Applications of Shellable Complexes to Distributed Computing

Abstract

This talk describes recent Joint work with Sergio Rajsbaum [3]. For models of concurrent computation in which processes may fail by crashing, each possible computation can be characterized as a simplicial complex, a geometric structure constructed by “gluing together” simplexes in a regular manner [6]. Informally, a complex is k-connected if it has no “holes” in dimension k or lower. It is known that if the complex corresponding to every such computation is k-connected, then one cannot solve (k+1)-set agreement [4,5,6]. A simplicial complex is shellable if it can be constructed by gluing a sequence of n-simplexes to one another along (n− 1)-faces only. Shellable complexes have been studied in the combinatorial topology literature [1,2,7] because they have many nice combinatorial properties. We can exploit these properties complexes to derive new and remarkably succinct tight (or nearly tight) lower bounds both on the connectivity of the associated complexes, and on solutions to the k-set agreement task in these models. We consider a round-by-round model of computation, where we view each round as a map carrying simplexes to complexes. The principal insight is that if the single-round complex is shellable, then multi-round compositions preserve connectivity under certain easily-checkable conditions. These are theorems of combinatorial topology, independent of any model of computation. We then show that for many classical models of computation, such as the synchronous, asynchronous, and semi-synchronous message-passing models, along with the asynchronous read-write memory model, each single-round complex is indeed shellable, so it becomes a straightforward exercise to derive tight (or nearly tight) bounds on when and if one can solve k-set agreement. For asynchronous shared-memory models in which processes have access to “black-box” objects that solve consensus or k-set agreement, matters are a little more complicated. The single-round complex, while not itself shellable, is a simple union of shellable complexes, with a shellable nerve, and the same consequences follow. Moreover, our results apply not just to the usual wait-free or t-resilient failure models, but to general adversary schedulers that can cause certain subsets of processes to fail, perhaps in a non-uniform way.