Convex hull computation in SciPy - Time & Space Complexity

Time Complexity: Convex hull computation

O(n log n)

Understanding Time Complexity

We want to understand how the time to find the convex hull changes as we add more points.

How does the work grow when the input size grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.spatial import ConvexHull
import numpy as np

points = np.random.rand(1000, 2)  # 1000 points in 2D
hull = ConvexHull(points)
    

This code computes the convex hull of 1000 random points in 2D space.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Checking and comparing points to build the hull boundary.
• How many times: Depends on the number of input points and the hull size; the algorithm processes points multiple times to find the outer boundary.

How Execution Grows With Input

As the number of points increases, the time to compute the hull grows a bit faster than the number of points.

Input Size (n)	Approx. Operations
10	About 30 operations
100	About 700 operations
1000	About 10,000 operations

Pattern observation: The operations grow roughly proportional to n log n, which means it grows faster than n but slower than n squared.

Final Time Complexity

Time Complexity: O(n log n)

This means the time to find the convex hull grows a bit faster than the number of points, but not too fast.

Common Mistake

[X] Wrong: "Computing the convex hull takes the same time no matter how many points there are."

[OK] Correct: More points mean more comparisons and checks, so the work grows as the input grows.

Interview Connect

Understanding how the convex hull algorithm scales helps you explain your approach clearly and shows you know how input size affects performance.

Self-Check

"What if the points were already sorted or arranged in a circle? How would the time complexity change?"

Practice

1. What does the scipy.spatial.ConvexHull function compute for a set of points?

easy

A. The smallest convex shape enclosing all points

B. The average distance between points

C. The maximum value among the points

D. The sum of all point coordinates

Solution

1. Step 1: Understand the purpose of ConvexHull

   The ConvexHull function finds the smallest convex polygon that contains all given points.

2. Step 2: Compare options with definition

   Only The smallest convex shape enclosing all points describes this smallest convex shape enclosing all points.

3. Final Answer:

   The smallest convex shape enclosing all points -> Option A

4. Quick Check:

   Convex hull = smallest convex shape [OK]

Hint: Convex hull wraps points tightly in a convex boundary [OK]

Common Mistakes:

• Confusing convex hull with average or sum calculations
• Thinking it finds maximum or minimum coordinate values
• Assuming it returns all points instead of boundary points

2. Which of the following is the correct way to import the ConvexHull class from scipy?

easy

A. from scipy.spatial import ConvexHull

B. import scipy.ConvexHull

C. from scipy import ConvexHull

D. import ConvexHull from scipy.spatial

Solution

1. Step 1: Recall correct import syntax in Python

   To import a class from a submodule, use 'from module.submodule import Class'.

2. Step 2: Match with scipy.spatial.ConvexHull

   The correct import is 'from scipy.spatial import ConvexHull'. Other options have syntax errors or wrong paths.

3. Final Answer:

   from scipy.spatial import ConvexHull -> Option A

4. Quick Check:

   Correct import = from scipy.spatial import ConvexHull [OK]

Hint: Use 'from module.submodule import Class' for importing classes [OK]

Common Mistakes:

• Using 'import scipy.ConvexHull' which is invalid syntax
• Trying 'import ConvexHull from ...' which is wrong order
• Importing directly from 'scipy' without 'spatial'

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

import numpy as np
from scipy.spatial import ConvexHull
points = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0.5, 0.5]])
hull = ConvexHull(points)
print(hull.vertices)

medium

A. [1 2 3]

B. [0 1 2 3 4]

C. [4]

D. [0 1 2 3]

Solution

1. Step 1: Identify points forming the convex hull

   The points at indices 0,1,2,3 form the square boundary; point 4 is inside.

2. Step 2: Understand hull.vertices output

   hull.vertices returns indices of points forming the hull boundary, excluding interior points.

3. Final Answer:

   [0 1 2 3] -> Option D

4. Quick Check:

   Vertices exclude inside points = [0 1 2 3] [OK]

Hint: Hull vertices exclude points inside the shape [OK]

Common Mistakes:

• Including interior points in hull vertices
• Expecting all points as hull vertices
• Misunderstanding output as coordinates instead of indices

4. The following code throws an error. What is the likely cause?

import numpy as np
from scipy.spatial import ConvexHull
points = np.array([[0, 0], [1, 1]])
hull = ConvexHull(points)
print(hull.vertices)

medium

A. Missing import statement

B. Points are not in ascending order

C. Not enough points to form a convex hull

D. Points array has wrong data type

Solution

1. Step 1: Check minimum points needed for convex hull

   Convex hull in 2D requires at least 3 points to form a polygon.

2. Step 2: Analyze given points

   Only 2 points are given, so ConvexHull cannot compute a polygon, causing an error.

3. Final Answer:

   Not enough points to form a convex hull -> Option C

4. Quick Check:

   Minimum 3 points needed for 2D hull [OK]

Hint: Convex hull needs at least 3 points in 2D [OK]

Common Mistakes:

• Assuming any number of points works
• Thinking order of points matters for hull computation
• Ignoring error message about input size

5. You have a set of 2D points representing city locations. You want to find the smallest fence enclosing all cities. Which steps correctly use scipy.spatial.ConvexHull to find the fence points?

hard

A. Import ConvexHull, sort points by x-coordinate, use hull.simplices to get fence points

B. Import ConvexHull, create numpy array of points, compute hull, use hull.vertices to get fence points

C. Import ConvexHull, compute hull, use hull.area to get fence points

D. Import ConvexHull, compute hull, use hull.points to get fence points

Solution

1. Step 1: Prepare data and compute hull

   Import ConvexHull, convert city coordinates to numpy array, then compute hull = ConvexHull(points).

2. Step 2: Extract fence points using hull.vertices

   hull.vertices gives indices of points forming the convex hull boundary (fence).

3. Final Answer:

   Import ConvexHull, create numpy array of points, compute hull, use hull.vertices to get fence points -> Option B

4. Quick Check:

   hull.vertices = fence points [OK]

Hint: Use hull.vertices for boundary points, not hull.area or hull.points [OK]

Common Mistakes:

• Using hull.area which is a number, not points
• Using hull.points which is all points, not boundary
• Sorting points manually instead of using hull.vertices