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Knapsack problem using fixed tuple
This document discusses the 0/1 knapsack problem and how it can be solved using backtracking. It begins with an introduction to backtracking and the difference between backtracking and branch and bound. It then discusses the knapsack problem, giving the definitions of the profit vector, weight vector, and knapsack capacity. It explains how the problem is to find the combination of items that achieves the maximum total value without exceeding the knapsack capacity. The document constructs state space trees to demonstrate solving the knapsack problem using backtracking and fixed tuples. It concludes with examples problems and references.
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Introduction to backtracking and its application to solve the 0/1 knapsack problem through generating feasible solutions.
Comparison of Backtracking and Branch & Bound methodologies regarding approach, application, and efficiency.
Advantages of backtracking include simplicity, efficient state management, and intuitive trial-error approach.
Disadvantages include high costs in function calls, inefficiency in branched states, and large space requirements.
Various applications of backtracking, such as N Queen Problem and 0/1 Knapsack Problem.
Details of the knapsack problem including formulation, combinations of items, and computational running time.
Applications in fields like cryptography and mathematics, along with advantages and disadvantages of solutions.
Examples illustrating the knapsack problem with solutions using a fixed and variable length tuple approach.
A practice question on the knapsack problem to apply learned concepts.
References used for the presentation content including online resources and literature.
Closing remarks and acknowledgments from the presentation.
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Knapsack problem using fixed tuple
- 1. LOGO 11 0/1 Knapsack Problem0/1Knapsack Problem (fixed tuple) using(fixed tuple) using BACKTRACKINGBACKTRACKING Submitted to: Submitted by:Submitted to: Submitted by: Mrs. Deepti Shrimali Krati KatyalMrs. Deepti Shrimali Krati Katyal MCA 3rd SemMCA 3rd Sem
- 2. Topics Covered âą Introduction âąDifference between Backtracking and Branch & Bound âą Advantages & Disadvantages âą Application âą Knapsack Problem âą Application âą Advantages & Disadvantages âą Problem âą Reference
- 3. IntroductionIntroduction In this seminarwe will learn about an optimization technique which is known as backtracking and solve the 0/1 knapsack problem using fixed tuple. Backtracking is a recursive method for building up feasible solutions one at a time. It provides a simple recursive method of generating all possible n-tuples. After each n-tuple is generated, it is checked for feasibility. If feasible, the solution incurred is compared to the current optimal solution
- 4. Difference Between BacktrackingDifferenceBetween Backtracking & Branch and Bound& Branch and BoundBACKTRACKING BRANCH & BOUND It is used to find all possible solutions available to the problem. It is used to solve optimization problem. It traverse tree by DFS (Depth First Search). It may traverse the tree in any manner, DFS or BFS. It realizes that it has made a bad choice & undoes the last choice by backing up. It realizes that it already has a better optimal solution that the pre- solution leads to so it abandons that pre- solution. It search the state space tree until it found a solution. It completely searches the state space tree to get optimal solution. It involves feasibility function It involves bounding function
- 5. Advantages: 1) Simple toimplement. 2) State changes are stored in stack, meaning we do not need to concern ourselves about them. 3) Intuitive approach of trial and error. 4) Code size is usually small. Advantages & Disadvantages of Backtracking
- 6. Disadvantages: 1) Multiple functioncalls are expensive. 2) Inefficient when there is lots of branching from one state. 3) Requires large amount of space as the each function state needs to be stored on system stack.
- 7. ApplicationApplication This approach isused for a number of problems like: âąN Queen Problem âąSorting âąSum Of Subset Problem âą0/1 Knapsack Problem âąSolving a Maze âąColoring a Map âąSolving a Puzzel
- 8. Knapsack ProblemKnapsack Problem Aknapsack problem consist of profit vector P = (p1,p2,p3, âŠ.,pn), a weight vector W = (w1,w2,w3,âŠ.wn) a capacity C of the Knapsack. Problem: How to pack the knapsack to achieve maximum total value of packed items? Problem, in other words, is to find ââ ââ †Ti i Ti i Wwb subject tomax
- 9. âą This problemis called a â0-1â problem, because each item must be entirely accepted or rejected. âą Since there are n items, there are 2n possible combinations of items. âą We go through all combinations and find the one with maximum value and with total weight less or equal to W. âą Running time will be O(2n ). âą This problem is sove by state space diagram.
- 10. State space tree: Leftchild is always xi= 1 in our formulation Right child is always xi= 0.
- 11. âą Cryptography âą AppliedMathematics âą Complexity Theory âą Computer science Application of Knapsack ProblemApplication of Knapsack Problem
- 12. Advantage: ⹠Will findan optimal solution if an optimal solution exists. Disadvantage: ⹠Computationally very demanding: O ( n · 2 n ). ⹠Very memory intensive. Number of nodes and links both grow with 2 n, node label grows with n.
- 13. Q1:Q1: In thisproblem n=4, profit P = (2,1,4,3) , weight of knapsack W = (5,3,7,6) and capacity of knapsack is C=11. SOL: Search tree of fixed length tuple approach is: ProblemsProblems The solution state are 17 and 19 representing the solution (1,0,0,1) and (0,1,1,0) respectively with the same profit 5
- 14. Search tree ofvariable length tuple approach is: The solution state are 8 and 9 representing the solution (1,4) and (2,3) respectively with the same profit 5.
- 15. Q2:Q2: In thisproblem n=4, profit P = (3,5,6,10) , weight of knapsack W = (2,3,4,5) and capacity of knapsack is C=8. Practice QuestionPractice Question
- 16. ReferenceReference âą www.google.comwww.google.com âą www.encyclopedia.comwww.encyclopedia.com âąwww.wikipedia.comwww.wikipedia.com âą Design Method and Analysis of Algorithm By S.K Basu.
- 17.