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Online algorithms

叶德仕yedeshi@zju.edu.cn

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Online & offline

Traditional theoretical analysis is concerned with off-line problems where the complete input is given and we look for a time-efficient algorithm. In on-line problems the input is not known in advance but instead it is revealed during the operation of the algorithm.

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Simple Example(Ski-Rental Problem) Karp92

Rental: $100/tourPurchase: $1,000 (can be used forever)Question: Buy or Rental?Inputs: 11111….. (1: new ski tour)You do not know how many more 1’s are coming.You have to decide buy or rental at each input 1.

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Three Online Algorithms

1. Always rental.2. But at the first 1.3. Buy at the 10th 1 (rental before

then)Your action must be decided at each1 (tour) uniquely by the algorithm

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CR (Competitive Ratio)

1001000nr

Cost of algorithm Cost of optimal offline algorithm

AlgOpt

Alg r Opt c

r

Competitive Analysis

Rental:$100Buy: $1,000

1. Always rental:

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CR (Competitive Ratio)

Cost of algorithm Cost of optimal offline algorithm

AlgOpt

Alg r Opt c

r

Competitive Analysis

Rental:$100Buy: $1,000

1000 10.0100

r 2. Buy at the beginning

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CR (Competitive Ratio)

Cost of algorithm Cost of optimal offline algorithm

AlgOpt

rAlg Opt c

r

Competitive Analysis

Rental:$100Buy: $1,000

100 9 1000 1.91000

r 3. Buy before

the 10th tour

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General lower bound

Thm: There is no online algorithm can achieve competitive ratio less than 2 -1/B.Proof. The adversary decides the duration T.

1) If buy before B - 1 ， stops T.2) If never buy, increase T3) Once buy B between B – 1 and 2B, stops T.

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Ski rental problem

A skier must decide every day she goes skiing whether to rent or buy skis, unless or until she decides to buy them. The cost to rent skis for a day is 1 unit, while the cost to buy skis is B units. The skiier doesn’t know how many days she will go on skiing before she gets tired of it (or breaks a leg). Call this (unknown) number of days T.

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Analysis

Claim: The competitive ratio 2 − 1/B.

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Randomized Ski rental

Suppose that the cost B of buying skis is 3 units.Suppose that our skier buys the skis on Day 1,2,3 with probabilities p1, p2, p3 respectively, where p1 + p2 + p3 = 1. If the adversary decides that the skier will give up skiing after T days The expected total cost will be:

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Randomized Ski rental

The expected competitive ratio is 27/19 = 1.42105Proof. Let p1 = 4/19, p2 = 6/19, p3 = 9/19