Evaluation of Algorithms for the List Update Problem

1. Evaluation of Algorithms for the List Update Problem Suporn Pongnumkul R. Ravi Kedar Dhamdhere

2. Overview • List Update Problem • Competitive Analysis • Average Case Analysis • Our Hybrid Model • Setup of our experiment • Results from our experiment • Conjecture

3. L: y w z x v u List Update Problem • Self-organizing sequential search • Unsorted list • Received a sequence of requests • Cost of accessing the ith element of the list is i.

4. L: y w z x v u List Update Example

5. L: y w z x v u List Update Example

6. L: y w z x v u List Update Example

7. L: y w z x v u List Update Example

8. L: y w z x v u List Update Example

9. L: y w z x v u List Update Example

10. L: y v w z x u List Update Example

11. L: y v w z x u List Update Example

12. L: y v w z x u List Update Example

13. L: y v w z x u List Update Example

14. L: y v w z x u List Update Example

15. L: y v z w x u List Update Example

16. L: y v w z x u List Update Example

17. Competitive Analysis • Definition: An analysis in which the performance of an online algorithm is compared to the best that could have been achieved if all the inputs had been known in advance.

18. A: Our online algorithm CA() OPT: Optimal Offline algorithm COPT() A is c-competitive if a CA() ≤ c COPT() + a for all request sequences  Competitive Ratio

19. Move-to-Front (MTF) • When an element is accessed, move it to the front of the list. • Theorem: [Sleator, Tarjan, 1985] MTF has competitive ratio 2 against optimal offline algorithm.

20. Average Case Analysis • Assume each request comes from a fixed probability distribution, independent of previous requests. Suppose the ith item has probability pi. Design algorithms to minimize the expected cost. • Optimal strategy is to keep the list sorted in non-increasing order of pi.

21. STAT = Static List • List is sorted in non-increasing order of the probabilities • Never moves anything • Good for when we have a good estimate of the probability distribution.

22. Need a new model? • Most real-world settings don't behave either like a discrete distribution, or like a worst-case one. • Can we design an algorithm that does well in both typical and worst-case? • How could we analyze such algorithms?

23. Hybrid Model • Assume a fixed probability distribution, • For each request, with probability , let the adversary change the request. •  Average Case Analysis •  Competitive Analysis •  Known probability distribution with uncertainty.

24. Hybrid Algorithm? • Parameterized by  • Matches best average case performance when  is low, and matches best competitive ratio when  is high.

25. Move-From-Back-Epsilon • List initially sorted in non-increasing order of probabilities. • When an element x is accessed, promote it past others that have probability up to px + .

26. L: y w z x v u MFBE Example ( = 0.2) Prob py = 0.5 pw = 0.2 pz = 0.15 px = 0.1 pv = 0.03 pu = 0.02 :v y z y

27. L: y w z x v u MFBE Example ( = 0.2) Prob py = 0.5 pw = 0.2 pz = 0.15 px = 0.1pv = 0.03 pu = 0.02 :x v y v

28. L: y w z x v u MFBE Example ( = 0.2) Prob py = 0.5 pw = 0.2 pz = 0.15 px = 0.1pv = 0.03 pu = 0.02 :x v y v

29. L: y x w z v u MFBE Example ( = 0.2) Prob py = 0.5 px = 0.1pw = 0.2 pz = 0.15 pv = 0.03 pu = 0.02 :x v y v

30. L: y x w z v u MFBE Example ( = 0.2) Prob py = 0.5 px = 0.1 pw = 0.2 pz = 0.15 pv = 0.03pu = 0.02 :x v y v

31. L: y x w z v u MFBE Example ( = 0.2) Prob py = 0.5 px = 0.1 pw = 0.2 pz = 0.15 pv = 0.03pu = 0.02 :x v y v

32. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5 pv = 0.03px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :x v y v

33. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xv y v

34. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xv y v

35. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xv y v

36. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5 pv = 0.03px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xvy v

37. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5 pv = 0.03px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xvy v

38. L: y v x w z u MFBE Example ( = 0.2) Prob py = 0.5 pv = 0.03px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02 :xvy v

39. Difficulties in Proofs • It’s not easy to understand the behavior of OPT. • OPT can be computed by Dynamic programming • Trivial way = O((n!)2m) • Improvement = O((2n)(n!)m) [Reingold, Westbrook, 1996]

40. Our Experiment • Motivation: To see the behavior of algorithms in our hybrid model. • Measurement: We measure the performance of an online algorithm by the average competitive ratio.

41. Our Experiment • Variables in our experiment • Type of List Update Algorithm (MTF, STAT, MFBE) • Type of Probability Distribution • Type of Adversary • Epsilon: 

42. Our Experiment • We generate a request sequence of length 100, with a chosen probability distribution. • Then, with probability  let the adversary change the request sequence.

43. Our Experiment • Record the cost incurred by the online algorithm = CostA() • Use Dynamic Programming to find optimum cost of that request sequence = CostOPT(). • Competitive Ratio = CostA()/CostOPT() • Repeat this 100 times to find the average competitive ratio.

44. Distribution • Geometric Distribution: • P[i] / 1/2i • Uniform Distribution: • P[i] = 1/n, n = length of the list • Zipfian Distribution (Zipf(2)): • P[i] / 1/i2

45. L: y w z x v u Cruel Adversary • This is an adaptive adversary • Looks at the current list and request for the last item in the list.

46. Cruel Adversary, Geometric Distribution, n=6

47. Cruel Adversary, Geometric Distribution, n=6

48. Reversed Geometric Adversary • This adversary chooses elements randomly, according to the geometric distribution on the reversed STAT order. • This adversary requests small probability items more frequently. • Oblivious Adversary = doesn’t look at the current list

49. Reversed Geometric Adversary, Geometric Distribution, n=6

50. Uniform Adversary • This adversary requests elements from the list uniformly at random. • Oblivious Adversary • With this adversary, the sorted order of elements in the combined probability distribution doesn’t change.