Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 33

Write the dynamic programming algorithm for the 0 -1 Knapsack problem.

Short Answer

Expert verified
Initialize a table with 'items+1' rows and 'knapsack capacity+1' columns, filling the first row and column with zeroes. Then fill the rest of the table for each item and each weight capacity, taking into consideration the weight of the current item. Finally, the maximum profit or optimal solution is stored at the last cell of the table, representing the full capacity and all items.

Step by step solution

01

Title - Initialize a table

First, create an array or table with rows equal to the number of items plus one, and columns equal to the weight of the knapsack plus one. Now, initialize the first row and first column of this array to zeroes. The row zero represents zero items and the column zero indicates zero capacity of knapsack. This signifies that if there are no items or the knapsack capacity is zero, then the maximum profit possible is zero.
02

Title - Populate the Remaining Table

For each item, start from one up to the total number of items. For each weight capacity, start from one up to the total knapsack capacity. If the weight of the current item is greater than the current weight capacity, then copy the maximum profit from the previous item into the current position. Else, take the maximum of either the value of the previous item or the sum of the value of the current item and the maximum profit from the remaining weight capacity (which is obtained from the previous row). Continue doing this until the last item and last weight capacity is reached.
03

Title - Retrieve the Optimal Solution

The optimal solution or the maximum profit that can be achieved is stored at the last row and the last column of the table. This is because this cell represents considering all items and the total weight capacity of the knapsack. Therefore, the value at this position will be our answer.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that a binary tree corresponding to an optimal binary prefix code must be full. A full binary tree is a binary tree such that every node is either a leaf or it has two children.

Use a greedy approach to write an algorithm for the Traveling Salesperson problem. Show that your algorithm does not always find a minimumlength tour.

Suppose we assign \(n\) persons to \(n\) jobs. Let \(C_{i j}\) be the cost of assigning the th person to the jth job. Use a greedy approach to write an algorithm that finds an assignment that minimizes the total cost of assigning all \(n\) persons to all \(n\) jobs. Analyze your algorithm and show the results using order notation.

Suppose we minimize the average time to store \(n\) files of lengths \(l_{1}, l_{2}, \dots, l_{n}\) on a tape. If the probability of requesting file \(k\) is given by \(p_{k}\), the expected access time to load these \(n\) files in the order \(k_{1}, k_{2}, \ldots, k_{n}\) is given by the formula \(T_{\text {average}}=C \sum_{f=1}^{n}\left(p_{k_{f}} \sum_{i=1}^{f} l_{k_{i}}\right)\) The constant \(C\) represents parameters such as the speed of the drive and the recording density. a. In what order should a greedy approach store these files to guarantee minimum average access time? b. Write the algorithm that stores the files, analyze your algorithm, and show the results using order notation.

Use a greedy approach to construct an optimal binary search tree by considering the most probable key, Key \(_{k}\), for the root, and constructing the left and right subtrees for \(K e y_{1}, K e y_{2}, \ldots\) Keyk-1 and Keyk+1, Keyk+2,..., Keyn a. Assuming the keys are already sorted, what is the worst-case time complexity of this apporach? Justify your answer. b. Use an example to show that this greedy approach does not always find an optimal binary search tree.
See all solutions

Recommended explanations on Computer Science Textbooks

Computer Systems

Read Explanation

Computer Network

Read Explanation

Issues in Computer Science

Read Explanation

Computer Organisation and Architecture

Read Explanation

Problem Solving Techniques

Read Explanation

Theory of Computation

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free