Find subset with maximum sum under given condition


Given values[] and labels[] of n items and a positive integer limit, We need to choose a subset of these items in such a way that the number of the same type of label in the subset should be <= limit and sum of values are maximum among all possible subset choices.

Examples:

Input: values[] = [5, 3, 7, 1, 2],
       labels[] = [5, 7, 7, 7, 6],
       limit = 2
Output: 17
Explanation:
You can select first, second, third 
and Fifth values.
So, there is 1 value of the label 5 -> {5},
    2 value of the label 7 -> {3, 7} ,
    1 value of the label 6 -> {2}.
Final subset = {5, 3, 7, 2}
Sum  = 5 + 3 + 7 + 2 = 17.

Input: values[] = [9, 8, 7, 6, 5],
       labels[] = [5, 7, 7, 7, 6],
       limit = 2
Output: 29

Approach: The idea is to use a Multimap and Hashmap to solve this problem.

  • We will store all the values and the respective labels as a pair in the Multimap.
  • In Multimap the {values, labels} pairs are sorted in increasing order. So, we will traverse the Multimap in reverse to get the pairs in decreasing order.
  • Now, we will add the values in our answer and store the occurrence of each label in Hashmap to check if the number of occurrence is <= limit.

Below is the implementation of the above approach:

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// C++ program to Find subset with
// maximum sum under given condition.
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the maximum
// sum of the subset
int MaxSumSubset(vector<int>& values, 
                 vector<int>& labels,
                 int n, int limit)
{
    int res = 0;
    multimap<int, int> s;
    unordered_map<int, int> map;
  
    if (n == 0)
    {
        return 0;
    }
  
    // Pushing the pair into 
    // the multimap
    for (int i = 0; i < n; i++)
    {
        s.insert({ values[i], 
                  labels[i] });
    }
  
    // Traversing the multimap 
    // in reverse
    for (auto it = s.rbegin();
         it != s.rend() && n > 0; it++)
    {
  
        if (++map[it->second] <= limit)
        {
            res += it->first;
            n--;
        }
    }
    return res;
}
// Driver code
int main()
{
    vector<int> values = { 5, 3, 7, 1, 2 };
    vector<int> labels = { 5, 7, 7, 7, 6 };
      
    int n = sizeof(values) / sizeof(values[0]);
    int limit = 2;
      
    cout << MaxSumSubset(values, labels,
                         n, limit);
    return 0;
}

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Output:

17

Time Complexity: O(N), where N is the length of the array.

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