Minimum Cost to Connect Two Groups of Points

This page explains Java solution to problem Minimum Cost to Connect Two Groups of Points using Bit Masking.

Problem Statement

You are given two groups of points where the first group has size1 points, the second group has size2 points, and size1 >= size2.

The cost of the connection between any two points are given in an size1 x size2 matrix where cost[i][j] is the cost of connecting point i of the first group and point j of the second group. The groups are connected if each point in both groups is connected to one or more points in the opposite group. In other words, each point in the first group must be connected to at least one point in the second group, and each point in the second group must be connected to at least one point in the first group.

Return the minimum cost it takes to connect the two groups.

Input: cost = [[15, 96], [36, 2]]
Output: 17
Explanation: The optimal way of connecting the groups is:
1--A
2--B
This results in a total cost of 17.

Input: cost = [[1, 3, 5], [4, 1, 1], [1, 5, 3]]
Output: 4
Explanation: The optimal way of connecting the groups is:
1--A
2--B
2--C
3--A
This results in a total cost of 4.
Note that there are multiple points connected to point 2 in the first group and point A in the second group. This does not matter as there is no limit to the number of points that can be connected. We only care about the minimum total cost.

Input: cost = [[2, 5, 1], [3, 4, 7], [8, 1, 2], [6, 2, 4], [3, 8, 8]]
Output: 10

Solution

package com.vc.hard;

import java.util.*;

class MinimumCostToConnectTwoGroupsOfPoints {
    public int connectTwoGroups(List<List<Integer>> cost) {
        int size1 = cost.size();
        int size2 = cost.get(0).size();

        int[] dp = new int[(1 << size2)];
        Arrays.fill(dp, Integer.MAX_VALUE);
        dp[0] = 0;

        //Iterate over all the elements from group1
        for(int group1 = 0; group1 < size1; group1++) {
            int[] current = new int[(1 << size2)];
            Arrays.fill(current, Integer.MAX_VALUE);

            //For each element in a group1 there are 2^group2 possibilities
            //Because each element from group1 can connect to one or more element from group2
            for(int group2 = 0; group2 < size2; group2++) {

                //Iterate over all 2^group2 possibilities
                for(int mask = 0; mask < (1 << size2); mask++) {
                    int newMask = mask | (1 << group2);

                    //If group2 element is not connected in a current mask,
                    //You can derive newMask from current mask
                    if((mask & (1 << group2)) == 0) {
                        if(current[mask] != Integer.MAX_VALUE) {
                            current[newMask] = Math.min(current[newMask],
                                    current[mask] + cost.get(group1).get(group2));
                        }
                    }

                    //Check if newMask value can be reduced further by using previous row state
                    if(dp[mask] != Integer.MAX_VALUE) {
                        current[newMask] = Math.min(current[newMask],
                                dp[mask] + cost.get(group1).get(group2));
                    }
                }
            }
            dp = current;
        }

        return dp[(1 << size2) - 1];
    }
}

Time Complexity

O(NM * 2^M) Where
N is number of elements in a group1
M is number of elements in a group2

Space Complexity

O(2^M) Where
M is number of elements in a group2