Proving time bounds for randomized distributed algorithms

Abstract

A method of analyzing time bounds for randomized distributed algorithms is presented, in the context of a new and general framework for describing and reasoning about randomized algorithms. The method consists of proving auxiliary statements of the form U t ! p U 0 , which means that whenever the algorithm begins in a state in set U, with probability p, it will reach a state in set U 0 within time t. The power of the method is illustrated by its use in proving a constant upper bound on the expected time for some process to reach its critical region, in Lehmann and Rabin's Dining Philosophers algorithm.

Quantitative Program Logic and Performance in Probabilistic Distributed Algorithms

“…This treatment differs from other approaches to performance analysis of probabilistic algorithms [11,3,6] in that we do not refer explicitly to the distribution over computation paths; neither do we factor out the nondeterminism as a first step nor do we analyse the behaviour of the composed system: instead we use compositionality of local properties thus simplifying our formal reasoning. Other approaches using expectations [19,8] do not treat nondeterminism and thus are not applicable to distributed algorithms like this at all.…”

“…In this view the role of nondeterminism is to allow an arbitrary selection over some range of probability distributions; however the agent making that selection can only influence the weights of the probabilistic transitions, not their actual resolution once those weights have been picked. That behaviour is precisely the kind exhibited by an 'adversary scheduler' assumed of many distributed algorithms [11]. We shall discuss schedulers in more detail in Sec.…”

“…Locality has particular significance here since in practice it reduces system-wide analysis to the analysis of a single component in isolation from the others, vastly simplifying arguments. And finally, it is worth mentioning that though our primary theme is proof, our estimated performance of the random dining philosophers [10] is still lower than some other published analyses [11].…”

“…Operationally we model probabilistic sequential programs [14,7] (compare also [3,11]) as functions from (initial) state to sets of distributions over (final) states. Intuitively that describes a computation proceeding in two (indivisible) stages: a nondeterministic choice, immediately followed by a probabilistic choice, where the probability refers to the possible final states reachable from a given initial state.…”

“…The precise decomposition used for the analysis of the dining philosophers follows that of the correctness proof (appearing elsewhere [17]), and it gives a total expected time to first eating of no more than 33 (compare 63 of [11]). The details of those calculations are set out elsewhere [12].…”