Nearest Neighbors

Euclidean Distance

Compute the Euclidean distance between two points as √Σ(aᵢ−bᵢ)². A single loop accumulates the squared difference per dimension; math.sqrt at the end. Library: np.linalg.norm(np.array(a) - np.array(b)) — single call, same formula. RESULT: distance (rounded).

By hand

a=[1,2,3], b=[4,6,3]. Per dimension: (4−1)²=9, (6−2)²=16, (3−3)²=0. sq_sum=25, dist=√25=5.0.

naive.py
Replay: real traced execution (multi-file project)
import math
a = [1, 2, 3]
b = [4, 6, 3]
n = len(a)
sq_sum = 0.0
for i in range(n):
    diff = a[i] - b[i]
    sq_sum = sq_sum + diff * diff
dist = math.sqrt(sq_sum)
print('RESULT:', round(dist, 4))

  1. import math

    1import math2a = [1, 2, 3]

  2. a ← [1, 2, 3]

    1import math2a = [1, 2, 3]3b = [4, 6, 3]
    values this step[1, 2, 3]a

  3. b ← [4, 6, 3]

    2a = [1, 2, 3]3b = [4, 6, 3]4n = len(a)
    values this step[4, 6, 3]b

  4. n ← 3

    3b = [4, 6, 3]4n = len(a)5sq_sum = 0.0
    values this step3n

  5. sq_sum ← 0.0

    4n = len(a)5sq_sum = 0.06for i in range(n):
    values this step0.0sq_sum

  6. i ← 0

    5sq_sum = 0.06for i in range(n):7    diff = a[i] - b[i]
    values this step0i

  7. diff ← -3

    6for i in range(n):7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff
    values this step-3diff

  8. sq_sum ← 9.0

    7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff9dist = math.sqrt(sq_sum)
    values this step0.0 → 9.0sq_sum

  9. i ← 1

    5sq_sum = 0.06for i in range(n):7    diff = a[i] - b[i]
    values this step0 → 1i

  10. diff ← -4

    6for i in range(n):7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff
    values this step-3 → -4diff

  11. sq_sum ← 25.0

    7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff9dist = math.sqrt(sq_sum)
    values this step9.0 → 25.0sq_sum

  12. i ← 2

    5sq_sum = 0.06for i in range(n):7    diff = a[i] - b[i]
    values this step1 → 2i

  13. diff ← 0

    6for i in range(n):7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff
    values this step-4 → 0diff

  14. sq_sum = sq_sum + diff * diff

    7    diff = a[i] - b[i]8    sq_sum = sq_sum + diff * diff9dist = math.sqrt(sq_sum)

  15. for i in range(n):

    5sq_sum = 0.06for i in range(n):7    diff = a[i] - b[i]

  16. dist ← 5.0

    8    sq_sum = sq_sum + diff * diff9dist = math.sqrt(sq_sum)10print('RESULT:', round(dist, 4))
    values this step5.0dist

  17. stdout ← RESULT: 5.0

    9dist = math.sqrt(sq_sum)10print('RESULT:', round(dist, 4))
    values this stepRESULT: 5.0stdout

With NumPy

np.linalg.norm computes the L2 norm of the difference vector — identical to the loop formula for Euclidean distance.

library.py 
import numpy as np
from dalib.display import set_display
set_display()

a = [1, 2, 3]
b = [4, 6, 3]
dist = float(np.linalg.norm(np.array(a) - np.array(b)))
print('a:', a)
print('b:', b)
print('RESULT:', round(dist, 4))

a: [1, 2, 3]
b: [4, 6, 3]
RESULT: 5.0

Implementation notes

  • Euclidean distance generalizes the Pythagorean theorem to n dimensions. For 2D points it reduces to √((x₂−x₁)²+(y₂−y₁)²).
  • The loop accumulates diff*diff (explicit multiply) so each squared term appears as a distinct step in the trace, keeping the squaring operation visible rather than folded into a single expression.
  • Scale sensitivity: dimensions with large magnitudes dominate. Standardize features before computing distances — see standardize-features (ch01).
  • Cross-reference: knn-classify-majority (this chapter) applies this distance to find nearest neighbors.