Depth-size trade-offs for parallel prefix computation
Journal of Algorithms
Size-time complexity of Boolean networks for prefix computations
Journal of the ACM (JACM)
Characterization of associative operations with prefix circuits of constant depth and linear size
SIAM Journal on Computing
Fast Addition of Large Integers
IEEE Transactions on Computers
Tight bounds on expected time to add correctly and add mostly correctly
Information Processing Letters
Circuit complexity: from the worst case to the average case
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
The Average Case Complexity of the Parallel Prefix Problem
ICALP '94 Proceedings of the 21st International Colloquium on Automata, Languages and Programming
Some New Results on Average Worst Case Carry
IEEE Transactions on Computers
Efficient Addition on Field Programmable Gate Arrays
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
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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.