Optimum location of point to minimize total distance
Last Updated :
25 Jun, 2022
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.
Example:
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 until low and high become approximately equal.
C++
// 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-coordinate
value as X */
double compute(point p[], int n, line l, double X)
{
double res = 0;
// calculating Y of chosen 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 until 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
else
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;
}
// A Java program to find optimum location
// and total cost
class GFG {
static double sq(double x) { return ((x) * (x)); }
static int EPS = (int)(1e-6) + 1;
static int N = 5;
// structure defining a point
static class point {
int x, y;
point() {}
public point(int x, int y)
{
this.x = x;
this.y = y;
}
};
// structure defining a line of ax + by + c = 0 form
static class line {
int a, b, c;
public line(int a, int b, int c)
{
this.a = a;
this.b = b;
this.c = c;
}
};
// method to get distance of point (x, y) from point p
static double dist(double x, double y, point p)
{
return Math.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-coordinate
value as X */
static double compute(point p[], int n, line l,
double X)
{
double res = 0;
// calculating Y of chosen 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
static double findOptimumCostUtil(point p[], int n,
line l)
{
double low = -1e6;
double high = 1e6;
// loop until 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
else
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
static double findOptimumCost(int points[][], line l)
{
point[] p = new point[N];
// converting 2D array input to point array
for (int i = 0; i < N; i++)
p[i] = new point(points[i][0], points[i][1]);
return findOptimumCostUtil(p, N, l);
}
// Driver Code
public static void main(String[] args)
{
line l = new line(1, -1, -3);
int points[][] = { { -3, -2 },
{ -1, 0 },
{ -1, 2 },
{ 1, 2 },
{ 3, 4 } };
System.out.println(findOptimumCost(points, l));
}
}
// This code is contributed by Rajput-Ji
# A Python3 program to find optimum location
# and total cost
import math
class Optimum_distance:
# Class defining a point
class Point:
def __init__(self, x, y):
self.x = x
self.y = y
# Class defining a line of ax + by + c = 0 form
class Line:
def __init__(self, a, b, c):
self.a = a
self.b = b
self.c = c
# Method to get distance of point
# (x, y) from point p
def dist(self, x, y, p):
return math.sqrt((x - p.x) ** 2 +
(y - p.y) ** 2)
# Utility method to compute total distance
# all points when choose point on given
# line has x-coordinate value as X
def compute(self, p, n, l, x):
res = 0
y = -1 * (l.a*x + l.c) / l.b
# Calculating Y of chosen point
# by line equation
for i in range(n):
res += self.dist(x, y, p[i])
return res
# Utility method to find minimum total distance
def find_Optimum_cost_untill(self, p, n, l):
low = -1e6
high = 1e6
eps = 1e-6 + 1
# Loop until difference between low
# and high become less than EPS
while((high - low) > eps):
# mid1 and mid2 are representative x
# co-ordiantes of search space
mid1 = low + (high - low) / 3
mid2 = high - (high - low) / 3
dist1 = self.compute(p, n, l, mid1)
dist2 = self.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
else:
low = mid1
# Compute optimum distance cost by
# sending average of low and high as X
return self.compute(p, n, l, (low + high) / 2)
# Method to find optimum cost
def find_Optimum_cost(self, p, l):
n = len(p)
p_arr = [None] * n
# Converting 2D array input to point array
for i in range(n):
p_obj = self.Point(p[i][0], p[i][1])
p_arr[i] = p_obj
return self.find_Optimum_cost_untill(p_arr, n, l)
# Driver Code
if __name__ == "__main__":
obj = Optimum_distance()
l = obj.Line(1, -1, -3)
p = [ [ -3, -2 ], [ -1, 0 ],
[ -1, 2 ], [ 1, 2 ],
[ 3, 4 ] ]
print(obj.find_Optimum_cost(p, l))
# This code is contributed by Sulu_mufi
// C# program to find optimum location
// and total cost
using System;
class GFG {
static double sq(double x) { return ((x) * (x)); }
static int EPS = (int)(1e-6) + 1;
static int N = 5;
// structure defining a point
public class point {
public int x, y;
public point() {}
public point(int x, int y)
{
this.x = x;
this.y = y;
}
};
// structure defining a line
// of ax + by + c = 0 form
public class line {
public int a, b, c;
public line(int a, int b, int c)
{
this.a = a;
this.b = b;
this.c = c;
}
};
// method to get distance of
// point (x, y) from point p
static double dist(double x, double y, point p)
{
return Math.Sqrt(sq(x - p.x) + sq(y - p.y));
}
/* Utility method to compute total distance
of all points when choose point on
given line has x-coordinate value as X */
static double compute(point[] p, int n, line l,
double X)
{
double res = 0;
// calculating Y of chosen 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
static double findOptimumCostUtil(point[] p, int n,
line l)
{
double low = -1e6;
double high = 1e6;
// loop until 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
else
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
static double findOptimumCost(int[, ] points, line l)
{
point[] p = new point[N];
// converting 2D array input to point array
for (int i = 0; i < N; i++)
p[i] = new point(points[i, 0], points[i, 1]);
return findOptimumCostUtil(p, N, l);
}
// Driver Code
public static void Main(String[] args)
{
line l = new line(1, -1, -3);
int[, ] points = { { -3, -2 },
{ -1, 0 },
{ -1, 2 },
{ 1, 2 },
{ 3, 4 } };
Console.WriteLine(findOptimumCost(points, l));
}
}
// This code is contributed by 29AjayKumar
<script>
// A JavaScript program to find optimum location
// and total cost
function sq(x)
{
return x*x;
}
let EPS = (1e-6) + 1;
let N = 5;
// structure defining a point
class point
{
constructor(x,y)
{
this.x=x;
this.y=y;
}
}
// structure defining a line of ax + by + c = 0 form
class line
{
constructor(a,b,c)
{
this.a = a;
this.b = b;
this.c = c;
}
}
// method to get distance of point (x, y) from point p
function dist(x,y,p)
{
return Math.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-coordinate
value as X */
function compute(p,n,l,X)
{
let res = 0;
// calculating Y of chosen point by line equation
let Y = -1 * (l.c + l.a * X) / l.b;
for (let i = 0; i < n; i++)
res += dist(X, Y, p[i]);
return res;
}
// Utility method to find minimum total distance
function findOptimumCostUtil(p,n,l)
{
let low = -1e6;
let high = 1e6;
// loop until difference between low and high
// become less than EPS
while ((high - low) > EPS) {
// mid1 and mid2 are representative x
// co-ordiantes of search space
let mid1 = low + (high - low) / 3;
let mid2 = high - (high - low) / 3;
let dist1 = compute(p, n, l, mid1);
let 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
else
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
function findOptimumCost(points,l)
{
let p = new Array(N);
// converting 2D array input to point array
for (let i = 0; i < N; i++)
p[i] = new point(points[i][0], points[i][1]);
return findOptimumCostUtil(p, N, l);
}
// Driver Code
let l = new line(1, -1, -3);
let points= [[ -3, -2 ],
[ -1, 0 ],
[ -1, 2 ],
[ 1, 2 ],
[ 3, 4 ]];
document.write(findOptimumCost(points, l));
// This code is contributed by rag2127
</script>
Time Complexity: O(n2)
Auxiliary Space: O(n)
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