The Average Time Complexity to Compute Prefix Functions in Processor Networks

Abstract

We analyze the average time complexity of evaluating all prefixes of an input vector over a given algebraic structure h\Sigma; . As a computational model networks of finite controls are used and a complexity measure for the average delay of such networks is introduced. Based on this notion, we then define the average case complexity of a computational problem for arbitrary strictly positive input distributions. We give a complete characterization of the average complexity of prefix functions with respect to the underlying algebraic structure h\Sigma; \Omega\Gamma resp. the corresponding Moore-machine M . By considering a related reachability problem for finite automata it is shown that the complexity only depends on two properties of M , called confluence and diffluence. We prove optimal lower bounds for the average case complexity. Furthermore, a network design is presented that achieves the optimal delay for all prefix functions and all inputs of a given length while keeping the netw..