Adaptability and the Usefulness of Hints (Extended Abstract)

Abstract

In this paper we study the problem of designing algorithms in situations where there is some information concerning the typical conditions that are encountered when the respective problem is solved. The basic goal is to assure efficient performance in the typical case, while satisfying the correctness requirements in every case. We introduce adaptability, a new measure for the quality of an algorithm, which generalizes competitive analysis and related frameworks. This new notion applies to sequential, parallel and distributed algorithms alike. In a nutshell, a "hint" function conveys certain information about the environment in which the algorithm operates. Adaptability compares the performance of the algorithm against the "specialist"-an algorithm specifically tuned to the particular hint value. From this perspective, finding that no single algorithm can adapt to all possible hint values is not necessarily a negative result, provided that a family of specialists can be constructed. Our case studies provide examples of both kinds. We first consider the "ancient" problem of on-line scheduling of jobs in m identical processors, and show that no algorithm can fully adapt to the natural hint of largest job size. In particular, for the cases m = 2, 3, we present specialists that beat the general lower bound for on-line makespan. In the domain of distributed computing, we analyze the Distributed Consensus problem under several hint functions. To fulfill the requirements of one of the cases we consider, we present the first consensus algorithm that is simultaneously optimal in number of processors, early-stopping property (that is, it runs in time proportional to the actual number of faults), and total number of communicated bits. In our new parlance, the algorithm adapts to the number of faults hint with both running time and communication.Work partly done while the author was visiting the Centrum voor Wiskunde en Informatica (CWI), Amsterdam, NL, and at the IBM T.

Condition-based consensus solvability: a hierarchy of conditions and efficient protocols

“…) case (wi = ) then return(wi) (7) (wi = ) then repeat Xi ← collect(W ); (8) if (∃ j : Xi[j] = ⊥, ) then return(Xi[j]) endif (9) until (⊥ / ∈ Xi) endrepeat; (10) [a1, . .…”

“…then if P [d] (currenti) then return(S(currenti)) endif; (7) prev d vali ← d; prev nb of boti ← # ⊥ (currenti) (8) endif; (9) if richi = yes then v ← v.right else v ← v.left endif (10) endwhile; (11) [0]…”

“…This approach has been abstracted in the notion of unreliable failure detectors [11]. There have also been studies of hybrid approaches, like combining failure detection with randomization [2,26], or more general approaches for designing algorithms in situations where there is some information about the typical conditions that are encountered when the respective problem is solved [7].…”

Conditions on input vectors for consensus solvability in asynchronous distributed systems

“…Agreement problems with a restricted set of inputs vectors were considered in Taubenfeld et al [1994] and Taubenfeld and Moran [1996], where possibility and impossibility results in a shared memory system and a hierarchy of problems that can be solved with up to f failures but not for ( f + 1) failures are introduced. More generally, an approach for designing algorithms in situations where there is some information about the typical conditions that are encountered when the respective problem is solved is presented in Berman and Garay [1998]. In this article, the consensus problem in a synchronous setting is analyzed taking as parameter the difference between the number of 0's and 1's in the input vector.…”

The Bancomat problem: An example of resource allocation in a partitionable asynchronous system

“…We s h o w that this approach has a low cushion. 4. Processes in a connected component share state and a process informs the other processes when it has taken an action.…”