N-set Consensus When Inputs Are Restricted Zvi Avidor N-set Consensus When Inputs Are Restricted

A protocol solving the k-set consensus problem [13] has to provide three properties: k-agreement non-faulty processes has to decide on at most k values, validity every decision value must be an input value of some process, and termination each non-faulty process must stop with a decision value after some nite number of steps. It has been proved [8] that this problem can be solved using only read / write operations in the presence of f crash failures if and only if f < k. One way to subvert this impossibility result is to restrict the possible assignments of input values to processes. This thesis presents a new problem, DS;DL-restricted k-set consensus, which limits the possible input vectors to a set DS , and requires termination of non-faulty processes only when the initial con guration is taken from a smaller set, DL. A characterization of the nite sets that allow a wait-free solution of DS;DL-restricted n-set consensus in a system with n + 1 processes, using only read and write operations is proved. Using this characterization, a polynomial algorithm is provided, which decides if two sets of input vectors, DS and DL, enable solving DS ;DL-restricted n-set consensus or not. 1 Abbreviations and Notations Sn | Simplex of rank n+ 1. sn | a simplex of rank n + 1 from a subdivision facev(Sn) | The set Sn n fvg. v Tm | The m+ 1-simplex, Tm [ fvg. hp; i | A vertex representing a process p and its local state . V al(v) | Process' state , when v = hp; i for some p. Proc(v) | The process p, when v = hp; i. O | Output complex. (Sn) | Standard chromatic subdivision of Sn. DS | Set of input vectors ensuring validity and agreement DL | Set of input vectors ensuring termination K(I) | Input complex induced by a set of input vectors I 2 Chapter