# Efficient distance calculation between N points and a reference in numpy/scipy

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### Question :

Efficient distance calculation between N points and a reference in numpy/scipy

I just started using scipy/numpy. I have an 100000*3 array, each row is a coordinate, and a 1*3 center point. I want to calculate the distance for each row in the array to the center and store them in another array. What is the most efficient way to do it?

I would take a look at `scipy.spatial.distance.cdist`:

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

``````import numpy as np
import scipy

a = np.random.normal(size=(10,3))
b = np.random.normal(size=(1,3))

dist = scipy.spatial.distance.cdist(a,b) # pick the appropriate distance metric
``````

`dist` for the default distant metric is equivalent to:

``````np.sqrt(np.sum((a-b)**2,axis=1))
``````

although `cdist` is much more efficient for large arrays (on my machine for your size problem, `cdist` is faster by a factor of ~35x).

I would use the sklearn implementation of the euclidean distance. The advantage is the usage of the more efficient expression by using Matrix multiplication:

``````dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y)
``````

A simple script would look like this:

``````import numpy as np

x = np.random.rand(1000, 3)
y = np.random.rand(1000, 3)

dist = np.sqrt(np.dot(x, x)) - (dot(x, y) + dot(x, y)) + dot(y, y)
``````

The advantage of this approach has been nicely described in the sklearn documentation:
http://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.euclidean_distances.html#sklearn.metrics.pairwise.euclidean_distances

I am using this approach to crunch large datamatrices (10000, 10000) with some minor modifications like using the np.einsum function.

You can also use the development of the norm (similar to remarkable identities). This is probably the most efficent way to compute the distance of a matrix of points.

Here is a code snippet that I originally used for a k-Nearest-Neighbors implementation, in Octave, but you can easily adapt it to numpy since it only uses matrix multiplications (the equivalent is numpy.dot()):

``````% Computing the euclidian distance between each known point (Xapp) and unknown points (Xtest)
% Note: we use the development of the norm just like a remarkable identity:
% ||x1 - x2||^2 = ||x1||^2 + ||x2||^2 - 2*<x1,x2>
[napp, d] = size(Xapp);
[ntest, d] = size(Xtest);

A = sum(Xapp.^2, 2);
A = repmat(A, 1, ntest);

B = sum(Xtest.^2, 2);
B = repmat(B', napp, 1);

C = Xapp*Xtest';

dist = A+B-2.*C;
``````

This might not answer your question directly, but if you are after all permutations of particle pairs, I’ve found the following solution to be faster than the pdist function in some cases.

``````import numpy as np

L   = 100       # simulation box dimension
N   = 100       # Number of particles
dim = 2         # Dimensions

# Generate random positions of particles
r = (np.random.random(size=(N,dim))-0.5)*L

# uti is a list of two (1-D) numpy arrays
# containing the indices of the upper triangular matrix
uti = np.triu_indices(100,k=1)        # k=1 eliminates diagonal indices

# uti[0] is i, and uti[1] is j from the previous example
dr = r[uti[0]] - r[uti[1]]            # computes differences between particle positions
D = np.sqrt(np.sum(dr*dr, axis=1))    # computes distances; D is a 4950 x 1 np array
``````

See this for a more in-depth look on this matter, on my blog post.

You may need to specify a more detailed manner the distance function you are interested of, but here is a very simple (and efficient) implementation of Squared Euclidean Distance based on `inner product` (which obviously can be generalized, straightforward manner, to other kind of distance measures):

``````In []: P, c= randn(5, 3), randn(1, 3)
In []: dot(((P- c)** 2), ones(3))
Out[]: array([  8.80512,   4.61693,   2.6002,   3.3293,  12.41800])
``````

Where `P` are your points and `c` is the center.

``````#is it true, to find the biggest distance between the points in surface?

from math import sqrt

n = int(input( "enter the range : "))
x = list(map(float,input("type x coordinates: ").split()))
y = list(map(float,input("type y coordinates: ").split()))
maxdis = 0
for i in range(n):
for j in range(n):
print(i, j, x[i], x[j], y[i], y[j])
dist = sqrt((x[j]-x[i])**2+(y[j]-y[i])**2)
if maxdis < dist:

maxdis = dist
print(" maximum distance is : {:5g}".format(maxdis))
``````