Greedy Approximate Algorithm for K Centers Problem

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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 the distances between them, how to place 2 ATMs among these 4 cities so that the maximum distance of a city to an ATM is minimized. 

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 that is never worse than twice the optimal solution. The greedy solution works only if the distances between cities follow Triangular Inequality (The distance between two points is always smaller than the 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)] 

Example (k = 3 in the above-shown Graph):

1. Let the first arbitrarily picked vertex be 0. 
2. The next vertex is 1 because 1 is the farthest vertex from 0. 
3. Remaining cities are 2 and 3. Calculate their distances from already selected centers (0 and 1). The greedy algorithm basically calculates the 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, city 2 is picked as the value corresponding to 2 is maximum. 

Note that the greedy algorithm doesn't give the best solution for k = 2 as this is just an approximate algorithm with a bound as twice 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 the Greedy algorithm is 2*OPT. 
The proof can be done using contradiction. 

1. Assume that the distance from the furthest point to all centers is > 2·OPT. 
2. This means that distances between all centers are also > 2·OPT. 
3. We have k + 1 points with distances > 2·OPT between every pair. 
4. Each point has a center of the optimal solution with distance <= OPT to it. 

1. There exists a pair of points with the same center X in the optimal solution (pigeonhole principle: k optimal centers, k+1 points) 
2. The distance between them is at most 2·OPT (triangle inequality) which is a contradiction. 

Implementation:

// C++ 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

// Java program for the above approach
import java.util.*;

class GFG{

static 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;
}

static void selectKcities(int n, int weights[][], 
                          int k)
{
    int[] dist = new int[n];
    ArrayList<Integer> centers = new ArrayList<>();
    for(int i = 0; i < n; i++) 
    {
        dist[i] = Integer.MAX_VALUE;
    }

    // Index of city having the
    // maximum distance to it's
    // closest center
    int max = 0;
    for(int i = 0; i < k; i++)
    {
        centers.add(max);
        for(int j = 0; j < n; j++)
        {
            
            // Updating the distance
            // of the cities to their
            // closest centers
            dist[j] = Math.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
    System.out.println(dist[max]);

    // Printing the cities that
    // were chosen to be made
    // centers
    for(int i = 0; i < centers.size(); i++) 
    {
        System.out.print(centers.get(i) + " ");
    }
    System.out.print("\n");
}

// Driver Code
public static void main(String[] args)
{
    int n = 4;
    int[][] weights = new int[][]{ { 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);
}
}

// This code is contributed by nspatilme

# Python3 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.

using System;
using System.Collections.Generic;

public class GFG{
    
    static 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;
    }
     
    static void selectKcities(int n, int[,] weights,
                              int k)
    {
        int[] dist = new int[n];
        List<int> centers = new List<int>();
        for(int i = 0; i < n; i++)
        {
            dist[i] = Int32.MaxValue;
        }
     
        // Index of city having the
        // maximum distance to it's
        // closest center
        int max = 0;
        for(int i = 0; i < k; i++)
        {
            centers.Add(max);
            for(int j = 0; j < n; j++)
            {
                 
                // Updating the distance
                // of the cities to their
                // closest centers
                dist[j] = Math.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
        Console.WriteLine(dist[max]);
     
        // Printing the cities that
        // were chosen to be made
        // centers
        for(int i = 0; i < centers.Count; i++)
        {
            Console.Write(centers[i] + " ");
        }
        Console.Write("\n");
    }
     
    // Driver Code
    static public void Main (){
        
        int n = 4;
    int[,] weights = new int[,]{ { 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);
        
    }
}

// This code is contributed by avanitrachhadiya2155.

<script>
// Javascript program for the above approach
    
    function maxindex(dist,n)
    {
        let mi = 0;
    for(let i = 0; i < n; i++)
    {
        if (dist[i] > dist[mi])
            mi = i;
    }
    return mi;
    }
    
    function selectKcities(n,weights,k)
    {
        let dist = new Array(n);
    let centers = [];
    for(let i = 0; i < n; i++)
    {
        dist[i] = Number.MAX_VALUE;
    }
 
    // Index of city having the
    // maximum distance to it's
    // closest center
    let max = 0;
    for(let i = 0; i < k; i++)
    {
        centers.push(max);
        for(let j = 0; j < n; j++)
        {
             
            // Updating the distance
            // of the cities to their
            // closest centers
            dist[j] = Math.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
    document.write(dist[max]+"<br>");
 
    // Printing the cities that
    // were chosen to be made
    // centers
    for(let i = 0; i < centers.length; i++)
    {
        document.write(centers[i] + " ");
    }
    document.write("<br>");
    }
    
    // Driver Code
    let n = 4;
    let weights = [ [ 0, 4, 8, 5 ],
              [ 4, 0, 10, 7 ],
              [ 8, 10, 0, 9 ],
              [ 5, 7, 9, 0 ] ]
    let k = 2
    selectKcities(n, weights, k)
    
    // This code is contributed by unknown2108
</script>

5
0 2 

Time Complexity: O(n*k), as we are using nested loops to traverse n*k times.
Auxiliary Space: O(n+k), as we are using extra space for the array center.

kartik

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Article Tags :

• Graph
• Greedy
• DSA

Practice Tags :

• Graph
• Greedy