Amortized efficiency of list update and paging rules
Communications of the ACM
Competitive algorithms for on-line problems
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
An optimal on-line algorithm for metrical task system
Journal of the ACM (JACM)
Competitive paging with locality of reference
Selected papers of the 23rd annual ACM symposium on Theory of computing
Randomized algorithms
A competitive analysis of the list update problem with lookahead
Theoretical Computer Science
Online computation and competitive analysis
Online computation and competitive analysis
Speed is as powerful as clairvoyance
Journal of the ACM (JACM)
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
The Online TSP Against Fair Adversaries
INFORMS Journal on Computing
Average Case and Smoothed Competitive Analysis of the Multi-Level Feedback Algorithm
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Smoothed analysis of binary search trees
Theoretical Computer Science
A regularization approach to metrical task systems
ALT'10 Proceedings of the 21st international conference on Algorithmic learning theory
Smoothed performance guarantees for local search
ESA'11 Proceedings of the 19th European conference on Algorithms
Price fluctuations: to buy or to rent
WAOA'09 Proceedings of the 7th international conference on Approximation and Online Algorithms
Hi-index | 5.23 |
Borodin et al. (J. ACM 39 (1992) 745) introduced metrical task systems, a framework to model a large class of online problems. Metrical task systems can be described as follows. We are given a graph G = (V, E) with n nodes and a positive edge length λ(e) for every edge e ∈ E. An online algorithm resides in G and has to service a sequence of tasks that arrive online. A task τ specifies for each node v; ∈ V a request cost r (v) ∈ R0+ ∪ {∞}. If the algorithm resides in node u before the arrival of task τ, the cost to service task τ in node v is equal to the shortest path distance from u to v plus the request cost r(v). The objective is to service all tasks at minimum total cost. Borodin et al. showed that every deterministic online algorithm has a competitive ratio of at least 2n - 1, independent of the underlying metric. Moreover, they presented an online work function algorithm (WFA) that achieves this competitive ratio.We present a smoothed competitive analysis of WFA. That is, given an adversarial task sequence, we randomly perturb the request costs and analyze the competitive ratio of WFA on the perturbed sequence. Here, we are mainly interested in the asymptotic behavior of WFA. Our analysis reveals that the smoothed competitive ratio of WFA is much better than O(n) and that it depends on several topological parameters of the underlying graph G, such as the minimum edge length λmin, the maximum degree Δ, the edge diameter emax, etc. For example, if the ratio between the maximum and the minimum edge length of G is bounded by a constant, the smoothed competitive ratio of WFA is O(emax (λmin/σ + log(Δ))) and O(√nċ (λmin/sigma + log(Δ))), where σ denotes the standard deviation of the smoothing distribution. That is, already for perturbations with σ= Θ(λmin) the competitive ratio reduces to O(log(n)) on a clique and to O(√/n) on a line. Furthermore, we provide lower bounds on the smoothed competitive ratio of any deterministic algorithm. We prove two general lower bounds that hold independently of the underlying metric. Moreover, we show that our upper bounds are asymptotically tight for a large class of graphs.We also provide the first average case analysis of WFA. We prove that WFA has O(log(Δ)) expected competitive ratio if the request costs are chosen randomly from an arbitrary non-increasing distribution with standard deviation σ = Θ(λmin).