Count of subsets with sum equal to X
source link: https://www.geeksforgeeks.org/count-of-subsets-with-sum-equal-to-x/
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Given an array arr[] of length N and an integer X, the task is to find the number of subsets with sum equal to X.
Examples:
Input: arr[] = {1, 2, 3, 3}, X = 6
Output: 3
All the possible subsets are {1, 2, 3},
{1, 2, 3} and {3, 3}Input: arr[] = {1, 1, 1, 1}, X = 1
Output: 4
Approach: A simple approach is to solve this problem by generating all the possible subsets and then checking whether the subset has the required sum. This approach will have exponential time complexity. However, for smaller values of X and array elements, this problem can be solved using dynamic programming.
Let’s look at the recurrence relation first.
dp[i][C] = dp[i + 1][C – arr[i]] + dp[i + 1][C]
Let’s understand the states of the DP now. Here, dp[i][C] stores the number of subsets of the sub-array arr[i…N-1] such that their sum is equal to C.
Thus, the recurrence is very trivial as there are only two choices i.e. either consider the ith element in the subset or don’t.
Below is the implementation of the above approach:
- Python3
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// C++ implementation of the approach #include <bits/stdc++.h> using namespace std; #define maxN 20 #define maxSum 50 #define minSum 50 #define base 50 // To store the states of DP int dp[maxN][maxSum + minSum]; bool v[maxN][maxSum + minSum]; // Function to return the required count int findCnt(int* arr, int i, int required_sum, int n) { // Base case if (i == n) { if (required_sum == 0) return 1; elsereturn 0; } // If the state has been solved before // return the value of the state if (v[i][required_sum + base]) return dp[i][required_sum + base]; // Setting the state as solved v[i][required_sum + base] = 1; // Recurrence relation dp[i][required_sum + base] = findCnt(arr, i + 1, required_sum, n) + findCnt(arr, i + 1, required_sum - arr[i], n); return dp[i][required_sum + base]; } // Driver code int main() { int arr[] = { 3, 3, 3, 3 }; int n = sizeof(arr) / sizeof(int); int x = 6; cout << findCnt(arr, 0, x, n); return 0; } 6
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