Double Knapsack | Dynamic Programming
Given an array ‘arr’ containing the weight of ‘N’ distinct items, and two knapsacks that can withstand ‘W1’ and ‘W2’ weights, the task is to find the sum of the largest subset of the array ‘arr’, that can be fit in the two knapsacks. It’s not allowed to break any items in two, i.e an item should be put in one of the bags as a whole.
Examples:
Input : arr[] = {8, 3, 2}
W1 = 10, W2 = 3
Output : 13
First and third objects go in the first knapsack. The second object goes in the second knapsack. Thus, the total weight becomes 13.
Input : arr[] = {8, 5, 3}
W1 = 10, W2 = 3
Output : 11
Solution:
A recursive solution is to try out all the possible ways of filling the two knapsacks and choose the one giving the maximum weight.
To optimize the above idea, we need to determine the states of DP, that we will build up our solution upon. After little observation, we can determine that this can be represented in three states (i, w1_r, w2_r). Here ‘i’ means the index of the element we are trying to store, w1_r means the remaining space of the first knapsack, and w2_r means the remaining space of the second knapsack. Thus, the problem can be solved using a 3-dimensional dynamic-programming with a recurrence relation
DP[i][w1_r][w2_r] = max( DP[i + 1][w1_r][w2_r],
arr[i] + DP[i + 1][w1_r - arr[i]][w2_r],
arr[i] + DP[i + 1][w1_r][w2_r - arr[i]])The explanation for the above recurrence relation is as follows:
For each ‘i’, we can either:
- Don’t select the item ‘i’.
- Fill the item ‘i’ in first knapsack.
- Fill the item ‘i’ in second knapsack.
Below is the implementation of the above approach:
C++
// C++ implementation of the above approach#include <bits/stdc++.h>#define maxN 31#define maxW 31using namespace std;// 3D array to store// states of DPint dp[maxN][maxW][maxW];// w1_r represents remaining capacity of 1st knapsack// w2_r represents remaining capacity of 2nd knapsack// i represents index of the array arr we are working onint maxWeight(int* arr, int n, int w1_r, int w2_r, int i){ // Base case if (i == n) return 0; if (dp[i][w1_r][w2_r] != -1) return dp[i][w1_r][w2_r]; // Variables to store the result of three // parts of recurrence relation int fill_w1 = 0, fill_w2 = 0, fill_none = 0; if (w1_r >= arr[i]) fill_w1 = arr[i] + maxWeight(arr, n, w1_r - arr[i], w2_r, i + 1); if (w2_r >= arr[i]) fill_w2 = arr[i] + maxWeight(arr, n, w1_r, w2_r - arr[i], i + 1); fill_none = maxWeight(arr, n, w1_r, w2_r, i + 1); // Store the state in the 3D array dp[i][w1_r][w2_r] = max(fill_none, max(fill_w1, fill_w2)); return dp[i][w1_r][w2_r];}// Driver codeint main(){ // Input array int arr[] = { 8, 2, 3 }; // Initializing the array with -1 memset(dp, -1, sizeof(dp)); // Number of elements in the array int n = sizeof(arr) / sizeof(arr[0]); // Capacity of knapsacks int w1 = 10, w2 = 3; // Function to be called cout << maxWeight(arr, n, w1, w2, 0); return 0;} |
Java
// Java implementation of the above approachclass GFG{ static int maxN = 31; static int maxW = 31; // 3D array to store // states of DP static int dp [][][] = new int[maxN][maxW][maxW]; // w1_r represents remaining capacity of 1st knapsack // w2_r represents remaining capacity of 2nd knapsack // i represents index of the array arr we are working on static int maxWeight(int arr [] , int n, int w1_r, int w2_r, int i) { // Base case if (i == n) return 0; if (dp[i][w1_r][w2_r] != -1) return dp[i][w1_r][w2_r]; // Variables to store the result of three // parts of recurrence relation int fill_w1 = 0, fill_w2 = 0, fill_none = 0; if (w1_r >= arr[i]) fill_w1 = arr[i] + maxWeight(arr, n, w1_r - arr[i], w2_r, i + 1); if (w2_r >= arr[i]) fill_w2 = arr[i] + maxWeight(arr, n, w1_r, w2_r - arr[i], i + 1); fill_none = maxWeight(arr, n, w1_r, w2_r, i + 1); // Store the state in the 3D array dp[i][w1_r][w2_r] = Math.max(fill_none, Math.max(fill_w1, fill_w2)); return dp[i][w1_r][w2_r]; } // Driver code public static void main (String[] args) { // Input array int arr[] = { 8, 2, 3 }; // Initializing the array with -1 for (int i = 0; i < maxN ; i++) for (int j = 0; j < maxW ; j++) for (int k = 0; k < maxW ; k++) dp[i][j][k] = -1; // Number of elements in the array int n = arr.length; // Capacity of knapsacks int w1 = 10, w2 = 3; // Function to be called System.out.println(maxWeight(arr, n, w1, w2, 0)); }}// This code is contributed by ihritik |
Python3
# Python3 implementation of the above approach# w1_r represents remaining capacity of 1st knapsack# w2_r represents remaining capacity of 2nd knapsack# i represents index of the array arr we are working ondef maxWeight(arr, n, w1_r, w2_r, i): # Base case if i == n: return 0 if dp[i][w1_r][w2_r] != -1: return dp[i][w1_r][w2_r] # Variables to store the result of three # parts of recurrence relation fill_w1, fill_w2, fill_none = 0, 0, 0 if w1_r >= arr[i]: fill_w1 = arr[i] + maxWeight(arr, n, w1_r - arr[i], w2_r, i + 1) if w2_r >= arr[i]: fill_w2 = arr[i] + maxWeight(arr, n, w1_r, w2_r - arr[i], i + 1) fill_none = maxWeight(arr, n, w1_r, w2_r, i + 1) # Store the state in the 3D array dp[i][w1_r][w2_r] = max(fill_none, max(fill_w1, fill_w2)) return dp[i][w1_r][w2_r]# Driver codeif __name__ == "__main__": # Input array arr = [8, 2, 3] maxN, maxW = 31, 31 # 3D array to store # states of DP dp = [[[-1] * maxW] * maxW] * maxN # Number of elements in the array n = len(arr) # Capacity of knapsacks w1, w2 = 10, 3 # Function to be called print(maxWeight(arr, n, w1, w2, 0)) # This code is contributed by Rituraj Jain |
C#
// C# implementation of the above approachusing System;class GFG{ static int maxN = 31; static int maxW = 31; // 3D array to store // states of DP static int [ , , ] dp = new int[maxN, maxW, maxW]; // w1_r represents remaining capacity of 1st knapsack // w2_r represents remaining capacity of 2nd knapsack // i represents index of the array arr we are working on static int maxWeight(int [] arr, int n, int w1_r, int w2_r, int i) { // Base case if (i == n) return 0; if (dp[i ,w1_r, w2_r] != -1) return dp[i, w1_r, w2_r]; // Variables to store the result of three // parts of recurrence relation int fill_w1 = 0, fill_w2 = 0, fill_none = 0; if (w1_r >= arr[i]) fill_w1 = arr[i] + maxWeight(arr, n, w1_r - arr[i], w2_r, i + 1); if (w2_r >= arr[i]) fill_w2 = arr[i] + maxWeight(arr, n, w1_r, w2_r - arr[i], i + 1); fill_none = maxWeight(arr, n, w1_r, w2_r, i + 1); // Store the state in the 3D array dp[i, w1_r, w2_r] = Math.Max(fill_none, Math.Max(fill_w1, fill_w2)); return dp[i, w1_r, w2_r]; } // Driver code public static void Main () { // Input array int [] arr = { 8, 2, 3 }; // Initializing the array with -1 for (int i = 0; i < maxN ; i++) for (int j = 0; j < maxW ; j++) for (int k = 0; k < maxW ; k++) dp[i, j, k] = -1; // Number of elements in the array int n = arr.Length; // Capacity of knapsacks int w1 = 10, w2 = 3; // Function to be called Console.WriteLine(maxWeight(arr, n, w1, w2, 0)); }}// This code is contributed by ihritik |
Javascript
<script>// Javascript implementation of// the above approachvar maxN = 31var maxW = 31// 3D array to store// states of DPvar dp = Array(maxN);for(var i=0;i<maxN;i++){ dp[i] = Array(maxW); for(var j =0; j<maxW; j++) { dp[i][j] = Array(maxW).fill(-1); }}// w1_r represents remaining// capacity of 1st knapsack// w2_r represents remaining// capacity of 2nd knapsack// i represents index of the array arr// we are working onfunction maxWeight(arr, n, w1_r, w2_r, i){ // Base case if (i == n) return 0; if (dp[i][w1_r][w2_r] != -1) return dp[i][w1_r][w2_r]; // Variables to store the result of three // parts of recurrence relation var fill_w1 = 0, fill_w2 = 0, fill_none = 0; if (w1_r >= arr[i]) fill_w1 = arr[i] + maxWeight(arr, n, w1_r - arr[i], w2_r, i + 1); if (w2_r >= arr[i]) fill_w2 = arr[i] + maxWeight(arr, n, w1_r, w2_r - arr[i], i + 1); fill_none = maxWeight(arr, n, w1_r, w2_r, i + 1); // Store the state in the 3D array dp[i][w1_r][w2_r] = Math.max(fill_none, Math.max(fill_w1, fill_w2)); return dp[i][w1_r][w2_r];}// Driver code// Input arrayvar arr = [8, 2, 3 ];// Number of elements in the arrayvar n = arr.length;// Capacity of knapsacksvar w1 = 10, w2 = 3;// Function to be calleddocument.write( maxWeight(arr, n, w1, w2, 0));</script> |
Output:
13
Time complexity: O(N*W1*W2)
Auxiliary Space: O(N*W1*W2))
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