Synchronization of 1-way connected processors

Abstract

We are given a network of n identical processors that work synchronously at discrete steps. At each time step every processor sends messages only to a given subset of its neighbouring processors and receives only from the remaining neighbours. The computation starts with one distinguished processor in a particular starting state and all other processors in a quiescent state. The problem is the following: to set all the processors in a given state for the first time and at the very same instant. This problem is known as the Firing Squad Synchronization Problem and was introduced by Moore in 1964. The usual formulation is given on cellular automata and it has been investigated on various topologies of networks of processors and for various kinds of communication. In this paper we present for the first time solutions that synchronize processors that communicate on 1-way links and are arranged in a ring or in a square whose rows and columns are rings. In particular we provide algorithms to synchronize both the two networks and prove that all such algorithms are optimal in time. In addition we show how to compose solutions to obtain new synchronizing solutions. In particular given two solutions in time t ₁(n) and t ₂ (n) we provide solutions in time t ₁(n)+t ₂(n)+d and t ₁(n)t ₂(n). Moreover, given a predicate P(n), a solution is given whose time is t ₁(n), if P(n) holds, and t ₂(n) otherwise. Finally, we give solutions which synchronize at a given time f(n), for f(n) equal to n ², n log n and 2ⁿ.