Outline Knapsack revisited How to output the optimal solution and how to prove correctness Longest Common Subsequence Maximum Independent Set on Trees Example 2 Knapsack Problem There is a knapsack that can hold items of total weight at most ID: 1027578
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1. Lecture 5 Dynamic Programming
2. OutlineKnapsack revisited: How to output the optimal solution, and how to prove correctness?Longest Common SubsequenceMaximum Independent Set on Trees.
3. Example 2 Knapsack ProblemThere is a knapsack that can hold items of total weight at most W. There are now n items with weights w1,w2,…, wn. Each item also has a value v1,v2,…,vn.Goal: Select some items to put into knapsack1. Total weight is at most W.2. Total value is as large as possible.Output: the set of items to put into the Knapsack
4. Recall: States and Transition functionExample: Capacity W = 4, 3 items with (weight, value) = (1, 2), (2, 3), (3, 4). States (sub-problems): What is the maximum value for a Knapsack with capacity j (= 0, 1, 2, 3, 4) and the first i (= 0, 1, 2, 3) items?Use a[i, j] to denote this maximum value, we have a[i, j] = a[i - 1, j - wi] + vi (item i in knapsack) a[i - 1, j] (item i not in knapsack) max
5. Dynamic Programming TableExample: Capacity W = 4, 3 items with (weight, value) = (1, 2), (2, 3), (3, 4). 01234000000102222202355302356+4
6. Outputting the SolutionRemember the choice (arrows) in the DP table.Solution = {1, 3}, value = 601234000000102222202355302356
7. Outputting the SolutionAnother Example (capacity = 3)Solution = {1, 2}, value = 501234000000102222202355302356
8. Pseudo-codeKnapsackInitialize a[i, 0] = 0, a[0, j] = 0 (Base Case)FOR i = 1 to n (Enumerate #items) FOR j = 1 to W (Enumerate capacity) a[i, j] = a[i-1, j] (Case 1: not using item i) IF j >= w[i] and a[i-1, j-w[i]] + v[i] > a[i,j] (if Case 2 is better) a[i, j] = a[i-1, j-w[i]] + v[i] (Case 2: using item i)Output(n, W)IF n = 0 or W = 0 THEN RETURNIF a[n, W] = a[n-1, W] THEN (Case 1) Output(n-1, W)ELSE (Case 2) Output(n-1, W-w[n]) Print(n)
9. Longest Common SubsequenceInput: two strings a[] = ‘ababcde’ and b[] = ‘abbecd’Subsequence: same definition as in LIS (can skip characters)E.g. ‘abac’ is a subsequence of a[], but not b[]‘abed’ is a subsequence of b[] but not a[]Goal: Find the length of the longest common subsequence (LCS)In this example: LCS = ‘abbcd’, length = 5.
10. Max Independent Set on TreesInput: a treeIndependent Set: Set of nodes that are not connected by any edgesGoal: Find an independent set of maximum size.
11. Max Independent Set on TreesInput: a treeIndependent Set: Set of nodes that are not connected by any edgesGoal: Find an independent set of maximum size.