# Closest distance between points in a list

In a course I'm doing, I was given the task of finding the closest pair of points among the given points. The program that passed all test cases was the following:

import math

if __name__ == '__main__':
n = int(input())
P = []
for i in range(n):
line = input().split()
x = int(line[0])
y = int(line[1])
p = (x, y)
P.append(p)
print(closest(P, n))

def square(x):
return x * x

def square_distance(p0, p1):
return square(p0[0] - p1[0]) + square(p0[1] - p1[1])

def closest(P, n):
P.sort() # sort by x coordinates
return math.sqrt(_closest_square_distance(P, n))

def _closest_square_distance(P, n):
if n == 2:
return square_distance(P[0], P[1])
if n == 3:
return min(square_distance(P[0], P[1]), square_distance(P[0], P[2]), square_distance(P[1], P[2]))

mid = n // 2
dl = _closest_square_distance(P[:mid], mid)
dr = _closest_square_distance(P[mid:], n - mid)

closest_square_distance = min(dl, dr)
closest_distance_so_far = math.sqrt(closest_square_distance)

mid_x = P[mid][0]
strip = []
strip_length = 0
for i in range(n):
p = P[i]
if abs(p[0] - mid_x) < closest_distance_so_far:
strip.append(p)
strip_length += 1

strip.sort(key=lambda x: x[1]) # sort strip by y coordinates

for i in range(strip_length):
js = min(strip_length, i + 7) # sufficient to compute next 6 neighbors
for j in range(i + 1, js):
ds = square_distance(strip[i], strip[j])
if ds < closest_square_distance:
closest_square_distance = ds

return closest_square_distance


This code was based on the algorithm found at https://en.wikipedia.org/wiki/Closest_pair_of_points_problem#Planar_case . What do you think I can do better? Is there a more efficient way to implement this algorithm? (In this one, I just tried to delay the sqrt computation as long as possible, and avoided to calculate len where I could). Is there an even more efficient algorithm?

### 1. Review

1. There are no docstrings. What do these functions do? What arguments do they take and what do they return?

2. You write:

I just tried to delay the sqrt computation as long as possible

This suggests that your cost model for Python is not quite right. In CPython, it's more expensive to call square than it to call sqrt:

>>> from timeit import timeit
>>> timeit('square(999)', 'from __main__ import square')
0.11685514450073242
>>> timeit('sqrt(999)', 'from math import sqrt')
0.057652950286865234


That's because most of the time executing a program in CPython is spent in the execution of bytecode. The actual cost of computing the square root is small by comparison.

Because bytecode is expensive, it sometimes pays (in CPython) to do more computation, if you can arrange for it to be done in the fast C implementation rather than in slow Python code. In this case I would consider using math.hypot, like this:

from math import hypot

def distance(p, q):
"Return the Euclidean distance between points p and q."
return hypot(p[0] - q[0], p[1] - q[1])


This is faster than calling square_distance:

>>> timeit('square_distance((1, 2), (3, 4))', 'from __main__ import square_distance')
0.4481959342956543
>>> timeit('distance((1, 2), (3, 4))', 'from __main__ import distance')
0.32285499572753906

3. In closest, the argument n must be the length of the list of points P. So it would be simpler to assign it yourself and avoid the requirement for the caller to pass it in (which is risky, because the caller might get it wrong).

4. The function closest has a side-effect: as well as computing the shortest distance, it sorts the list P. Unexpected side effects should be avoided — it is easier to understand code if it only has "local" effects on data structures. In this case, there are two alternatives: you could require the caller to sort the points before passing them in, or you could take a copy and sort the copy, like this:

def closest(P):
"Return the closest Euclidean distance between two points in the list P."
return _closest_distance(sorted(P), len(P))

5. When recursing:

dl = _closest_square_distance(P[:mid], mid)
dr = _closest_square_distance(P[mid:], n - mid)


the list slice P[:mid] has to copy out the contents of the slice. It would be cheaper to remember the endpoints of the slice you are working on and avoid the copying:

def _closest(P, start, stop):
# closest Euclidean distance between two points in the slice P[start:stop]
# handle base cases here

mid = (start + stop) // 2
dl = _closest(P, start, mid)
dr = _closest(P, mid, stop)

6. In the base cases, you could save some duplication by using itertools.combinations and writing:

if n <= 3:
return min(distance(p, q) for p, q in combinations(P, 2))

7. The base case logic could be extended to larger values of n. I find that for large random point sets, n <= 8 performs well.

8. The function:

lambda x: x[1]


is the same every time, so consider storing it in a global variable (or using operator.itemgetter(1)).

9. The code finds the points in the strip by iterating over all the points in P. But since you have the points sorted by their $x$ coordinate, you can find the endpoints of the strip using the bisect module.

10. The code does not take advantage of the fact that the two points being compared can only be closer than closest_distance if they were in different halves (if they were in the same half, then they have already been compared by the recursive calls and closest_distance already takes them into account).

### 2. Revised code

from bisect import bisect_left, bisect_right
from itertools import combinations, islice
from math import hypot
from operator import itemgetter

def distance(p, q):
"Return the Euclidean distance between points p and q."
return hypot(p[0] - q[0], p[1] - q[1])

_Y_COORD = itemgetter(1)

def closest_distance(P):
"Return the closest Euclidean distance between two points in the list P."
P = sorted(P)
PX = [x for x, _ in P]

def _closest(start, stop):
if stop - start <= 8:
return min(distance(p, q)
for p, q in combinations(P[start:stop], 2))

mid = (start + stop) // 2
dist = min(_closest(start, mid), _closest(mid, stop))

mid_x = PX[mid]
left = bisect_left(PX, mid_x - dist, lo=start, hi=mid)
right = bisect_right(PX, mid_x + dist, lo=mid, hi=stop)
strip = sorted(P[mid:right], key=_Y_COORD)
strip_y = list(map(_Y_COORD, strip))

for p in P[left:mid]:
y = p[1]
i = bisect_left(strip_y, y - dist)
j = bisect_right(strip_y, y + dist)
assert j - i <= 6
for q in strip[i:j]:
dist = min(dist, distance(p, q))

return dist

return _closest(0, len(P))


In general, it's safe to assume that using built-in functions/features in Python will be faster than doing it yourself.

Lists always know their size, so len(list_obj) is O(1). (Ref here). List comprehensions are faster than for loops. (Ref here).

For example, I would change:

strip = []
strip_length = 0
for i in range(n):
p = P[i]
if abs(p[0] - mid_x) < closest_distance_so_far:
strip.append(p)
strip_length += 1


Into:

strip = [
p
for p in P[:n]
if abs(p[0] - mid_x) < closest_distance_so_far
]
strip_length = len(strip)


It eliminates some fluff, includes only what is important, and uses built-in features.

The only other thing I see with a quick look would be to inline the square() function; although functions are fairly low overhead, this might improve performance a wee bit.

You can use the timeit module to test various changes to the code.