Optimum location of point to minimize total distance


Given a set of points as and a line as ax+by+c = 0. We need to find a point on given line for which sum of distances from given set of points is minimum.


In above figure optimum location of point of x - y - 3 = 0 line 
is (2, -1), whose total distance with other points is 20.77, 
which is minimum obtainable total distance.

If we take one point on given line at infinite distance then total distance cost will be infinite, now when we move this point on line towards given points the total distance cost starts decreasing and after some time, it again starts increasing which reached to infinite on the other infinite end of line so distance cost curve looks like a U-curve and we have to find the bottom value of this U-curve.
As U-curve is not monotonically increasing or decreasing we can’t use binary search for finding bottom most point, here we will use ternary search for finding bottom most point, ternary search skips one third of search space at each iteration, you can read more about ternary search here.
So solution proceeds as follows, we start with low and high initialized as some smallest and largest values respectively, then we start iteration, in each iteration we calculate two mids, mid1 and mid2, which represent 1/3rd and 2/3rd position in search space, we calculate total distance of all points with mid1 and mid2 and update low or high by comparing these distance cost, this iteration continues untill low and high become approximately equal.

//  C/C++ program to find optimum location and total cost
#include <bits/stdc++.h>
using namespace std;
#define sq(x) ((x)*(x))
#define EPS 1e-6
#define N 5

//  structure defining a point
struct point
    int x, y;
    point()  {}
    point(int x, int y) : x(x), y(y)   {}

//  structure defining a line of ax + by + c = 0 form
struct line
    int a, b, c;
    line(int a, int b, int c) : a(a), b(b), c(c) {}

//  method to get distance of point (x, y) from point p
double dist(double x, double y, point p)
    return sqrt(sq(x - p.x) + sq(y - p.y));

/*  Utility method to compute total distance all points
    when choose point on given line has x-cordinate
    value as X   */
double compute(point p[], int n, line l, double X)
    double res = 0;

    //  calculating Y of choosen point by line equation
    double Y = -1 * (l.c + l.a*X) / l.b;
    for (int i = 0; i < n; i++)
        res += dist(X, Y, p[i]);

    return res;

//  Utility method to find minimum total distance
double findOptimumCostUtil(point p[], int n, line l)
    double low = -1e6;
    double high = 1e6;

    // loop untill difference between low and high
    // become less than EPS
    while ((high - low) > EPS)
        // mid1 and mid2 are representative x co-ordiantes
        // of search space
        double mid1 = low + (high - low) / 3;
        double mid2 = high - (high - low) / 3;

        double dist1 = compute(p, n, l, mid1);
        double dist2 = compute(p, n, l, mid2);

        // if mid2 point gives more total distance,
        // skip third part
        if (dist1 < dist2)
            high = mid2;

        // if mid1 point gives more total distance,
        // skip first part
            low = mid1;

    // compute optimum distance cost by sending average
    // of low and high as X
    return compute(p, n, l, (low + high) / 2);

//  method to find optimum cost
double findOptimumCost(int points[N][2], line l)
    point p[N];

    //  converting 2D array input to point array
    for (int i = 0; i < N; i++)
        p[i] = point(points[i][0], points[i][1]);

    return findOptimumCostUtil(p, N, l);

//  Driver code to test above method
int main()
    line l(1, -1, -3);
    int points[N][2] = {{-3, -2}, {-1, 0}, {-1, 2},
                                   {1, 2}, {3, 4}};
    cout << findOptimumCost(points, l) << endl;
    return 0;



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