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CS 345 Data Mining Online algorithms Search advertising
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Online algorithms box3 Classic model of algorithms square6 You get to see the entire input, then compute some function of it square6 In this context, “offline algorithm” box3 Online algorithm square6 You get to see the input one piece at a time, and need to make irrevocable decisions along the way box3 How is this different from the data stream model?
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Example: Bipartite matching 1 2 3 4 a b c d Girls Boys
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Example: Bipartite matching 1 2 3 4 a b c d M = {(1,a),(2,b),(3,d)} is a matching Cardinality of matching = |M| = 3 Girls Boys
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Example: Bipartite matching 1 2 3 4 a b c d Girls Boys M = {(1,c),(2,b),(3,d),(4,a)} is a perfect matching
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Matching Algorithm box3 Problem: Find a maximum-cardinality matching square6 A perfect one if it exists box3 There is a polynomial-time offline algorithm (Hopcroft and Karp 1973) box3 But what if we don’t have the entire graph upfront?
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Online problem box3 Initially, we are given the set Boys box3 In each round, one girl’s choices are revealed box3 At that time, we have to decide to either: square6 Pair the girl with a boy square6 Don’t pair the girl with any boy box3 Example of application: assigning tasks to servers
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Online problem 1 2 3 4 a b c d (1,a) (2,b) (3,d)
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Greedy algorithm box3 Pair the new girl with any eligible boy square6 If there is none, don’t pair girl box3 How good is the algorithm?
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Competitive Ratio box3 For input I, suppose greedy produces matching M greedy while an optimal matching is M opt Competitive ratio = min all possible inputs I (|M greedy |/|M opt |)
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Analyzing the greedy algorithm box3 Consider the set G of girls matched in M opt but not in M greedy box3 Then it must be the case that every boy
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