Distance computation (distance.cdist) in SciPy - Step-by-Step Execution

Concept Flow - Distance computation (distance.cdist)

Input: Two sets of points A and B

↓

Select distance metric (e.g., Euclidean)

↓

Compute pairwise distances between each point in A and each point in B

↓

Output: Distance matrix with shape (len(A), len(B))

We start with two groups of points, pick a distance type, then calculate distances between every pair from the two groups, producing a matrix of distances.

Execution Sample

SciPy

import numpy as np
from scipy.spatial import distance

A = np.array([[0, 0], [1, 1]])
B = np.array([[1, 0], [2, 2]])

D = distance.cdist(A, B, 'euclidean')
print(D)

This code calculates Euclidean distances between points in A and B, outputting a matrix of distances.

Execution Table

Step	Action	Points from A	Points from B	Distance Computed	Distance Matrix State
1	Start with points A and B	[[0,0],[1,1]]	[[1,0],[2,2]]	-	Empty matrix (2x2)
2	Compute distance between A[0] and B[0]	[0,0]	[1,0]	sqrt((1-0)^2+(0-0)^2)=1.0	[[1.0, ?], [?, ?]]
3	Compute distance between A[0] and B[1]	[0,0]	[2,2]	sqrt((2-0)^2+(2-0)^2)=2.8284	[[1.0, 2.8284], [?, ?]]
4	Compute distance between A[1] and B[0]	[1,1]	[1,0]	sqrt((1-1)^2+(0-1)^2)=1.0	[[1.0, 2.8284], [1.0, ?]]
5	Compute distance between A[1] and B[1]	[1,1]	[2,2]	sqrt((2-1)^2+(2-1)^2)=1.4142	[[1.0, 2.8284], [1.0, 1.4142]]
6	All pairs computed	-	-	-	Final distance matrix completed

💡 All pairs of points from A and B have been processed, distance matrix fully computed.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
D	empty 2x2 matrix	[[1.0, ?], [?, ?]]	[[1.0, 2.8284], [?, ?]]	[[1.0, 2.8284], [1.0, ?]]	[[1.0, 2.8284], [1.0, 1.4142]]	[[1.0, 2.8284], [1.0, 1.4142]]

Key Moments - 3 Insights

Why does the output matrix have shape (2, 2) instead of (4, 4)?

Because the distance matrix rows correspond to points in A (2 points) and columns to points in B (2 points), so shape is (len(A), len(B)) as shown in execution_table rows 2-5.

How is the Euclidean distance calculated between two points?

It uses the formula sqrt((x2 - x1)^2 + (y2 - y1)^2), as shown in the 'Distance Computed' column in execution_table rows 2-5.

What happens if we change the metric from 'euclidean' to 'cityblock'?

The distances would be computed as sum of absolute differences instead of Euclidean, changing the values in the distance matrix but keeping the same shape and process.

Visual Quiz - 3 Questions

Look at the execution table, what is the distance between A[1] and B[1] at step 5?

A 1.0

B 1.4142

C 2.8284

D 0.0

💡 Hint

Check the 'Distance Computed' column at step 5 in the execution_table.

At which step does the distance matrix become fully filled?

A Step 4

B Step 6

C Step 5

D Step 3

💡 Hint

Look at the 'Distance Matrix State' column; the matrix becomes fully filled (no ? left) at step 5.

If we add one more point to B, how would the distance matrix shape change?

A It would have one more column

B It would have one more row

C It would stay the same size

D It would become a 3D matrix

💡 Hint

Distance matrix shape is (len(A), len(B)) as explained in key_moments and concept_flow.

Concept Snapshot

distance.cdist(A, B, metric)
- Computes pairwise distances between points in A and B
- Returns matrix shape (len(A), len(B))
- metric='euclidean' is default for straight-line distance
- Supports many metrics like 'cityblock', 'cosine'
- Useful for comparing sets of points quickly

Full Transcript

This visual execution shows how scipy.spatial.distance.cdist computes distances between two sets of points. We start with two arrays A and B, each holding points in 2D space. The function calculates the distance between every point in A and every point in B using the chosen metric, here Euclidean distance. The output is a matrix where each row corresponds to a point in A and each column to a point in B. The matrix entries are the distances. Step by step, we see each pair's distance computed and placed in the matrix until all pairs are done. This helps understand how cdist works internally and what the output means.

Practice

1.

What does the scipy.spatial.distance.cdist function do?

easy

A. Generates random points within a specified range.

B. Calculates the sum of all points in a single array.

C. Computes distances between two sets of points and returns a matrix of distances.

D. Sorts points in ascending order based on their coordinates.

Solution

1. Step 1: Understand the function purpose

   cdist calculates distances between two groups of points, not sums or sorting.

2. Step 2: Identify the output type

   The function returns a matrix showing distances from each point in the first group to each point in the second.

3. Final Answer:

   Computes distances between two sets of points and returns a matrix of distances. -> Option C

4. Quick Check:

   cdist = distance matrix [OK]

Hint: Remember: cdist = cross distances between two point sets [OK]

Common Mistakes:

• Thinking cdist sums points instead of distances
• Confusing cdist with sorting or random generation
• Assuming cdist returns a single distance value

2.

Which of the following is the correct way to import and use cdist from scipy.spatial.distance?

?

easy

A. from scipy import cdist
result = cdist(A, B, metric='euclidean')

B. import scipy.distance.cdist
result = cdist(A, B, metric='euclidean')

C. import cdist from scipy.spatial.distance
result = cdist(A, B, metric='euclidean')

D. from scipy.spatial.distance import cdist
result = cdist(A, B, metric='euclidean')

Solution

1. Step 1: Check correct import syntax

   The correct import is from scipy.spatial.distance import cdist. Other forms are invalid or cause errors.

2. Step 2: Confirm usage syntax

   Calling cdist(A, B, metric='euclidean') is correct to compute Euclidean distances.

3. Final Answer:

   from scipy.spatial.distance import cdist result = cdist(A, B, metric='euclidean') -> Option D

4. Quick Check:

   Correct import and call = from scipy.spatial.distance import cdist result = cdist(A, B, metric='euclidean') [OK]

Hint: Use 'from scipy.spatial.distance import cdist' to import [OK]

Common Mistakes:

• Using incorrect import statements causing ImportError
• Trying to import cdist directly from scipy
• Using wrong syntax like 'import cdist from ...'

3.

Given the code below, what is the output?

import numpy as np
from scipy.spatial.distance import cdist

A = np.array([[0, 0], [1, 1]])
B = np.array([[1, 0], [2, 2]])
result = cdist(A, B, metric='euclidean')
print(result)

medium

A. [[1. 2.82842712]
[1. 1.41421356]]

B. [[1.41421356 2.23606798]
[1. 1.41421356]]

C. [[1. 1.41421356]
[1.41421356 2.23606798]]

D. [[0. 1.41421356]
[1. 2.23606798]]

Solution

1. Step 1: Calculate distances from points in A to points in B

   Distances: - From [0,0] to [1,0] = sqrt((1-0)^2 + (0-0)^2) = 1.0 - From [0,0] to [2,2] = sqrt((2-0)^2 + (2-0)^2) ≈ 2.8284 - From [1,1] to [1,0] = sqrt((1-1)^2 + (0-1)^2) = 1.0 - From [1,1] to [2,2] = sqrt((2-1)^2 + (2-1)^2) ≈ 1.4142

2. Step 2: Form the distance matrix

   Matrix rows correspond to points in A, columns to points in B: [[1.0, 2.8284], [1.0, 1.4142]]

3. Final Answer:

   [[1. 2.82842712] [1. 1.41421356]] -> Option A

4. Quick Check:

   Euclidean distances matrix = [[1. 2.82842712] [1. 1.41421356]] [OK]

Hint: Calculate each pair distance carefully, row by row [OK]

Common Mistakes:

• Mixing row and column order in the output matrix
• Using wrong distance formula (e.g., Manhattan instead of Euclidean)
• Rounding errors causing confusion

4.

What is wrong with the following code snippet?

import numpy as np
from scipy.spatial.distance import cdist

A = np.array([[0, 0], [1, 1]])
B = np.array([1, 0])
result = cdist(A, B, metric='euclidean')
print(result)

medium

A. The metric 'euclidean' is not supported by cdist.

B. B should be a 2D array, but it is 1D, causing a shape error.

C. cdist requires both inputs to be lists, not numpy arrays.

D. The import statement for cdist is incorrect.

Solution

1. Step 1: Check input array shapes

   cdist expects both inputs to be 2D arrays representing sets of points. Here, B is 1D, which is invalid.

2. Step 2: Understand error cause

   Passing 1D array B causes a shape mismatch error because cdist tries to interpret rows as points.

3. Final Answer:

   B should be a 2D array, but it is 1D, causing a shape error. -> Option B

4. Quick Check:

   Input arrays must be 2D for cdist [OK]

Hint: Always use 2D arrays for both inputs in cdist [OK]

Common Mistakes:

• Passing 1D arrays instead of 2D arrays
• Assuming cdist accepts lists only
• Using unsupported metric names

5.

You have two sets of points representing locations of stores and customers:

stores = np.array([[0, 0], [5, 5], [10, 10]])
customers = np.array([[1, 1], [6, 6], [8, 8], [12, 12]])

Using cdist, how can you find the closest store for each customer?

hard

A. Compute the distance matrix with cdist(customers, stores) and find the index of the minimum distance in each row.

B. Compute the distance matrix with cdist(stores, customers) and find the index of the maximum distance in each column.

C. Use cdist(stores, customers) and sum all distances to find closest stores.

D. Calculate Euclidean distance manually for each pair without using cdist.

Solution

1. Step 1: Compute distance matrix from customers to stores

   Use cdist(customers, stores) to get distances where each row is a customer and each column is a store.

2. Step 2: Find closest store for each customer

   For each customer (row), find the index of the minimum distance value. This index corresponds to the closest store.

3. Final Answer:

   Compute the distance matrix with cdist(customers, stores) and find the index of the minimum distance in each row. -> Option A

4. Quick Check:

   Closest store = min distance per customer row [OK]

Hint: cdist(customers, stores), then argmin per row finds closest store [OK]

Common Mistakes:

• Swapping order of inputs causing wrong distance matrix shape
• Using max instead of min to find closest
• Summing distances instead of finding minimum