The Average Time Complexity to Compute Prefix Functions in Processor Networks
We analyze the average time complexity of evaluating all prefixes of an input vector over a given algebraic structure 〈 Σ,⊗〉. 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 〈 Σ, ⊗〉 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 network size linear. It differs substantially from the known constructions for the worst case.
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