" a Combinatorial Characterization of the Distributed Tasks

Fischer, Lynch and Paterson showed in a fundamental paper that achieving a distributed agreement for N > 1 processors is impossible in the presence of ohe faulty processor. This' resulf was hiter extended by Wolfstahl and Moran who showed that it holdS for any task with a connected input graph and a ~connected decision graph (where a vertex in the input [decision] graph is an N -tuple of input [decision] values of the processors, and there is an edge connecting two vertices if and only if they differ in exactly one component). In this paper we extend thilt latter.result, and in fact we ~t the exact borderline between solvable and unsolvable tasks, by giving a necessary and sufficient condition ror a task to be solvable in the presence of a faulty processor. We present a universal protocol which solves any task which is found to be solvable by our condition. Using our characterization, we derive a novel technique to prove lower bounds on the number of messages that must be sent due to processor failure; specifically, we show. that for each fixed N > 2 there exist distributed tasks for N processors that can be solved in the presence of a faulty processor, but any protOCOl that solves them must send arbitrarily many messages in the worst case. T ec hn io n C om pu te r Sc ie nc e D ep ar tm en t T eh ni ca l R ep or t C S0 48 5 19 87