Agreement In the Presence of Faults, On Networks of Bounded Degree

We present networks of bounded degree and a fully polynomial almost everywhere agreement scheme which tolerate, with high probability, randomly located faulty processors, where processors fail independently with some constant probability. 1 Introduction Dwork et. al. [DPPU] introduced the notion of almost everywhere agreement. They exhibited a communication network of bounded degree connecting N processors along with a polynomial protocol enabling all but O(t) of the correct processors to reach agreement in the presence of up to t = O(N= log N) faulty processors. Further work by Berman and Garay [BG1, BG2] improved the efficiency of these protocols. Handling more faults is a considerably more difficult problem. Note that in networks of bounded degree the distance between most pairs of nodes is\Omega\Gamma/36 N ). Therefore, even if the faulty processors are picked randomly, each independently with probability ffl ?? 1= log N , then with high probability most communication paths betwee..