Identifying points with the smallest Euclidean distance

I have a collection of n dimensional points and I want to find which 2 are the closest. The best I could come up for 2 dimensions is:

from numpy import *
myArr = array( [[1, 2],
                [3, 4],
                [5, 6],
                [7, 8]] )

n = myArr.shape[0]
cross = [[sum( ( myArr[i] - myAr开发者_如何学编程r[j] ) ** 2 ), i, j]
         for i in xrange( n )
         for j in xrange( n )
         if i != j
         ]

print min( cross )

which gives

[8, 0, 1]

But this is too slow for large arrays. What kind of optimisation can I apply to it?

RELATED:

Euclidean distance between points in two different Numpy arrays, not within

Try scipy.spatial.distance.pdist(myArr). This will give you a condensed distance matrix. You can use argmin on it and find the index of the smallest value. This can be converted into the pair information.

There's a whole Wikipedia page on just this problem, see: http://en.wikipedia.org/wiki/Closest_pair_of_points

Executive summary: you can achieve O(n log n) with a recursive divide and conquer algorithm (outlined on the Wiki page, above).

You could take advantage of the latest version of SciPy's (v0.9) Delaunay triangulation tools. You can be sure that the closest two points will be an edge of a simplex in the triangulation, which is a much smaller subset of pairs than doing every combination.

Here's the code (updated for general N-D):

import numpy
from scipy import spatial

def closest_pts(pts):
    # set up the triangluataion
    # let Delaunay do the heavy lifting
    mesh = spatial.Delaunay(pts)

    # TODO: eliminate reduncant edges (numpy.unique?)
    edges = numpy.vstack((mesh.vertices[:,:dim], mesh.vertices[:,-dim:]))

    # the rest is easy
    x = mesh.points[edges[:,0]]
    y = mesh.points[edges[:,1]]

    dists = numpy.sum((x-y)**2, 1)
    idx = numpy.argmin(dists)

    return edges[idx]
    #print 'distance: ', dists[idx]
    #print 'coords:\n', pts[closest_verts]

dim = 3
N = 1000*dim
pts = numpy.random.random(N).reshape(N/dim, dim)

Seems closely O(n):

There is a scipy function pdist that will get you the pairwise distances between points in an array in a fairly efficient manner:

http://docs.scipy.org/doc/scipy/reference/spatial.distance.html

that outputs the N*(N-1)/2 unique pairs (since r_ij == r_ji). You can then search on the minimum value and avoid the whole loop mess in your code.

Perhaps you could proceed along these lines:

In []: from scipy.spatial.distance import pdist as pd, squareform as sf
In []: m= 1234
In []: n= 123
In []: p= randn(m, n)
In []: d= sf(pd(p))
In []: a= arange(m)
In []: d[a, a]= d.max()
In []: where(d< d.min()+ 1e-9)
Out[]: (array([701, 730]), array([730, 701]))

With substantially more points you need to be able to somehow utilize the hierarchical structure of your clustering.

How fast is it compared to just doing a nested loop and keeping track of the shortest pair? I think creating a huge cross array is what might be hurting you. Even O(n^2) is still pretty quick if you're only doing 2 dimensional points.

The accepted answer is OK for small datasets, but its execution time scales as n**2. However, as pointed out by @payne, an optimal solution can achieve n*log(n) computation time scaling.

This optial solution can be obtained using sklearn.neighbors.BallTree as follows.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.neighbors import BallTree as tree

n = 10
dim = 2
xy = np.random.uniform(size=[n, dim])

# This solution is optimal when xy is very large
res = tree(xy)
dist, ids = res.query(xy, 2)
mindist = dist[:, 1]  # second nearest neighbour
minid = np.argmin(mindist)

plt.plot(*xy.T, 'o')
plt.plot(*xy[ids[minid]].T, '-o')

This procedure scales well for very large sets of xy values and even for large dimensions dim (altough the example illustrates the case dim=2). The resulting output looks like this

An identical solution can be obtained using scipy.spatial.cKDTree, by replacing the sklearn import with the following Scipy one. Note however that cKDTree, unlike BallTree, does not scale well for high dimensions

from scipy.spatial import cKDTree as tree