Given an integer array W[] consisting of weights of the items and some queries consisting of capacity C of knapsack, for each query find maximum weight we can put in the knapsack. Value of C does’t exceed a certain integer C_MAX.
Examples:
Input: W[] = {3, 8, 9} q = {11, 10, 4}
Output:
11
9
3
If C = 11: select 3 + 8 = 11
If C = 10: select 9
If C = 4: select 3Input: W[] = {1, 5, 10} q = {6, 14}
Output:
6
11
Its recommended that you go through this article on 0-1 knapsack before attempting this problem.
Naive approach: For each query, we can generate all possible subsets of weight and choose the one that has maximum weight among all those subsets that fits in the knapsack. Thus, answering each query will take exponential time.
Efficient approach: We will optimise answering each query using dynamic programming.
0-1 knapsack is solved using 2-D DP, one state ‘i’ for current index(i.e select or reject) and one for remaining capacity ‘R’.
Recurrence relation is
dp[i][R] = max(arr[i] + dp[i + 1][R – arr[i]], dp[i + 1][R])
We will pre-compute the the 2-d array dp[i][C] for every possible value of ‘C’ between 1 to C_MAX in O(C_MAX*i).
Using this, pre-computation we can answer each queries in O(1) time.
Below is the implementation of the above approach:
C++
// C++ implementation of the approach#include <bits/stdc++.h>#define C_MAX 30#define max_arr_len 10using namespace std; // To store states of DPint dp[max_arr_len][C_MAX + 1]; // To check if a state has// been solvedbool v[max_arr_len][C_MAX + 1]; // Function to compute the statesint findMax(int i, int r, int w[], int n){ // Base case if (r < 0) return INT_MIN; if (i == n) return 0; // Check if a state has // been solved if (v[i][r]) return dp[i][r]; // Setting a state as solved v[i][r] = 1; dp[i][r] = max(w[i] + findMax(i + 1, r - w[i], w, n), findMax(i + 1, r, w, n)); // Returning the solved state return dp[i][r];} // Function to pre-compute the states// dp[0][0], dp[0][1], .., dp[0][C_MAX]void preCompute(int w[], int n){ for (int i = C_MAX; i >= 0; i--) findMax(0, i, w, n);} // Function to answer a query in O(1)int ansQuery(int w){ return dp[0][w];} // Driver codeint main(){ int w[] = { 3, 8, 9 }; int n = sizeof(w) / sizeof(int); // Performing required // pre-computation preCompute(w, n); int queries[] = { 11, 10, 4 }; int q = sizeof(queries) / sizeof(queries[0]); // Perform queries for (int i = 0; i < q; i++) cout << ansQuery(queries[i]) << endl; return 0;} |
Java
// Java implementation of the approachclass GFG{ static int C_MAX = 30; static int max_arr_len = 10; // To store states of DP static int dp [][] = new int [max_arr_len][C_MAX + 1]; // To check if a state has // been solved static boolean v[][]= new boolean [max_arr_len][C_MAX + 1]; // Function to compute the states static int findMax(int i, int r, int w[], int n) { // Base case if (r < 0) return Integer.MIN_VALUE; if (i == n) return 0; // Check if a state has // been solved if (v[i][r]) return dp[i][r]; // Setting a state as solved v[i][r] = true; dp[i][r] = Math.max(w[i] + findMax(i + 1, r - w[i], w, n), findMax(i + 1, r, w, n)); // Returning the solved state return dp[i][r]; } // Function to pre-compute the states // dp[0][0], dp[0][1], .., dp[0][C_MAX] static void preCompute(int w[], int n) { for (int i = C_MAX; i >= 0; i--) findMax(0, i, w, n); } // Function to answer a query in O(1) static int ansQuery(int w) { return dp[0][w]; } // Driver code public static void main (String[] args) { int w[] = new int []{ 3, 8, 9 }; int n = w.length; // Performing required // pre-computation preCompute(w, n); int queries[] = new int [] { 11, 10, 4 }; int q = queries.length; // Perform queries for (int i = 0; i < q; i++) System.out.println(ansQuery(queries[i])); }} // This code is contributed by ihritik |
Python3
# Python3 implementation of the approach import numpy as npimport sys C_MAX = 30max_arr_len = 10 # To store states of DP dp = np.zeros((max_arr_len,C_MAX + 1)); # To check if a state has # been solved v = np.zeros((max_arr_len,C_MAX + 1)); INT_MIN = -(sys.maxsize) + 1 # Function to compute the states def findMax(i, r, w, n) : # Base case if (r < 0) : return INT_MIN; if (i == n) : return 0; # Check if a state has # been solved if (v[i][r]) : return dp[i][r]; # Setting a state as solved v[i][r] = 1; dp[i][r] = max(w[i] + findMax(i + 1, r - w[i], w, n), findMax(i + 1, r, w, n)); # Returning the solved state return dp[i][r]; # Function to pre-compute the states # dp[0][0], dp[0][1], .., dp[0][C_MAX] def preCompute(w, n) : for i in range(C_MAX, -1, -1) : findMax(0, i, w, n); # Function to answer a query in O(1) def ansQuery(w) : return dp[0][w]; # Driver code if __name__ == "__main__" : w = [ 3, 8, 9 ]; n = len(w) # Performing required # pre-computation preCompute(w, n); queries = [ 11, 10, 4 ]; q = len(queries) # Perform queries for i in range(q) : print(ansQuery(queries[i])); # This code is coontributed by AnkitRai01 |
C#
// C# implementation of the approachusing System; class GFG{ static int C_MAX = 30; static int max_arr_len = 10; // To store states of DP static int [ , ] dp = new int [max_arr_len, C_MAX + 1]; // To check if a state has // been solved static bool [ , ] v = new bool [max_arr_len, C_MAX + 1]; // Function to compute the states static int findMax(int i, int r, int [] w, int n) { // Base case if (r < 0) return Int32.MinValue; if (i == n) return 0; // Check if a state has // been solved if (v[i, r]) return dp[i, r]; // Setting a state as solved v[i, r] = true; dp[i, r] = Math.Max(w[i] + findMax(i + 1, r - w[i], w, n), findMax(i + 1, r, w, n)); // Returning the solved state return dp[i, r]; } // Function to pre-compute the states // dp[0][0], dp[0][1], .., dp[0][C_MAX] static void preCompute(int []w, int n) { for (int i = C_MAX; i >= 0; i--) findMax(0, i, w, n); } // Function to answer a query in O(1) static int ansQuery(int w) { return dp[0, w]; } // Driver code public static void Main () { int []w = new int []{ 3, 8, 9 }; int n = w.Length; // Performing required // pre-computation preCompute(w, n); int [] queries = new int [] { 11, 10, 4 }; int q = queries.Length; // Perform queries for (int i = 0; i < q; i++) Console.WriteLine (ansQuery(queries[i])); }} // This code is contributed by ihritik |
11 9 3
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