345 CS Data 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 decisions along the way
Similar to data stream models
Example: Bipartite matching
1 2 3 Girls 4 a b c d Boys
Example: Bipartite matching
1 2 3 Girls 4 a b c d Boys
M = {(1,a),(2,b),(3,d)} is a matching Cardinality of matching = |M| = 3
Example: Bipartite matching
1 2 3 Girls 4 a b c d Boys
M = {(1,c),(2,b),(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, 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 a b c d (1,a) (2,b) (3,d)
Greedy algorithm
Pair the new girl 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 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|/|Mopt| 1/2
Worst-case scenario
1 2 3 4 a b c d (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 clickthrough 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 later adopted by Google with some changes
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 queries 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 to 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 expected revenue per click
Complications (2)
Each advertiser has a 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 eligible an advertiser for 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, Vazirani, 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 =
Analyzing BALANCE (1)
Consider simple case: two advertisers, P and Q, each with budget B (assume B 1) Assume optimal solution exhausts both advertisers budgets BALANCE must exhaust at least one advertisers budget
OPT = 2B
If not, we can allocate more queries Assume BALANCE exhausts Qs budget, but aloocates x queries fewer than the optimal BAL = 2B - x
Analyzing Balance
B Queries allocated to A1 in optimal solution Queries allocated to A2 in optimal solution A1 A2
x y A1 A2 x
B
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
Analyzing BALANCE (2)
Three types of queries: (A) P is the only bidder (B) Q is the only bidder (C) P and Q both bid Since Qs budget is exhausted but Ps is not, and we couldnt allocate x queries, they must be of type C
Analyzing BALANCE (3)
BALANCE allocates at least x Type C queries to Q
In the Optimal, these were assigned to P
Consider the last Type C query assigned to Q
At this point, Qs leftover budget was greater than Ps So Ps allocation was at least x
So we have BAL B + x
Analyzing BALANCE (4)
We now have: BAL = 2B x BAL B + x The minimum value of BAL is obtained when x = B/2 BAL = 3B/2 OPT = 2B So BAL/OPT = 3/4
General Result
In the general case, worst competitive ratio of BALANCE is 11/e = approx. 0.63 Interestingly, no online algorithm has a better competitive ratio Wont go through the details here, but lets see the worst case that gives this ratio
Worst case for BALANCE
N advertisers, each with budget B N 1 NB queries appear in N rounds of B queries each Round 1 queries: bidders A1, A2, , AN Round 2 queries: bidders A2, A3, , AN Round i queries: bidders Ai, , AN Optimum allocation: allocate round i queries to Ai
Optimum revenue NB
BALANCE allocation
A1 A2 A3 AN-1 AN
B/(N-2) B/(N-1) B/N
After k rounds, sum of allocations to each of bins Ak,,AN is Sk = Sk+1 = = SN = 1i kB/(N-i+1) If we find the smallest k such that Sk B, then after k rounds we cannot allocate any queries to any advertiser
BALANCE analysis
B/1 B/2 B/3 B/(N-k+1) B/(N-1)
S2 Sk = B
B/N
S1
1/1
1/2
1/3 1/(N-k+1) 1/(N-1)
S2 Sk = 1
1/N
S1
BALANCE analysis
Fact: Hn = 1 i n1/i = approx. log(n) for large n
Result due to Euler
1/1 1/2 1/3 1/(N-k+1) 1/(N-1)
log(N) log(N)-1 Sk = 1
1/N
Sk = 1 implies HN-k = log(N)-1 = log(N/e) N-k = N/e k = N(1-1/e)
BALANCE analysis
So after the first N(1-1/e) rounds, we cannot allocate a query to any advertiser Revenue = BN(1-1/e) Competitive ratio = 1-1/e
General version of problem
Arbitrary bids, budgets Consider query q, advertiser i
Bid = xi Budget = bi
BALANCE can be terrible
Consider two advertisers A1 and A2 A1: x1 = 1, b1 = 110 A2: x2 = 10, b2 = 100
Generalized BALANCE
Arbitrary bids; consider query q, bidder i
Bid = xi Budget = bi Amount spent so far = mi Fraction of budget left over fi = 1-mi/bi Define i(q) = xi(1-e-fi)
Allocate query q to bidder i with largest value of i(q) Same competitive ratio (1-1/e)