Skip to content
Related Articles

Related Articles

K Centers Problem | Set 1 (Greedy Approximate Algorithm)
  • Difficulty Level : Hard
  • Last Updated : 22 Mar, 2021
GeeksforGeeks - Summer Carnival Banner

Given n cities and distances between every pair of cities, select k cities to place warehouses (or ATMs or Cloud Server) such that the maximum distance of a city to a warehouse (or ATM or Cloud Server) is minimized. 
For example consider the following four cities, 0, 1, 2 and 3 and distances between them, how do place 2 ATMs among these 4 cities so that the maximum distance of a city to an ATM is minimized. 
 

kcenters1

There is no polynomial time solution available for this problem as the problem is a known NP-Hard problem. There is a polynomial time Greedy approximate algorithm, the greedy algorithm provides a solution which is never worse that twice the optimal solution. The greedy solution works only if the distances between cities follow Triangular Inequality (Distance between two points is always smaller than sum of distances through a third point). 
The 2-Approximate Greedy Algorithm: 
1) Choose the first center arbitrarily. 
2) Choose remaining k-1 centers using the following criteria. 
       Let c1, c2, c3, … ci be the already chosen centers. Choose 
       (i+1)’th center by picking the city which is farthest from already 
       selected centers, i.e, the point p which has following value as maximum 
                 Min[dist(p, c1), dist(p, c2), dist(p, c3), …. dist(p, ci)] 
 

greedyAlgo

Example (k = 3 in the above shown Graph) 
a) Let the first arbitrarily picked vertex be 0. 
b) The next vertex is 1 because 1 is the farthest vertex from 0. 
c) Remaining cities are 2 and 3. Calculate their distances from already selected centers (0 and 1). The greedy algorithm basically calculates following values. 
        Minimum of all distanced from 2 to already considered centers 
        Min[dist(2, 0), dist(2, 1)] = Min[7, 8] = 7 
        Minimum of all distanced from 3 to already considered centers 
        Min[dist(3, 0), dist(3, 1)] = Min[6, 5] = 5 
        After computing the above values, the city 2 is picked as the value corresponding to 2 is maximum. 
Note that the greedy algorithm doesn’t give best solution for k = 2 as this is just an approximate algorithm with bound as twice of optimal. 
Proof that the above greedy algorithm is 2 approximate. 
Let OPT be the maximum distance of a city from a center in the Optimal solution. We need to show that the maximum distance obtained from Greedy algorithm is 2*OPT. 
The proof can be done using contradiction. 
a) Assume that the distance from the furthest point to all centers is > 2·OPT. 
b) This means that distances between all centers are also > 2·OPT. 
c) We have k + 1 points with distances > 2·OPT between every pair. 
d) Each point has a center of the optimal solution with distance <= OPT to it. 
e) There exists a pair of points with the same center X in the optimal solution (pigeonhole principle: k optimal centers, k+1 points) 
f) The distance between them is at most 2·OPT (triangle inequality) which is a contradiction. 



C++




// CPP program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
int maxindex(int* dist, int n)
{
    int mi = 0;
    for (int i = 0; i < n; i++) {
        if (dist[i] > dist[mi])
            mi = i;
    }
    return mi;
}
 
void selectKcities(int n, int weights[4][4], int k)
{
    int* dist = new int[n];
    vector<int> centers;
    for (int i = 0; i < n; i++) {
        dist[i] = INT_MAX;
    }
 
    // index of city having the
    // maximum distance to it's
    // closest center
    int max = 0;
    for (int i = 0; i < k; i++) {
        centers.push_back(max);
        for (int j = 0; j < n; j++) {
 
            // updating the distance
            // of the cities to their
            // closest centers
            dist[j] = min(dist[j], weights[max][j]);
        }
 
        // updating the index of the
        // city with the maximum
        // distance to it's closest center
        max = maxindex(dist, n);
    }
 
    // Printing the maximum distance
    // of a city to a center
    // that is our answer
    cout << endl << dist[max] << endl;
 
    // Printing the cities that
    // were chosen to be made
    // centers
    for (int i = 0; i < centers.size(); i++) {
        cout << centers[i] << " ";
    }
    cout << endl;
}
 
// Driver Code
int main()
{
    int n = 4;
    int weights[4][4] = { { 0, 4, 8, 5 },
                          { 4, 0, 10, 7 },
                          { 8, 10, 0, 9 },
                          { 5, 7, 9, 0 } };
    int k = 2;
 
    // Function Call
    selectKcities(n, weights, k);
}
// Contributed by Balu Nagar

Python3




#CPP program for the above approach
def maxindex(dist, n):
    mi = 0
    for i in range(n):
        if (dist[i] > dist[mi]):
            mi = i
    return mi
 
def selectKcities(n, weights, k):
    dist = [0]*n
    centers = []
 
    for i in range(n):
        dist[i] = 10**9
         
    # index of city having the
    # maximum distance to it's
    # closest center
    max = 0
    for i in range(k):
        centers.append(max)
        for j in range(n):
 
            # updating the distance
            # of the cities to their
            # closest centers
            dist[j] = min(dist[j], weights[max][j])
 
        # updating the index of the
        # city with the maximum
        # distance to it's closest center
        max = maxindex(dist, n)
 
    # Printing the maximum distance
    # of a city to a center
    # that is our answer
    # print()
    print(dist[max])
 
    # Printing the cities that
    # were chosen to be made
    # centers
    for i in centers:
        print(i, end = " ")
 
# Driver Code
if __name__ == '__main__':
    n = 4
    weights = [ [ 0, 4, 8, 5 ],
              [ 4, 0, 10, 7 ],
              [ 8, 10, 0, 9 ],
              [ 5, 7, 9, 0 ] ]
    k = 2
 
    # Function Call
    selectKcities(n, weights, k)
 
# This code is contributed by mohit kumar 29.
Output
5
0 2 

Source: 
http://algo2.iti.kit.edu/vanstee/courses/kcenter.pdf 
This article is contributed by Harshit. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up
Recommended Articles
Page :