On the advice complexity of the k-server problem

Abstract

The model of advice complexity offers an alternative measurement allowing for a more fine-grained analysis of the hardness of online problems than standard competitive analysis. Here, one measures the amount of information an online algorithm is lacking about the yet unrevealed parts of the input. This model was successfully used for many online problems including the k-server problem. We extend the analysis of the k-server problem by giving a lower bound on the advice necessary to obtain optimality, and upper bounds on online algorithms for the general and the Euclidean case. For the general case, we improve the previous results by an exponential factor; for the Euclidean case, we achieve a constant competitive ratio using constantly many advice bits per request. Furthermore, for many online problems, we show how lower bounds on the advice complexity can be transformed into lower bounds on the expected competitive ratio of randomized online algorithms.

Online Bin Packing with Advice of Small Size

“…The online advice model was first introduced by Böckenhauer et al [7,6] and by Emek et al [12]. Both papers were inspired by the work of Dobrev et al [11].…”

“…In the model of Böckenhauer et al, the advice is written on a read-only tape prior to the algorithm's execution, and the algorithm can read advice bits from that tape at will. The advice complexity has established itself as a prolific sub-field of online computation, and many online problems have been studied under the setting of online computation with advice (e.g., metrical task systems [12], job shop scheduling [7,19], the k-server problem [12,6,20], knapsack [5], buffer reordering management [1], and list update [9]).…”

Disjoint Path Allocation with Sublinear Advice

“…. , Alg 2 b } [3,4]. From Corollary 2, we know that, for every such deterministic algorithm Alg i , the fraction of good instances from I, and hence the fraction of instances on which Alg i has a competitive ratio of at most (h + 2)/(2(1 + h/f )), is at most…”

“…In this setting, the advice is not read from a tape, but it is supplied in every time step, and the number of advice bits is the same in every time step. Both models have so far been used to study a large number of problems, including the paging problem [4,14], the k-server problem [3,11,12,16], or metrical task systems [11]. One of the first online problems studied in the model of computing with advice as we use it in this paper was the DPA problem [4].…”

Reordering buffer management with advice

“…Both models are inspired by the model of online computation with advice originally proposed by Dobrev et al [13], wherein the oracle could provide information without a cost by using an empty string as the advice for a request, and the focus was on optimality. Several results have been given in the semi-online model [13,11,20,21,10]. For example, in [13,11], the authors explore the number of bits of advice required for deterministic and randomized paging algorithms, algorithms for the DiffServ problem, algorithms for a special case of the job shop scheduling problem, and algorithms for the disjoint path allocation problem, to be 1-competitive.…”