Dynamic Programming Algorithm for Knapsack Problem

This document presents a dynamic programming approach to solve the knapsack problem with accompanying code examples. The implementation includes file handling capabilities that read input parameters from text files, making it particularly accessible for beginners to study and comprehend. The core algorithm employs a bottom-up approach where we construct a DP table storing maximum values achievable for different weight capacities. Key functions include parsing input data, initializing the DP matrix, and implementing the recurrence relation: dp[i][w] = max(dp[i-1][w], values[i] + dp[i-1][w-weights[i]]). Dynamic programming proves especially valuable for knapsack problems as it efficiently handles varying capacities and item quantities while guaranteeing optimal solutions. Mastering this algorithm will significantly enhance your programming skills and provide deeper understanding of dynamic programming concepts, including state transition, optimal substructure, and memoization techniques.