CODEWITHSUNDEEP
algorithmmatlabgeometry
saastn
saastn

Reputation: 6015

How to find a unique set of closest pairs of points?

A and B are sets of m and n points respectively, where m<=n. I want to find a set of m unique points from B, named C, where the sum of distances between all [A(i), C(i)] pairs is the minimal.

To solve this without uniqueness constraint I can just find closest points from B to each point in A:

m = 5; n = 8; dim = 2;
A = rand(m, dim);
B = rand(n, dim);
D = pdist2(A, B);
[~, I] = min(D, [], 2);
C2 = B(I, :);

Where there may be repeated elements of B present in C. Now the first solution is brute-force search:

minSumD = inf;
allCombs = nchoosek(1:n, m);
for i = 1:size(allCombs, 1)
    allPerms = perms(allCombs(i, :));
    for j = 1:size(allPerms, 1)
        ind = sub2ind([m n], 1:m, allPerms(j, :));
        sumD = sum(D(ind));
        if sumD<minSumD
            minSumD = sumD;
            I = allPerms(j, :);
        end
    end
end
C = B(I, :);

I think C2 (set of closest points to each A(i)) is pretty much alike C except for its repeated points. So how can I decrease the computation time?

Upvotes: 2

Views: 2418

Answers (1)

David Eisenstat
David Eisenstat

Reputation: 65427

Use a variant of the Hungarian algorithm, which computes a minimum/maximum weight perfect matching. Create n-m dummy points for the unused B points to match with (or, if you're willing to put in more effort, adapt the Hungarian algorithm machinery to non square matrices).

Upvotes: 3

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