Day 33: The Knapsack Problem
The Knapsack Problem #
Welcome to Day 33 of our 60 Days of Coding Algorithm Challenge! Today, we’ll explore the Knapsack Problem, a fundamental problem in combinatorial optimization and a classic example of dynamic programming.
What is the Knapsack Problem? #
The Knapsack Problem is an optimization problem where we need to select items from a set, each with a weight and a value, to maximize the total value while keeping the total weight within a given limit.
There are two main variants of the Knapsack Problem:
- 0/1 Knapsack Problem: Each item can be included only once or not at all.
- Unbounded Knapsack Problem: Each item can be included any number of times.
We’ll focus on the 0/1 Knapsack Problem in this lesson.
Problem Statement #
Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack.
Naive Recursive Approach #
Let’s start with a naive recursive implementation:
1def knapsack_naive(W, wt, val, n):
2 # Base Case
3 if n == 0 or W == 0:
4 return 0
5
6 # If weight of the nth item is more than Knapsack capacity W,
7 # then this item cannot be …
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