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CS 345 Data Mining Online algorithms Search advertising

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

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Competitive Ratio 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 Consider the set G of girls matched in M opt  but not in  M greedy Then it must be the case that every boy adjacent to  girls in G is already matched in M greedy There must be at least |G| such boys Otherwise the optimal algorithm could not have matched all  the G girls Therefore  |M greedy ¸  |G| = |M opt  - M greedy | |M greedy |/|M opt ¸  1/2

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Worst-case scenario 1 2 3 4 a b c (1,a) (2,b) d

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