CS 170 Discussion 7: Greedy Algorithms and Huffman Encoding, Lecture notes of Algorithms and Programming

Download CS 170 Discussion 7: Greedy Algorithms and Huffman Encoding and more Lecture notes Algorithms and Programming in PDF only on Docsity! CS 170 DISCUSSION 7 GREEDY ALGORITHMS Raymond Chan UC Berkeley Fall 17 Raymond Chan, UC Berkeley Fall 2017 APPROXIMATING SET COVER • See board or notes/video Raymond Chan, UC Berkeley Fall 2017 GREEDY IS AT LEAST BETTER THAN OPTIMAL • Suppose you have an optimal solution S* and greedy solution S. • Look at the first value in which greedy and optimal differ. Usually talk about an ordering. • Prove that at this point, the greedy solution is at least as better as the optimal solution. And by induction, the greedy solution is better at each differing value. • Prove optimality via contradiction. Assume greedy is not optimal. Use above point to derive a contradiction. • Discussion today: Meeting Scheduling Raymond Chan, UC Berkeley Fall 2017 APPROACH 1 VS 2 • Use exchange argument (1) instead of contradictory proofs (2) because there could be multiple optimal solutions. • Both require induction. • Nuanced difference. Exchange argument takes some optimal ordering and swaps elements in the ordering that violate greedy heuristic. We will reach greedy without increasing cost. • Approach 2 compares specific differing values in greedy and optimal solutions. Choosing the greedy value at that point is at least as good as optimal. Raymond Chan, UC Berkeley Fall 2017 HUFFMAN ENCODING • Motivation: Given symbols and their respective probabilities, can we encode the symbols to prevent ambiguity and minimize length of longest encoding? • Prefix-free encoding: no codeword can be a prefix of another codeword. • Use full binary tree. Nodes have 0 or 2 children. • Leaves are symbols. • Internal nodes (v) have 2 children (u, w) with probability (or frequency) equal to sum of children’s probabilities. f(v) = f(u) + f(w)