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Lecture topic-BruteForce

Maximum design and analysis of algorithms chapter 3

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Unformatted text preview: ck problem a Given a knapsack with maximum capacity W, and Given and a set S consisting of n items a Each item i has some weight wi and benefit value vi Each a Problem: How to pack the knapsack to achieve How maximum total value of packed items? maximum Design and Analysis of Algorithms – Chapter 3 51 0-1 Knapsack problem: a picture Weight vi 2 3 3 4 4 5 8 9 This is a knapsack Max weight: W = 20 wi 5 Items W = 20 Benefit value 10 Design and Analysis of Algorithms – Chapter 3 52 0-1 Knapsack problem a Problem, in other words, is to find max ∑ vi subject to i∈T a a ∑w ≤W i∈T i The problem is called a “0-1” problem, problem, because each item must be entirely accepted or rejected. rejected. In the “Fractional Knapsack Problem,” we can ,” take fractions of items. Design and Analysis of Algorithms – Chapter 3 53 0-1 Knapsack problem: brute-force 0-1 approach approach Let’s first solve this problem with a Let’s straightforward algorithm straightforward a We go through all combinations (subsets) and We find the one with maximum value and with total weight less or equal to W Design and Analysis of Algorithms – Chapter 3 54 Example 2: Knapsack Problem Given n items: Given • weights: w1 w2 … wn weights: • values: v1 v2 … vn values: • a knapsack of capacity W knapsac...
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