Assume there is a set of N points named A. I'd like to select K points and make a subset of A, which is A', where the minimum distance between points are maximum over subsets.
- the distance between all points within A' is defined as y(x, x'), x∈A, x'∈A
- the minimum distance is defined as k(A') = min_{x∈A', x'∈A', x!=x'}(y(x,x'))
- C = argmin_A'(k(A')) where A'⊂A, |A'| = K
- N = 100
- K = 10
I want to solve and get set C. Is there an efficient algorithm to solve this?
The sample result is like the following, which is quite heavy to solve ; O(N^K).
import numpy as np
from matplotlib import pyplot as plt
from itertools import combinations
N=30
K=5
# generate sample point
mean = [0, 0]
cov = [[1, 0], [0, 1]]
rs=np.random.multivariate_normal(mean,cov, N)
def calc_min_dist(arr):
dist_arr = ((arr[None,:,:] - arr[:,None,:])**2).sum(axis=2)**.5
min_dist = min(dist_arr[dist_arr>0])
return min_dist
def min_points(arr, K):
max_value = 0
max_index = [range(K)]
index = list(range(len(arr)))
for sub_index in combinations(index, K):
sub_arr = arr[tuple([sub_index])]
dist = calc_min_dist(sub_arr)
if max_value < dist:
max_value = dist
max_index = sub_index
return max_value, max_index
# solve
min_value, sub_index = min_points(rs, K)
# plot result
fig, ax = plt.subplots()
ax.scatter(*rs.T)
ax.scatter(*rs[tuple([sub_index])].T)
