Time bounds for decision problems in the presence of timing uncertainty and failures

Abstract

Lower and upper bounds are proved for the time complexity of solving two decision problems in a distributed network in the presence of process failures and inexact information about time. It is assumed that the amount of (real) time between two consecutive steps of a nonfaulty process is at least c1and at most c2; thus, C=c2/c1is a measure of the timing uncertainty. It is also assumed that a message sent by a nonfaulty process is delivered within time at most d. Processes fail by not obeying the timing requirements on their steps and on messages they send (late timing failures). This paper presents a new stretching technique for deriving lower bounds in this model in the presence of timing failures. The combination of this technique with known bounds yields two new lower bounds for this model: 1. Any comparison based renaming algorithm has a worst-case running time of Ω(log n·Cd), in the presence of n−1 timing failures. 2.Any agreement algorithm has a worst-case running time of (formula presented), in the presence of f timing failures, for any f≥n/2. The paper also presents a general transformation from the synchronous model that tolerates timing failures; this transformation yields an O(log n. Cd) renaming algorithm (which is within a constant factor from the lower bound). It is also shown that when only crash failures have to be tolerated there is no need to incur an overhead of Cd for each round, by presenting an (formula presented) algorithm.