Question

The question is:

Given N points(in 2D) with x and y coordinates, find a point P (in N
given points) such that the sum of distances from other(N-1) points to
P is minimum.

This point is commonly known as Geometric Median. Is there any efficient algorithm to solve this problem, other than the naive O(N^2) one?

Answer 1

I solved something similar for a local online judge once using simulated annealing. That was the official solution as well and the program got AC.

The only difference was that the point I had to find did not have to be part of the N given points.

This was my C++ code, and N could be as large as 50000. The program executes in 0.1s on a 2ghz pentium 4.

// header files for IO functions and math
#include <cstdio>
#include <cmath>

// the maximul value n can take
const int maxn = 50001;

// given a point (x, y) on a grid, we can find its left/right/up/down neighbors
// by using these constants: (x + dx[0], y + dy[0]) = upper neighbor etc.
const int dx[] = {-1, 0, 1, 0};
const int dy[] = {0, 1, 0, -1};

// controls the precision - this should give you an answer accurate to 3 decimals
const double eps = 0.001;

// input and output files
FILE *in = fopen("adapost2.in","r"), *out = fopen("adapost2.out","w");

// stores a point in 2d space
struct punct
{
    double x, y;
};

// how many points are in the input file
int n;

// stores the points in the input file
punct a[maxn];

// stores the answer to the question
double x, y;

// finds the sum of (euclidean) distances from each input point to (x, y)
double dist(double x, double y)
{
    double ret = 0;

    for ( int i = 1; i <= n; ++i )
    {
        double dx = a[i].x - x;
        double dy = a[i].y - y;

        ret += sqrt(dx*dx + dy*dy); // classical distance formula
    }

    return ret;
}

// reads the input
void read()
{
    fscanf(in, "%d", &n); // read n from the first 

    // read n points next, one on each line
    for ( int i = 1; i <= n; ++i )
        fscanf(in, "%lf %lf", &a[i].x, &a[i].y), // reads a point
        x += a[i].x,
        y += a[i].y; // we add the x and y at first, because we will start by approximating the answer as the center of gravity

    // divide by the number of points (n) to get the center of gravity
    x /= n; 
    y /= n;
}

// implements the solving algorithm
void go()
{
    // start by finding the sum of distances to the center of gravity
    double d = dist(x, y);

    // our step value, chosen by experimentation
    double step = 100.0;

    // done is used to keep track of updates: if none of the neighbors of the current
    // point that are *step* steps away improve the solution, then *step* is too big
    // and we need to look closer to the current point, so we must half *step*.
    int done = 0;

    // while we still need a more precise answer
    while ( step > eps )
    {
        done = 0;
        for ( int i = 0; i < 4; ++i )
        {
            // check the neighbors in all 4 directions.
            double nx = (double)x + step*dx[i];
            double ny = (double)y + step*dy[i];

            // find the sum of distances to each neighbor
            double t = dist(nx, ny);

            // if a neighbor offers a better sum of distances
            if ( t < d )
            {
                update the current minimum
                d = t;
                x = nx;
                y = ny;

                // an improvement has been made, so
                // don't half step in the next iteration, because we might need
                // to jump the same amount again
                done = 1;
                break;
            }
        }

        // half the step size, because no update has been made, so we might have
        // jumped too much, and now we need to head back some.
        if ( !done )
            step /= 2;
    }
}

int main()
{
    read();
    go();

    // print the answer with 4 decimal points
    fprintf(out, "%.4lf %.4lf\n", x, y);

    return 0;
}

Then I think It’s correct to pick the one from your list that is closest to the (x, y) returned by this algorithm.

This algorithm takes advantage of what this wikipedia paragraph on the geometric median says:

However, it is straightforward to calculate an approximation to the
geometric median using an iterative procedure in which each step
produces a more accurate approximation. Procedures of this type can be
derived from the fact that the sum of distances to the sample points
is a convex function, since the distance to each sample point is
convex and the sum of convex functions remains convex. Therefore,
procedures that decrease the sum of distances at each step cannot get
trapped in a local optimum.

One common approach of this type, called
Weiszfeld’s algorithm after the work of Endre Weiszfeld,[4] is a form
of iteratively re-weighted least squares. This algorithm defines a set
of weights that are inversely proportional to the distances from the
current estimate to the samples, and creates a new estimate that is
the weighted average of the samples according to these weights. That
is,

The first paragraph above explains why this works: because the function we are trying to optimize does not have any local minimums, so you can greedily find the minimum by iteratively improving it.

Think of this as a sort of binary search. First, you approximate the result. A good approximation will be the center of gravity, which my code computes when reading the input. Then, you see if adjacent points to this give you a better solution. In this case, a point is considered adjacent if it as a distance of step away from your current point. If it is better, then it is fine to discard your current point, because, as I said, this will not trap you into a local minimum because of the nature of the function you are trying to minimize.

After this, you half the step size, just like in binary search, and continue until you have what you consider to be a good enough approximation (controlled by the eps constant).

The complexity of the algorithm therefore depends on how accurate you want the result to be.