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Online Algorithms

Online Algorithms. Amrinder Arora. Permalink: http://standardwisdom.com/softwarejournal/presentations/. Ski Rental Problem. Classic toy problem of rent/buy nature Rental cost = 1$, Buying cost = $k Algorithm Rent for k-1 days Buy skis on day #k

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Online Algorithms

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  1. Online Algorithms Amrinder Arora Permalink: http://standardwisdom.com/softwarejournal/presentations/

  2. Ski Rental Problem • Classic toy problem of rent/buy nature • Rental cost = 1$, Buying cost = $k • Algorithm • Rent for k-1 days • Buy skis on day #k • If you stop skiing during the first (k-1) days, it costs the same • If you stop skiing after day #k, your cost is $(2k-1) which is (2-1/k) times more than best possible ($k) • This is the best algorithm. (But we will look for a better one shortly.) http://en.wikipedia.org/wiki/Ski_rental_problem

  3. Agenda • What are online algorithms? (And why?) • How to analyze them? • Some example problems and algorithms • Categories of online algorithms

  4. What does “Online” mean? • “Online” means - Input arrives a little at a time, need instant response • E.g., stock market, paging • Question: what is a “good” algorithm?

  5. Applications • Variety of Applications • Scheduling • Memory Management • Routing • Sample Problems • Ski Rental • Job Scheduling • Paging • Ambulances • Graph Problems

  6. Analyzing Online Algorithms • Competitive Analysis • Probabilistic Analysis • Other/newer methods

  7. Competitive Analysis For any input, the cost of our online algorithm is never worse than c times the cost of the optimal offline algorithm.

  8. Probabilistic Analysis • Assume a distribution generating the input • Find an algorithm which minimizes the expected cost of the algorithm.

  9. Competitive Analysis • Input sequence: • Full knowledge optimum (also called offline optimum): • k-competitive if for allinput sequences : • Sometimes, to ignore boundary conditions

  10. 1966 – Online Era Begins! • We are given a set of m identical machines. • A sequence of jobs arrives online. • Each job must be scheduled immediately on one of m machines without knowledge of any future jobs. • The goal is to minimize the completion time of the last job • The must basic scheduling problem was introduced in 1966

  11. List Scheduling • Schedule the job on the least loaded machine. • Graham showed that the List Scheduling Algorithm is (2-1/m) competitive • This analysis is tight. • This is the optimal algorithm for m = 2 and m = 3.

  12. Fancier Results • In 1992 Bartalgave an 1.986-competitive "New Algorithms for an Ancient Scheduling Problem" • Karger in 1996 generalized the algorithm and proved an upper bound of 1.945 "A Better Algorithm for an Ancient Scheduling Problem" • The best algorithm known so far achieved a 1.923 competitive ration • Lower Bounds • Susanne Albers proved a bound of 1.852 • Current best lower bound of 1.88

  13. Extending Research • Many variants of basic problem is studied: • Jobs may be preempted • Jobs may be rejected at a penalty • Online algorithms may use randomization • In addition they are results for different machine types

  14. Paging Problem • Maintain a two level memory system, consisting of a small fast memory and a large slow memory • The goal is to serve a sequence of requests to memory pages so as to minimize number of page faults. • Hit: Page found

  15. Minimizing Paging Faults • On a fault evict a page from cache • Paging algorithm ≡ Eviction policy • Goal: minimize the number of page faults

  16. Paging Algorithms • LRU, FIFO, LIFO, Least freq use • LIFO, LFU not competititive • LRU, FIFO k-competitive. • Theorem: No deterministic online algorithm can be better than k-competitive

  17. Worst case • In the worst case page, the number of page faults on n requests is n. • E.g. cache of size 4, request sequence • p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12

  18. Compare to optimal p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 … is hard for everyone (i.e. 13 faults) p1 p2 p3 p4 p5 p1 p2 p3 p4 p5 p1 p2 p3 p4 … 8 faults Optimal algorithm knows the future

  19. k-Server problem • Ambulance location problem • Metrical Task Systems • Somewhat, ice cream vending problem

  20. Online Graph Theory • Online Minimum Spanning Tree • Online Graph Coloring • Online Shortest Paths

  21. Randomization • If ski rental costs $1, and buying costs $10, then with probability=0.5, buy after 8 days, and with probability=0.5, buy after 10 days.

  22. Randomization 3 kinds of adversaries • Oblivious (weak): Does not know the randomized results • Adaptive Online (medium): Knows the decision/random output of algorithm, but needs to make their decision first. • Adaptive Offline (strong): Knows everything – randomization does not help.

  23. Summary

  24. Extra Material • Adversary Approach to prove that 3n/2 – 2 comparisons are always needed to find both the minimum and the maximum given an unsorted array of n numbers: • Notes at: http://is.gd/csp1ab

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