Online Algorithms. - Lecture 4 -

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Ferdinand Strickland
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1 Online Algorithms - Lecture 4 -

2 Outline Quick recap.. The Cashing Problem Randomization in Online Algorithms Other views to Online Algorithms The Ski-rental problem The Parking Permit Problem 2

3 The Caching Problem Cache x x A request for a page x arrives If x is in cache we re happy! If x is in main memory need to bring x from main memory to cache Cache Miss Goal: serve all requests by evicting pages smartly from the cache so that the number of cache misses is minimum 3

4 An Algorithm for the Caching Problem Cache has capacity of k pages and main memory has capacity of N pages. Each page can be requested more than once. Algorithm: Remove the page which was used farthest in the past (called Least Recently Used or LRU algorithm) Competitive ratio: k 4

5 Analysis of LRU Proof: divide the request sequence into M rounds, such that a round contains a maximal sequence of exactly k requests. Ex: Let k=3. Suppose the request sequence is [1,2,1,3,3,2,1,4]. Then a round is: [1,2,1,3,3,2,1] 5

6 Analysis of LRU Cost of ALG: At most k cache misses can happen in each round cost of ALG M. k Cost of OPT: Since the cache has only k slots, OPT must have at least one cache miss after serving k distinct pages without cache misses cost of OPT M Competitive ratio: k Can we do better? 6

7 A lower bound for the Caching Problem Best competitive ratio any online deterministic algorithm can get is k Proof: Let N = k+1 The adversary always requests the page that is not in the algorithm cache. Then, the algorithm has at least k+1 cache misses (all requests are missed) OPT is to evict the page that is farthest into the future and so has cost 1 Competitive ratio is at least k 7

8 Randomization in Online Algorithms Algorithm can make random choices Three types of adversaries Oblivious Adaptive Online Adaptive Offline Adversary generates the sequence before any request is served by Alg Adversary generates according to Alg; must construct a solution online Adversary generates according to Alg; constructs a solution offline The competitive ratio in terms of the expected cost of the algorithm 8

9 Other views to Online Algorithms Resource Augmentation: more resources to the algorithm than OPT, e.g., more cache capacity Lookahead: algorithm knows finite number of future requests Max/Max ratio: algorithm s worst cost for any sequence of length n compared to OPT s worst cost for any sequence of length n 9

10 Other views to Online Algorithms Diffuse Adversary: Input sequence restricted to some probability distribution Online algorithms with advice: how much knowledge of the future is necessary to achieve a given competitive ratio 10

11 The Ski-rental Problem Suppose you need skis to go skiing in the winter. Every day, you have two options Either rent the skis for $1 or pay $B to buy the skis and use it forever You do not know in advance when will the season end Will you buy or rent the skis? 11

12 The Ski-rental Algorithm Online Algorithm: Buy the skis on day (B + 1) Competitive ratio: 2 Proof: Let x be the length of the skiing season. If x B, then both the online algorithm and the optimal offline algorithm pay the same: they both rent for x days; the competitive ratio will be: x/x = 1 If x > B, the optimal offline algorithm buys the skis on day 1 already; the competitive ratio will be: (B + B) / B = 2 This algorithm is optimal. Can we do better? 12

13 A lower bound for the Ski-rental Problem Best competitive ratio any online deterministic algorithm can get is 2 Proof: Consider an adversary that gives an instance so that the online algorithm gets the highest competitive ratio possible As soon as the online algorithm buys the skis, the adversary stops the season. Let s say the algorithm buys on day x+1. the algorithm pays x + B and the optimal pays at most B the competitive ratio will be (x + B) / B = 2 (minimum is attained when x = B) 13

14 Online Leasing Subcontracting Company 14

15 Parking Permit Problem sunny day walk rainy day drive 15

16 Parking Permit Problem Provide each rainy day with a valid permit while minimizing total cost K permit types 1 day permit 1 week permit 1 month permit 1 year permit A February permit can not be used in March 16

17 Bounds for the Parking Permit Problem Deterministic Θ K - competitive Randomized Θ log K - competitive 17

18 Simplified Parking Permit Problem Leases of same length do no overlap and all lease lengths are powers of two Constant factor loss in competitiveness how? 18

19 Online Algorithms - Lecture 4 -

Primality Testing. Public-Key Cryptography needs large prime numbers How can you tell if p is prime? Try dividing p by all smaller integers

Primality Testing. Public-Key Cryptography needs large prime numbers How can you tell if p is prime? Try dividing p by all smaller integers Primality Testing Public-Key Cryptography needs large prime numbers How can you tell if p is prime? Try dividing p by all smaller integers Exponential in p (number of bits to represent p) Improvement: