0 1 Knapsack Problem Geeksforgeeks Vdxeb

Bonisiwe Shabane

-Jan 20, 2026, 4:02 PM

0 1 knapsack problem geeksforgeeks vdxeb

Given n items where each item has some weight and profit associated with it and also given a bag with capacity W, [i.e., the bag can hold at most W weight in it]. The task is to put the items into the bag such that the sum of profits associated with them is the maximum possible. Note: The constraint here is we can either put an item completely into the bag or cannot put it at all [It is not possible to put a part of an item into the... Input: W = 4, profit[] = [1, 2, 3], weight[] = [4, 5, 1]Output: 3Explanation: There are two items which have weight less than or equal to 4. If we select the item with weight 4, the possible profit is 1. And if we select the item with weight 1, the possible profit is 3.

So the maximum possible profit is 3. Note that we cannot put both the items with weight 4 and 1 together as the capacity of the bag is 4. Input: W = 3, profit[] = [1, 2, 3], weight[] = [4, 5, 6]Output: 0 A simple solution is to consider all subsets of items and calculate the total weight and value of all subsets. Consider the only subsets whose total weight is smaller than W. From all such subsets, pick the subset with maximum value.

W3Schools offers a wide range of services and products for beginners and professionals, helping millions of people everyday to learn and master new skills. Enjoy our free tutorials like millions of other internet users since 1999 Explore our selection of references covering all popular coding languages Create your own website with W3Schools Spaces - no setup required Test your skills with different exercises Given a list of items, each with a weight and a value, determine the maximum value you can achieve without exceeding a given weight capacity (maxWeight).

You can only include each item once, and you must decide whether to include or exclude each item. Example Let's take an example where you have the following items: The goal is to maximize the total value of items in the knapsack without exceeding the weight capacity of 5. Approach 1: Recursive (Brute Force) Solution The first approach uses recursion to explore all possible combinations of including or excluding each item. The basic idea is to try both options for every item and take the maximum value from these choices. Approach 2: Memoization To optimize the recursive approach, we can use memoization.

This technique stores the results of subproblems, avoiding redundant calculations. The 0/1 Knapsack Problem is one of the most important and frequently asked Dynamic Programming (DP) problems in DSA interviews. It is used to test your understanding of decision making, optimization, and DP table construction. In simple words, this problem is about choosing the best items to put into a bag so that the total value is maximum, without exceeding the bag’s capacity. This article explains the 0/1 Knapsack problem in very simple language, step by step, from basics to interview-level understanding. The word knapsack means a bag.

In this problem: You are given a bag with a fixed capacity We discussed the fractional knapsack problem using the greedy approach, earlier in this tutorial. It is shown that Greedy approach gives an optimal solution for Fractional Knapsack. However, this chapter will cover 0-1 Knapsack problem using dynamic programming approach and its analysis. Unlike in fractional knapsack, the items are always stored fully without using the fractional part of them.

Its either the item is added to the knapsack or not. That is why, this method is known as the 0-1 Knapsack problem. Hence, in case of 0-1 Knapsack, the value of xi can be either 0 or 1, where other constraints remain the same. 0-1 Knapsack cannot be solved by Greedy approach. Greedy approach does not ensure an optimal solution in this method. In many instances, Greedy approach may give an optimal solution.

Problem Statement − A thief is robbing a store and can carry a maximal weight of W into his knapsack. There are n items and weight of ith item is wi and the profit of selecting this item is pi. What items should the thief take? Sarah Lee AI generated Llama-4-Maverick-17B-128E-Instruct-FP8 7 min read · June 13, 2025 The 0/1 Knapsack problem is a classic problem in combinatorial optimization. It involves finding the optimal way to pack a set of items, each with a weight and a value, into a knapsack of limited capacity.

The goal is to maximize the total value of the items in the knapsack without exceeding its capacity. In this article, we will explore how to solve the 0/1 Knapsack problem efficiently using dynamic programming and memoization techniques. The 0/1 Knapsack problem can be solved using a naive recursive approach. The idea is to consider each item one by one and decide whether to include it in the knapsack or not. The recursive formula for the 0/1 Knapsack problem can be defined as follows: Let $n$ be the number of items, $w_i$ be the weight of the $i^{th}$ item, $v_i$ be the value of the $i^{th}$ item, and $W$ be the capacity of the knapsack.

The maximum value that can be obtained with $i$ items and a capacity of $j$ is denoted by $V(i, j)$. Then, the recursive formula is given by: \[ V(i, j) = \max{V(i-1, j), V(i-1, j-w_i) + v_i} \]

0 1 Knapsack Problem Geeksforgeeks Vdxeb

People Also Search

Given N Items Where Each Item Has Some Weight And

So The Maximum Possible Profit Is 3. Note That We

W3Schools Offers A Wide Range Of Services And Products For

You Can Only Include Each Item Once, And You Must

This Technique Stores The Results Of Subproblems, Avoiding Redundant Calculations.