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CS 345 Data Mining Online algorithms Search advertising

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Online algorithms b Classic model of algorithms s You get to see the entire input, then compute some function of it s In this context, “offline algorithm” b Online algorithm s You get to see the input one piece at a time, and need to make irrevocable decisions along the way b How is this different from the data stream model?
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
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 b Problem: Find a maximum-cardinality matching s A perfect one if it exists b There is a polynomial-time offline algorithm (Hopcroft and Karp 1973) b But what if we don’t have the entire graph upfront?
Online problem b Initially, we are given the set Boys b In each round, one girl’s choices are revealed b At that time, we have to decide to either: s Pair the girl with a boy s Don’t pair the girl with any boy b 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)
Greedy algorithm b Pair the new girl with any eligible boy s If there is none, don’t pair girl b How good is the algorithm?

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Competitive Ratio b 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 |)
Analyzing the greedy algorithm b Consider the set G of girls matched in M

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## This note was uploaded on 01/31/2011 for the course CS 345 taught by Professor Dunbar,a during the Fall '07 term at UC Davis.

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