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1、CS 345Data Mining,Online algorithms Search advertising,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 deci

2、sions along the way Similar to data stream models,Example: Bipartite matching,1,2,3,4,a,b,c,d,Girls,Boys,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、,(3,d),(4,a) is a perfect matching,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 dont have the entire graph upfront?,Online problem,Initially,

4、 we are given the set Boys In each round, one girls choices are revealed At that time, we have to decide to either: Pair the girl with a boy Dont pair the girl with any boy Example of application: assigning tasks to servers,Online problem,1,2,3,4,(1,a),(2,b),(3,d),Greedy algorithm,Pair the new girl

5、with any eligible boy If there is none, dont pair girl How good is the algorithm?,Competitive Ratio,For input I, suppose greedy produces matching Mgreedy while an optimal matching is Mopt Competitive ratio = minall possible inputs I (|Mgreedy|/|Mopt|),Analyzing the greedy algorithm,Consider the set

6、G of girls matched in Mopt but not in Mgreedy Then it must be the case that every boy adjacent to girls in G is already matched in Mgreedy There must be at least |G| such boys Otherwise the optimal algorithm could not have matched all the G girls Therefore |Mgreedy| |G| = |Mopt - Mgreedy| |Mgreedy|/

7、|Mopt| 1/2,Worst-case scenario,1,2,3,4,(1,a),(2,b),History of web advertising,Banner ads (1995-2001) Initial form of web advertising Popular websites charged X$ for every 1000 “impressions” of ad Called “CPM” rate Modeled similar to TV, magazine ads Untargeted to demographically tageted Low clickthr

8、ough rates low ROI for advertisers,Performance-based advertising,Introduced by Overture around 2000 Advertisers “bid” on search keywords When someone searches for that keyword, the highest bidders ad is shown Advertiser is charged only if the ad is clicked on Similar model adopted by Google with som

9、e changes around 2002 Called “Adwords”,Ads vs. search results,Web 2.0,Performance-based advertising works! Multi-billion-dollar industry Interesting problems What ads to show for a search? If Im an advertiser, which search terms should I bid on and how much to bid?,Adwords problem,A stream of querie

10、s arrives at the search engine q1, q2, Several advertisers bid on each query When query qi arrives, search engine must pick a subset of advertisers whose ads are shown Goal: maximize search engines revenues Clearly we need an online algorithm!,Greedy algorithm,Simplest algorithm is greedy Its easy t

11、o see that the greedy algorithm is actually optimal!,Complications (1),Each ad has a different likelihood of being clicked Advertiser 1 bids $2, click probability = 0.1 Advertiser 2 bids $1, click probability = 0.5 Clickthrough rate measured historically Simple solution Instead of raw bids, use the

12、“expected revenue per click”,The Adwords Innovation,Advertiser,Bid,CTR,Bid * CTR,A,B,C,$1.00,$0.75,$0.50,1%,2%,2.5%,1 cent,1.5 cents,1.125 cents,The Adwords Innovation,Advertiser,Bid,CTR,Bid * CTR,A,B,C,$1.00,$0.75,$0.50,1%,2%,2.5%,1 cent,1.5 cents,1.125 cents,Complications (2),Each advertiser has a

13、 limited budget Search engine guarantees that the advertiser will not be charged more than their daily budget,Simplified model (for now),Assume all bids are 0 or 1 Each advertiser has the same budget B One advertiser per query Lets try the greedy algorithm Arbitrarily pick an eligible advertiser for

14、 each keyword,Bad scenario for greedy,Two advertisers A and B A bids on query x, B bids on x and y Both have budgets of $4 Query stream: xxxxyyyy Worst case greedy choice: BBBB_ Optimal: AAAABBBB Competitive ratio = Simple analysis shows this is the worst case,BALANCE algorithm MSVV,Mehta, Saberi, V

15、azirani, and Vazirani For each query, pick the advertiser with the largest unspent budget Break ties arbitrarily,Example: BALANCE,Two advertisers A and B A bids on query x, B bids on x and y Both have budgets of $4 Query stream: xxxxyyyy BALANCE choice: ABABBB_ Optimal: AAAABBBB Competitive ratio =

16、,Analyzing BALANCE,Consider simple case: two advertisers, A1 and A2, each with budget B (assume B 1) Assume optimal solution exhausts both advertisers budgets BALANCE must exhaust at least one advertisers budget If not, we can allocate more queries Assume BALANCE exhausts A2s budget,Analyzing Balance,Opt revenue = 2B Balance revenue = 2B-x = B+y,We have y x Balance revenue is minimum for x=y=B/2 Minimum Balance revenue = 3B/2 Competitive Ratio = 3/4,Queries allocated to A1 in optimal solution,Qu