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

Online Algorithms Definition: An algorithm that must process each input in turn, without detailed knowledge of future inputs. Examples: Ski Rental, List Update

Example Online Problem: Ski Rental Problem Scenario I am a skier. Each day, I have to either rent a pair of skis for $1 per day, or buy them for $T. BUT I don’t know when the ski season will end. What should I do?????

Strategies for Ski Rental Problem First Strategy: I will buy the skis on the first day. Second Strategy: I will keep renting the skis. What should I do??????

Wait! Why did I talk about paying a number of times more than I should have? Why not some amount of dollars more than I should have?

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.

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

Back to Ski Rental Problem Consider this algorithm: I’ll rent the skis for T days, and buy them on the T+1 st day. Claim: The competitive ratio of this algorithm is 2.

List Update Problem Self-organizing sequential search Unsorted list Received a sequence of requests Cost of accessing the i th element of the list is i. ywzxvu L:

List Update Problem (Cont.) Must access the requested item. We can reorganize the list. –We can move the requested element to any position closer to the front of the list for FREE. –Any other transposition costs 1.

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.

FREQUENCY-COUNT (FC) Maintain a frequency counter for each item, and keep the list in nonincreasing order of their frequencies. FC has a bad competitive ratio = Ω(n), where n is the length of the list. FC does well in typical cases.

Average Case Analysis Assume each request comes from a fixed probability distribution, independent of previous requests. Suppose the i th item has probability p i. Design algorithms to minimize the expected cost. Optimal strategy is to keep the list sorted in nonincresing order of p i. Fact: FC converges to optimal solution.

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?

Previous Work Diffuse Adversary –Beyond Competitive Analysis [Koutsoupias, Papadimitriou, 2000] Statistical Adversary –A Statistical Adversary for On-line Algorithms [Raghavan, 1991]

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.

Current Difficulties Cannot settle on a proper OPT to evaluate performance under this model. –Finite vs. infinite horizon. –Competitive ratio with optimal offline adversary: reduces to online case –Comparison with static list is inappropriate

Lead-Time Quotation Problem Scheduling Problem Each job takes 1 day to process. Everyday, I get some number of jobs. For each job j, I have to accept with a deadline (d j ), or reject it right away. Profit = l – d j. Goal: Maximize the total profit

Lead-Time Quotation Problem Competitive Ratio of Q-FRAC algorithm = [Keskinocak, Ravi, Tayur, 2001] We are working on –Average Case Analysis –Hybrid Model

Our goals Investigate the OPT for our List Update Hybrid Model. Average Case Analysis and Hybrid Model for Lead-time Quotation.

Questions?

Thank you !!!