contour plots in MATLAB - Time & Space Complexity

Time Complexity: contour plots

O(n^2)

Understanding Time Complexity

When we create contour plots in MATLAB, the program calculates many points to draw lines showing levels of a surface.

We want to understand how the time to draw these plots grows as the data size increases.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

% Create grid points
[x, y] = meshgrid(linspace(-2, 2, n), linspace(-2, 2, n));

% Define function values on grid
z = x.^2 + y.^2;

% Draw contour plot
contour(x, y, z, 10);
    

This code creates a grid of points, computes values for each point, and draws contour lines for 10 levels.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Calculating values for each point in the n-by-n grid.
• How many times: Once for each of the n² points.

How Execution Grows With Input

As the grid size n grows, the number of points grows by n squared, so the work grows quickly.

Input Size (n)	Approx. Operations
10	100
100	10,000
1000	1,000,000

Pattern observation: Doubling n makes the work about four times bigger because we have n by n points.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to create the contour plot grows roughly with the square of the grid size.

Common Mistake

[X] Wrong: "The time grows linearly with n because we just loop once over the data."

[OK] Correct: The data is a grid with n rows and n columns, so the total points are n times n, not just n.

Interview Connect

Understanding how plotting time grows helps you explain performance when working with large data sets or visualizations.

Self-Check

"What if we increased the number of contour levels from 10 to 100? How would the time complexity change?"

Practice

1. What does a contour plot in MATLAB primarily represent?

easy

A. A 3D surface plot

B. Lines connecting points with the same value

C. A bar chart of values

D. A scatter plot of points

Solution

1. Step 1: Understand contour plot purpose

   A contour plot shows curves that connect points with equal values on a grid.

2. Step 2: Compare with other plot types

   3D surface plots show height as a surface, bar charts show bars, scatter plots show points, so they differ.

3. Final Answer:

   Lines connecting points with the same value -> Option B

4. Quick Check:

   Contour plot = lines of equal value [OK]

Hint: Contour plots connect equal values with lines [OK]

Common Mistakes:

• Confusing contour plots with 3D surface plots
• Thinking contour plots show individual points
• Mixing contour plots with bar charts

2. Which of the following is the correct syntax to create a basic contour plot in MATLAB using matrices X, Y, and Z?

easy

A. contour(X, Y, Z)

B. plot(X, Y, Z)

C. contourplot(X, Y, Z)

D. surface(X, Y, Z)

Solution

1. Step 1: Recall MATLAB contour syntax

   The function to create contour plots is contour(X, Y, Z).

2. Step 2: Eliminate incorrect functions

   plot() is for 2D plots, contourplot() is not a MATLAB function, surface() creates 3D surfaces.

3. Final Answer:

   contour(X, Y, Z) -> Option A

4. Quick Check:

   Use contour() for contour plots [OK]

Hint: Use contour(X,Y,Z) to plot contours [OK]

Common Mistakes:

• Using plot() instead of contour()
• Typing contourplot() which is invalid
• Using surface() which makes 3D plots

3. What will be the output of the following MATLAB code?

X = 1:3; Y = 1:3; [Xg,Yg] = meshgrid(X,Y); Z = Xg.^2 + Yg.^2; contour(Xg, Yg, Z, 3);

medium

A. An error because contour needs vectors, not matrices

B. A 3D surface plot of Z values

C. A scatter plot of points (Xg, Yg, Z)

D. A contour plot with 3 contour lines representing levels of Z

Solution

1. Step 1: Understand meshgrid and Z calculation

   meshgrid creates grids Xg and Yg; Z is sum of squares, a smooth surface.

2. Step 2: Interpret contour call with 3 levels

   contour with 3 as last argument draws 3 contour lines at different Z levels.

3. Final Answer:

   A contour plot with 3 contour lines representing levels of Z -> Option D

4. Quick Check:

   contour with level count = contour lines count [OK]

Hint: Last argument in contour sets number of lines [OK]

Common Mistakes:

• Thinking contour creates 3D surface plots
• Believing contour requires vectors, not matrices
• Confusing contour with scatter plots

4. Identify the error in this MATLAB code snippet for contour plotting:

X = 1:5; Y = 1:5; Z = X + Y; contour(X, Y, Z);

medium

A. contour requires only two arguments

B. X and Y must be matrices, not vectors

C. Z must be a matrix matching grid size, not a vector

D. No error, code runs fine

Solution

1. Step 1: Check dimensions of X, Y, Z

   X and Y are vectors; Z is vector sum, not a matrix grid.

2. Step 2: Understand contour input requirements

   contour needs Z as matrix matching X, Y grid sizes; here Z is vector, causing error.

3. Final Answer:

   Z must be a matrix matching grid size, not a vector -> Option C

4. Quick Check:

   Z size mismatch causes error [OK]

Hint: Z must be matrix matching X,Y grid for contour [OK]

Common Mistakes:

• Passing vector Z instead of matrix
• Assuming contour accepts vector inputs
• Ignoring dimension mismatch errors

5. You want to plot contour lines only at specific Z values: 5, 10, and 15. Which MATLAB command correctly achieves this given matrices X, Y, and Z?

hard

A. contour(X, Y, Z, [5 10 15])

B. contour(X, Y, Z, 3)

C. contour(X, Y, Z, 'Levels', [5 10 15])

D. contour(X, Y, Z, 'LineLevels', [5 10 15])

Solution

1. Step 1: Recall contour level specification syntax

   To specify exact contour levels, pass a vector of levels as the fourth argument.

2. Step 2: Evaluate options for specifying levels

   contour(X, Y, Z, [5 10 15]) uses vector [5 10 15] correctly; contour(X, Y, Z, 3) sets number of lines, not levels; options C and D use invalid syntax.

3. Final Answer:

   contour(X, Y, Z, [5 10 15]) -> Option A

4. Quick Check:

   Use vector of levels as 4th argument [OK]

Hint: Pass vector of levels as 4th argument to contour [OK]

Common Mistakes:

• Using number instead of vector for levels
• Trying unsupported name-value pairs
• Confusing level count with specific levels