that type.

A good choice is ptrdiff_t , which represents differences between pointers. Other reasonable choices include uintptr_t and unsigned long .

QUESTION DATAREP-16D. Write a C expression to generate a (properly typed) pointer to the element referenced by the r_next field of ll1 .

(ll1*) (region + node1.r_next)

QUESTION DATAREP-16E. Propose a type for TYPE2 and give 1 sentence why you chose that type.

The same choices work; again ptrdiff_t is best.

QUESTION DATAREP-16F. Write a C expression to generate a (properly typed) pointer to the element referenced by the sr_next field of ll2 .

(ll2*) ((char*) &node2.sr_next + node2.sr_next)

DATAREP-17. Data representation: Allocation sizes

How much user-accessible space is allocated on the stack and/or the heap by each of the following statements? Assume x86-64.

QUESTION DATAREP-17A. union my_union { ... };

0; this declares the type , not any object

QUESTION DATAREP-17B. void* p = malloc(sizeof(char*));

16: 8 on the heap plus 8 on the stack

QUESTION DATAREP-17C. my_union u;

16 (on the stack)

QUESTION DATAREP-17D. my_union* up = &u;

8 (on the stack)

DATAREP-18. Data representation: ENIAC

Professor Kohler has been developing Eddie’s NIfty Awesome Computer (ENIAC). When he built the C compiler for ENIAC, he assigned the following sizes and alignments to C’s fundamental data types. (Assume that every other fundamental type has the same size and alignment as one of these.)

QUESTION DATAREP-18A. This set of sizes is valid: it obeys all the requirements set by C’s abstract machine. Give one different size assignment that would make the set as a whole invalid.

Some examples: sizeof(char) = 0 ; sizeof(char) = 2 ; sizeof(short) = 8 (i.e., longer than int ); sizeof(int) = 2 (though not discussed in class, turns out that C++ requires ints are at least 2 bytes big); etc.

QUESTION DATAREP-18B. What alignment must the ENIAC malloc guarantee?

For the following two questions, assume the following struct on the ENIAC:

QUESTION DATAREP-18C. What is sizeof(struct s) ?

f1 is 7 bytes. f2 is 16 bytes with 16-byte alignment, so add 9B padding. f3 is 4 bytes (and is already aligned). f4 is 8 bytes with 8-byte alignment, so add 4B padding. That adds up to 7 + 9 + 16 + 4 + 4 + 8 = 16 + 16 + 16 = 48 bytes. That’s a multiple of the structure’s alignment, which is 16, so no need for any end padding.

QUESTION DATAREP-18D. What is alignof(struct s) ?

The remaining questions refer to this structure definition:

Indicate for each statement whether the statement is always true, possibly true, or never true on the ENIAC.

QUESTION DATAREP-18E : sizeof(outer) > sizeof(inner) (Always / Possibly / Never)

QUESTION DATAREP-18F : sizeof(outer) is a multiple of sizeof(inner) (Always / Possibly / Never)

QUESTION DATAREP-18G : alignof(outer) > alignof(struct inner) (Always / Possibly / Never)

QUESTION DATAREP-18H : sizeof(outer) - sizeof(inner) < 4 (Always / Possibly / Never)

QUESTION DATAREP-18I : sizeof(outer) - sizeof(inner) > 32 (Always / Possibly / Never)

QUESTION DATAREP-18J : alignof(inner) == 2 (Always / Possibly / Never)

DATAREP-19. Undefined behavior

Which of the following expressions, instruction sequences, and code behaviors cause undefined behavior? For each question, write Defined or Undefined. (Note that the INT_MAX and UINT_MAX constants have types int and unsigned , respectively.)

QUESTION DATAREP-19A. INT_MAX + 1 (Defined / Undefined)

QUESTION DATAREP-19B. UINT_MAX + 1 (Defined / Undefined)

QUESTION DATAREP-19C.

(Defined / Undefined)

Defined (only C++ programs can have undefined behavior; the behavior of x86-64 instructions is always defined)

QUESTION DATAREP-19D. Failed memory allocation, i.e., malloc returns nullptr (Defined / Undefined)

QUESTION DATAREP-19E. Use-after-free (Defined / Undefined)

QUESTION DATAREP-19F. Here are two functions and a global variable:

C’s undefined behavior rules would allow an aggressive optimizing compiler to simplify the code generated for f . Fill in the following function with the simplest C code you can, under the constraint that an aggressive optimizing compiler might generate the same object code for f and f_simplified .

return i * 2;

DATAREP-20. Bit manipulation

It’s common in systems code to need to switch data between big-endian and little-endian representations. This is because networks represent multi-byte integers using big-endian representation, whereas x86-family processors store multi-byte integers using little-endian representation.

QUESTION DATAREP-20A. Complete this function, which translates an integer from big-endian representation to little-endian representation by swapping bytes. For instance, big_to_little(0x01020304) should return 0x04030201 . Your return statement must refer to the u.c array, and must not refer to x . This function is compiled on x86-64 Linux (as every function is unless we say otherwise).

return (u.c[ 0 ] << 24 ) | (u.c[ 1 ] << 16 ) | (u.c[ 2 ] << 8 ) | u.c[ 3 ];

QUESTION DATAREP-20B. Complete the function again, but this time write a single expression that refers to x (you may refer to x multiple times, of course).

return ((x & 0xFF ) << 24 ) | ((x & 0xFF00 ) << 8 ) | ((x & 0xFF0000 ) >> 8 ) | (x >> 24 );

QUESTION DATAREP-20C. Now write the function little_to_big , which will translate a little-endian integer into big-endian representation. You may introduce helper variables or even call big_to_little if that’s helpful.

return big_to_little (x);

DATAREP-21. Computer arithmetic

Bitwise operators and computer arithmetic can represent vectors of bits, which in turn are useful for representing sets . For example, say we have a function bit that maps elements to distinct bits; thus, bit(X) == (1 << i) for some i . Then a set {X0, X1, X2, …, Xn} can be represented as bit(X0) | bit(X1) | bit(X2) | … | bit(Xn) . Element Xi is in the set with integer representation z if and only if (bit(Xi) & z) != 0 .

QUESTION DATAREP-21A. What is the maximum number of set elements that can be represented in a single unsigned variable on an x86 machine?

QUESTION DATAREP-21B. Match each set operation with the C operator(s) that could implement that operation. (Complement is a unary operation.)

intersection & equality == complement ~ union | toggle membership ^

QUESTION DATAREP-21C. Complete this function, which should return the set difference between the sets with representations a and b . This is the set containing exactly those elements of set a that are not in set b .

Any of these work: return a & ~ b; return a - (a & b); return a & ~ (a & b);

QUESTION DATAREP-21D. Below we’ve given a number of C++ expressions, some of their values, and some of their set representations for a set of elements. For example, the first row says that the integer value of expression 0 is just 0, which corresponds to an empty set. Fill in the blanks. This will require figuring out which bits correspond to the set elements A , B , C , and D , and the values for the 32-bit int variables a , x , and s . No arithmetic operation overflows; abs(x) returns the absolute value of x (that is, x < 0 ? -x : x ).

Expression e Integer value Represented set 0 0 {} a == a 1 {A} (unsigned) ~a < (unsigned) a 1 {A} a < 0 1 {A} (1 << (s/2)) - 1 15 {A,B,C,D} a * a 4 {C} abs(a) 2 {D} x & (x - 1) 0 {} x - 1 3 {A,D} x 4 {C} s 8 {B}

DATAREP-22. Bit Tac Toe

Brenda Bitdiddle is implementing tic-tac-toe using bitwise arithmetic. (If you’re unfamiliar with tic-tac-toe, see below.) Her implementation starts like this:

Each position on the board is assigned a distinct bit.

Tic-tac-toe, also known as noughts and crosses, is a simple paper-and-pencil game for two players, X and O. The board is a 3x3 grid. The players take turns writing their symbol (X or O) in an empty square on the grid. The game is won when one player gets their symbol in all three squares in one of the rows, one of the columns, or one of the two diagonals. X goes first; played perfectly, the game always ends in a draw. You may access the Wikipedia page for tic-tac-toe .

QUESTION DATAREP-22A. Brenda’s current code doesn’t check whether a move reuses a position. Write a snippet of C code that returns –1 if an attempted move is reusing a position. This snippet will replace line 3.

Lots of people misinterpreted this to mean the player reused their own position and ignored the other player. That mistake was allowed with no points off. The code below checks whether any position was reused by either player. if ((b -> moves[XS] | b -> moves[OS]) & ttt_values[row][col]) { return - 1 ; } OR if ((b -> moves[XS] | b -> moves[OS] | ttt_values[row][col]) == (b -> moves[XS] | b -> moves[OS])) { return - 1 ; } OR if ((b -> moves[XS] + b -> moves[OS]) & ttt_values[row][col]) { return - 1 ; } OR if ((b -> moves[p] ^ ttt_values[row][col]) < b -> moves[p]) { return - 1 ; } etc.

QUESTION DATAREP-22B. Complete the following function. You may use the following helper function:

• int popcount(unsigned n) Return the number of 1 bits in n . (Stands for “population count”; is implemented on recent x86 processors by a single instruction, popcnt .)

For full credit, your code should consist of a single “ return ” statement with a simple expression, but for substantial partial credit write any correct solution.

return popcount (b -> moves[XS] | b -> moves[OS]);

QUESTION DATAREP-22C. Write a simple expression that, if nonzero, indicates that player XS has a win on board b across the main diagonal (has marks in positions 0,0 , 1,1 , and 2,2 ).

(b -> moves[XS] & 0x421 ) == 0x421

Lydia Davis notices Brenda’s code and has a brainstorm. “If you use different values,” she suggests, “it becomes easy to detect any win.” She suggests:

QUESTION DATAREP-22D. Repeat part A for Lydia’s values: Write a snippet of C code that returns –1 if an attempted move is reusing a position. This snippet will replace line 3 in Brenda’s code.

The same answers as for part A work.

QUESTION DATAREP-22E. Repeat part B for Lydia’s values: Use popcount to complete tictactoe_nmoves .

Either of: return popcount ((b -> moves[ 0 ] | b -> moves[ 1 ]) & 0x777 ); return popcount ((b -> moves[ 0 ] | b -> moves[ 1 ]) & 0x777000 );

QUESTION DATAREP-22F. Complete the following function for Lydia’s values. For full credit, your code should consist of a single “ return ” statement containing exactly two constants, but for substantial partial credit write any correct solution.

return (b -> moves[p] + 0x11111111 ) & 0x88888888 ; // Another amazing possibility (Allen Chen and others): return b -> moves[p] & (b -> moves[p] << 1 ) & (b -> moves[p] << 2 );

DATAREP-23. Memory and Pointers

Two processes are mapping a file into their address space. The mapped file contains an unsorted linked list of integers. As the processes cannot ensure that the file will be mapped at the same virtual address, they use relative pointers to link elements in the list. A relative pointer holds not an address, but an offset that user code can use to calculate a true address. Our processes define the offset as relative to the start of the file.

Thus, each element in the linked list is represented by the following structure:

offset == (size_t) -1 indicates the end of the list. Other offset values represent the position of the next item in the list, calculated relative to the start of the file.

QUESTION DATAREP-23A. Write a function to find an item in the list. The function's prototype is:

The mapped_file parameter is the address of the mapped file data; the list parameter is a pointer to the first node in the list; and the value parameter is the value for which we are searching. The function should return a pointer to the linked list element if the value appears in the list or nullptr if the value is not in the list.

ll_node * find_element ( void * mapped_file, ll_node * list, int value) { while ( 1 ) { if (list -> value == value) return list; if (list -> offset == (size_t) - 1 ) return NULL ; list = (ll_node * ) (( char * ) mapped_file + list -> offset); } }

DATAREP-24. Integer representation

Write the value of the variable or expression in each problem, using signed decimal representation.

For example, if we gave you:

• int i = 0xA;
• int j = 0xFFFFFFFF;

you would write A) 10 B) -1.

QUESTION DATAREP-24A. int i = 0xFFFF; (You may write this either in decimal or as an expression using a power of 2)

2 16 - 1 or 65535

QUESTION DATAREP-24B. short s = 0xFFFF; (You may write this either in decimal or as an expression using a power of 2)

QUESTION DATAREP-24C. unsigned u = 1 << 10;

1024 or 2 10

QUESTION DATAREP-24D. ⚠️ From WeensyOS: unsigned long l = PTE_P | PTE_U;

QUESTION DATAREP-24E. int j = ~0;

QUESTION DATAREP-24F. ⚠️ From WeensyOS: sizeof(x86_64_pagetable);

4096 or 2 12

QUESTION DATAREP-24G. Given this structure:

This expression: sizeof(ps);

TRICK QUESTION! 8

QUESTION DATAREP-24H. Using the structure above: sizeof(*ps);

QUESTION DATAREP-24I. unsigned char u = 0xABC;

0xBC == 11*16 + 12 == 160 + 16 + 12 == 188

QUESTION DATAREP-24J. signed char c = 0xABC;

0xBC has most-significant bit on, so the value as a signed char is less than zero. We seek x so that 0xBC + x == 0x100 . The answer is 0x44: 0xBC + 4 == 0xC0 , and 0xC0 + 0x40 == 0x100 . So −0x44 == −4*16 − 4 == −68 .

DATAREP-25. Data representation

In gdb, you observe the following values for a set of memory locations.

For each C expression below, write its value in hexadecimal. For example, if we gave you:

the answer would be 0xa0 .

Assume the following structure and union declarations and variable definitions.

QUESTION DATAREP-25A. cp[4] =

QUESTION DATAREP-25B. cp + 7 =

0x100001027

QUESTION DATAREP-25C. s1 + 1 =

0x100001038

QUESTION DATAREP-25D. s1->i =

0xd3c2b1a0 (-742215264)

QUESTION DATAREP-25E. sizeof(s1) =

QUESTION DATAREP-25F. &s2->s =

0x100001028

QUESTION DATAREP-25G. &u->s =

0x100001020

QUESTION DATAREP-25H. s1->l =

0x9f8e7d6c5b4a3928 (-6949479270644565720)

QUESTION DATAREP-25I. s2->s.s =

0xf201 (-3583)

QUESTION DATAREP-25J. u->l =

0x1706f5e4d3c2b1a0 (1659283875886707104)

DATAREP-26. Sizes and alignments

Here’s a test struct with n members. Assume an x86-64 machine, where each Ti either is a basic x86-64 type (e.g., int , char , double ) or is a type derived from such types (e.g., arrays, structs, pointers, unions, possibly recursively), and assume that ai ≤8 for all i .

In these questions, you will compare this struct with other structs that have the same members, but in other orders.

QUESTION DATAREP-26A. True or false: The size of struct test is minimized when its members are sorted by size. In other words, if s1 ≤ s2 ≤…≤ sn , then sizeof(struct test) is less than or equal to the struct size for any other member order.

If true, briefly explain your answer; if false, give a counterexample (i.e., concrete types for T1 , …,  Tn that do not minimize sizeof(struct test) ).

False. T1 = char , T2 = int , T3 = char[5]

QUESTION DATAREP-26B. True or false: The size of struct test is minimized when its members are sorted by alignment. In other words, if a1 ≤ a2 ≤…≤ an , then sizeof(struct test) is less than or equal to the struct size for any other member order.

If true, briefly explain your answer; if false, give a counterexample.

True. Padding only occurs between objects with different alignments, and is limited by the second alignment; sorting by alignment therefore minimizes padding.

QUESTION DATAREP-26C. True or false: The alignment of struct test is minimized when its members are sorted in increasing order by alignment. In other words, if a1 ≤ a2 ≤…≤ an , then alignof(struct test) is less than or equal to the struct alignment for any other member order.

True. It’s all the same; alignment is max alignment of every component, and is independent of order.

QUESTION DATAREP-26D. What is the maximum number of bytes of padding that struct test could contain for a given n ? The answer will be a pretty simple formula involving n . (Remember that ai ≤8 for all i .)

Alternating char and long gives the most padding, which is 7*(n/2) when n is even and 7*(n+1)/2 otherwise.

QUESTION DATAREP-26E. What is the minimum number of bytes of padding that struct test could contain for a given n ?

DATAREP-27. Undefined behavior

QUESTION DATAREP-27A. Sometimes a conforming C compiler can assume that a + 1 > a , and sometimes it can’t. For each type below, consider this expression:

and say whether the compiler:

• Must reject the expression as a type error.
• May assume that the expression is true (that a + (int) 1 > a for all a ).
• Must not assume that the expression is true.
• unsigned char a
• struct {int m;} a

1—May assume; 2—Must not assume; 3—May assume; 4—May assume (in fact due to integer promotion, this statement really is always true, even in mathematical terms); 5—Must reject.

QUESTION DATAREP-27B. The following code checks its arguments for sanity, but not well: each check can cause undefined behavior.

Rewrite these checks to avoid all undefined behavior. You will likely add one or more casts to uintptr_t . For full credit, write each check as a single comparison (no && or || , even though the current ptr_into_array check uses || ).

array_size check:

ptr_into_array check:

array_size check: (uintptr_t) array + 4 * array_size < (uintptr_t) array ptr_into_array check: (uintptr_t) ptr_into_array - (uintptr_t) array > 4 * array_size

QUESTION DATAREP-27C. In lecture, we discussed several ways to tell if a signed integer x is negative. One of them was the following:

But this is incorrect: it has undefined behavior. Correct it by adding two characters.

(x & (1UL << (sizeof(x) * CHAR_BIT - 1))) != 0

DATAREP-28. ⚠️ Memory errors and garbage collection ⚠️

⚠️ We didn’t discuss garbage collectors in class this year.

Recall that a conservative garbage collector is a program that can automatically free dynamically-allocated memory by detecting when that memory is no longer referenced. Such a GC works by scanning memory for currently-referenced pointers, starting from stack and global memory, and recursing over each referenced object until all referenced memory has been scanned. We built a conservative garbage collector in lecture datarep6.

QUESTION DATAREP-28A. An application program that uses conservative GC, and does not call free directly, will avoid certain errors and undefined behaviors. Which of the following errors are avoided? List all that apply.

• Use-after-free
• Double free
• Signed integer overflow
• Boundary write error
• Unaligned access

QUESTION DATAREP-28B. Write a C program that leaks unbounded memory without GC, but does not do so with GC. You should need less than 5 lines. (Leaking “unbounded” memory means the program will exhaust the memory capacity of any machine on which it runs.)

while ( 1 ) { ( void ) malloc( 1 ); }

QUESTION DATAREP-28C. Not every valid C program works with a conservative GC, because the C abstract machine allows a program to manipulate pointers in strange ways. Which of the following pointer manipulations might cause the conservative GC from class to inappropriately free a memory allocation? List all that apply.

Storing the pointer in a uintptr_t variable

Writing the pointer to a disk file and reading it back later

Using the least-significant bit of the pointer to store a flag:

Storing the pointer in textual form:

Splitting the pointer into two parts and storing the parts in an array:

DATAREP-29. Bitwise

QUESTION DATAREP-29A. Consider this C fragment:

Or, shorter:

Write a single expression that evaluates to the same value, but that does not use the conditional ?: operator. You will use the fact that a < b always equals 0 or 1. For full credit, do not use expensive operators (multiply, divide, modulus).

Examples: (a < b) * x , (-(uintptr_t) (a < b)) & x

QUESTION DATAREP-29B. This function returns one more than the index of the least-significant 1 bit in its argument, or 0 if its argument is zero.

This function runs in O ( B ) time, where B is the number of bits in an unsigned long . Write a version of ffs that runs instead in O (log  B ) time.

int ffs ( unsigned long x) { if ( ! x) { return 0 ; } int ans = 1 ; if ( ! (x & 0x00000000FFFFFFFFUL )) { ans += 32 ; x >>= 32 ; } if ( ! (x & 0x0000FFFF )) { ans += 16 ; x >>= 16 ; } if ( ! (x & 0x00FF )) { ans += 8 ; x >>= 8 ; } if ( ! (x & 0x0F )) { ans += 4 ; x >>= 4 ; } if ( ! (x & 0x3 )) { ans += 2 ; x >>= 2 ; } return ans + (x & 0x1 ? 0 : 1 ); }

DATAREP-30. Data representation

QUESTION DATAREP-30A. Write a type whose size is 19,404,329 times larger than its alignment.

char[19404329]

QUESTION DATAREP-30B. Consider a structure type T with N members, all of which have nonzero size. Assume that sizeof(T) == alignof(T) . What is N ?

QUESTION DATAREP-30C. What is a C type that obeys (T) -1 == (T) 255 on x86-64?

char (or unsigned char or signed char )

Parts D–G use this code. The architecture might or might not be x86-64.

Assume that (uintptr_t) s2 - (uintptr_t) s1 == 4 and *s1 > *s2 .

QUESTION DATAREP-30D. What is sizeof(a) ?

QUESTION DATAREP-30E. What is sizeof(unsigned) on this architecture?

QUESTION DATAREP-30F. Is this architecture big-endian or little-endian?

QUESTION DATAREP-30G. Might the architecture be x86-64?

DATAREP-31. Memory errors

Mavis Gallant is starting on her debugging memory allocator. She’s written code that aims to detect invalid frees, where a pointer passed to m61_free was not returned by an earlier m61_malloc .

Help her track down bugs.

QUESTION DATAREP-31A. What is sizeof(struct m61_metadata) ?

QUESTION DATAREP-31B. Give an m61_ function call (function name and arguments) that would cause both unsigned integer overflow and invalid memory accesses.

m61_malloc((size_t) -15) . This turns into malloc(1) and the dereference of meta->magic becomes invalid.

QUESTION DATAREP-31C. Give an m61_ function call (function name and arguments) that would cause integer overflow, but no invalid memory access within the m61_ functions . (The application might or might not make an invalid memory access later.)

m61_malloc((size_t) -1)

QUESTION DATAREP-31D. These functions have some potential null pointer dereferences. Fix one such problem, including the line number(s) where your code should go.

C3: if (p) { memset(p, 0, sz * count); }

QUESTION DATAREP-31E. Put a single line of C code in the blank. The resulting program should (1) be well-defined with no memory leaks when using default malloc / free / calloc , but (2) always cause undefined behavior when using Mavis’s debugging malloc / free / calloc .

free(nullptr);

QUESTION DATAREP-31F. A double free should print a different message than an invalid free. Write code so Mavis’s implementation does this; include the line numbers where the code should go.

F4: fprintf(stderr, meta->magic == 0xB0B0B0B0 ? "double free of %p" : "invalid free of %p", ptr) after F6: meta->magic = 0xB0B0B0B0;

DATAREP-32. Memory word search

QUESTION DATAREP-32A. What is decimal 61 in hexadecimal?

0x3d, 2 pts

The following is a partial dump of an x86-64 Linux program’s memory. Note that each byte is represented as a hexadecimal number. Memory addresses increase from top to bottom and from left to right.

QUESTION DATAREP-32B. What memory segment contains the memory dump?

Data segment (globals), 2 pts

For questions 3C–3I, give the last two digits of the address of the given object in this memory dump (so for the object located at address 0x6010a8, you would write “a8”). There’s a single best answer for every question, and all questions have different answers .

QUESTION DATAREP-32C. A long (64 bits) of value 61.

QUESTION DATAREP-32D. A long of value -61.

QUESTION DATAREP-32E. An int of value 61.

QUESTION DATAREP-32F. A pointer to an object in the heap.

QUESTION DATAREP-32G. A pointer to a local variable.

QUESTION DATAREP-32H. A pointer that points to an object present in the memory dump.

QUESTION DATAREP-32I. The first byte of a C string comprising at least 4 printable ASCII characters. (Printable ASCII characters have values from 0x20 to 0x7e.)

DATAREP-33. Architecture

QUESTION DATAREP-33A. Write a C++ expression that will evaluate to false on all typical 32-bit architectures and true on all typical 64-bit architectures.

3pts sizeof(void*) == 8

QUESTION DATAREP-33B. Give an integer value that has the same representation in big-endian and little-endian, and give its C++ type. Assume the same type sizes as x86-64.

2pts ex: int x = 0

QUESTION DATAREP-33C. Repeat question B, but with a different integer value.

2pts ex: int x = -1 :)

QUESTION DATAREP-33D. Complete this C++ function, which should return true iff it is called on a machine with little-endian integer representation. Again, you may assume the same type sizes as x86-64.

3pts bool is_little_endian () { union { int a; char c[ 4 ]; } u; u.a = 0 ; u.c[ 0 ] = 1 ; return u.a == 1 ; }

DATAREP-34. Undefined behavior

The following code is taken from the well-known textbook Mastering C Pointers (with some stylistic changes). Assume it is run on x86-64.

QUESTION DATAREP-34A. List all situations in which this program will not execute undefined behavior. (There is at least one.)

3pts If malloc returns nullptr .

QUESTION DATAREP-34B. True or false? The expression ++x could be replaced with ++y without changing the meaning of the program.

3pts True! The undefined behavior is the same either way.

Here’s an updated version of the program.

QUESTION DATAREP-34C. True or false? The expression *(x + y) could be replaced with x[y] without changing the meaning of the program.

QUESTION DATAREP-34D. True or false? The expression *(x + y) could be replaced by * (int*) ((uintptr_t) x + y) without changing the meaning of the program.

3pts False (address arithmetic isn’t the same for pointer arithmetic, except for pointers with type equivalent to char )

DATAREP-35. Odd alignments

QUESTION DATAREP-35A. Write a struct definition that contains exactly seven bytes of padding on x86-64.

3pts struct {long a; char b;}

QUESTION DATAREP-35B. Can an x86-64 struct comprise more than half padding? Give an example if so, or explain briefly why not.

3pts Yes. struct {char a; long b; char c;} has size 24 and 14 bytes of padding.

The remaining questions consider a new architecture called x86-64-rainbow , which is like x86-64 plus special support for a fundamental data type called color . A color is a three-byte data type with red, green, and blue components, kind of like this:

But unlike struct color on x86-64, which has size 3 and alignment 1, color on x86-64-rainbow has size 3 and alignment 3 . All the usual rules for C abstract machine sizes and alignments still hold.

QUESTION DATAREP-35C. What is the alignment of pointers returned by malloc on x86-64-rainbow?

QUESTION DATAREP-35D. Give an example of an x86-64-rainbow struct that ends with more than 16 bytes of padding.

2pts struct { long a[2]; color b[3]; } has alignment lcm(8, 3) = 24 and size 8 + 8 + 3 + 3 + 3 = 25, so it ends with 23 bytes of padding!

DATAREP-36. Debugging allocators

In problem set 1, you built a debugging allocator that could detect many kinds of error. Some students used exclusively internal metadata , where the debugging allocator’s data was stored within the same block of memory as the payload. Some students used external metadata , such as a separate hash table mapping payload pointers to metadata information.

QUESTION DATAREP-36A. Which kind of error mentioned by the problem set could not be detected by external metadata alone?

2pts Boundary write errors.

The problem set requires the m61 library to detect invalid frees, which includes double frees. However, double frees are so common that a special double-free error message can help users. Jia Tolentino wants to print such a message. Her initial idea is to keep a set of recently freed pointers:

QUESTION DATAREP-36B. What eviction policy is used for the recently_freed array?

2pts FIFO (round-robin).

Jia uses these helpers in m61_free as follows. (You may assume that is_invalid_free(ptr) returns true for every invalid or double free.)

She has not yet changed m61_malloc , though she knows she needs to. Help her evaluate whether her design achieves the following mandatory and desirable (that is, optional) requirements.

Note : As you saw in tests 32 and 33 in pset 1, aggressive attacks from users can corrupt metadata arbitrarily. You should assume for the following questions that such aggressive attacks do not occur. The user might free something that was never allocated, and might double-free something, but will never overwrite metadata.

QUESTION DATAREP-36C. Mandatory : The design must not keep unbounded, un-reusable state for pointers that have been freed. Write “Achieved” or “Not achieved” and explain briefly.

6 pts for C-E together. +5 if one serious error; +3 if two serious errors Achieved. The design only keeps space for 10 recently-freed pointers.

QUESTION DATAREP-36D. Desirable : The design should report a double-free as a double-free, not an invalid free. Write “Achieved” or “Not achieved” and explain briefly.

Not achieved. If a pointer falls off the list after 10 frees, Jia’s design will report a double-free of that pointer as an invalid free.

QUESTION DATAREP-36E. Mandatory : The design must never report valid frees as double-frees or invalid frees. Write “Achieved” or “Not achieved” and explain briefly.

Not achieved. A recently-freed pointer can be reused by malloc, but she hasn’t accounted for that. So if (1) a pointer is freed, (2) a malloc returns the same pointer, (3) the user frees that pointer again, that valid free will be reported as a double free.

QUESTION DATAREP-36F. Describe code to be added to m61_malloc that will achieve all mandatory requirements, if any are required.

3pts Here’s the cleanest solution to the 4E problem. Remove returned pointers from recently_freed . Add this code to malloc before return ptr : for ( int i = 0 ; i != 10 ; ++ i) { if (recently_freed[i] == ptr) { recently_freed[i] = nullptr ; } }

DATAREP-37. Representations

Your job in the following questions is to complete each C++ program so that it prints 61 to standard output when run. Your answer should fill in the blank _________ , and you may assume system calls execute without error. If a question cannot be answered so that the program reliably prints 61 with no undefined behavior, say so and explain briefly.

To compile these programs, use a command line such as c++ -std=gnu++17 -pthread x.cc .

QUESTION DATAREP-37A.

QUESTION DATAREP-37B.

61L or (long) 61 are best; 61 will cause a compiler warning, because it’s an int and the format expects a long .

QUESTION DATAREP-37C.

6 * 16 + 1 , 0x61 , 97

QUESTION DATAREP-37D.

157 , - x + 61 , 96 + 61

QUESTION DATAREP-37E.

char c[61];

QUESTION DATAREP-37F.

This cannot be done. struct sixtyfun has minimum alignment of 4 (because int has alignment 4), so its size must be a multiple of 4.

QUESTION DATAREP-37G.

Anything with size 20, such as char c[20]; or int x[5]; .

QUESTION DATAREP-37H.

int a = 255, b = 0, c = 61; int a = -1, b = 0 (or anything ≤31), c = 61; unsigned a = 0x10000, b = 12, c = 0xFF; int a = 32, b = 1, c = 63; There are many other possibilities!

QUESTION DATAREP-37I.

This cannot be done without undefined behavior, because y is computed using an uninitialized variable.

QUESTION DATAREP-37J.

Anything that adds 15 total to x[0] … x[2] . For example, 5

QUESTION DATAREP-37K.

{ "echo", "61", nullptr }

QUESTION DATAREP-37L.

dup2(pfd[0], 0); close(pfd[0]); close(pfd[1]); The sleep(3) , which is not waitpid , makes many other solutions work too. dup2(pfd[0], 0); works; pipe hygiene isn't important because the parent exits. printf("61\n"); almost works, but something weird happens on Mac OS X.

QUESTION DATAREP-37M.

waitpid(p, nullptr, 0)

QUESTION DATAREP-37N. Note that there are three blanks to fill in with code (you may also leave them empty).

#1, #3: nothing. #2: std::unique_lock<std::mutex> guard(m); If the lock is in #1 (and there is no unlock in #3), that causes deadlock.