We define general algebraic frameworks for shortest-distance problems based on the structure of semirings. We give a generic algorithm for finding single-source shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently classical shortest paths problems or to find the k-shortest distances in a directed graph. It can be used to solve single-source shortest-distance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity.

problem in a weighted automaton. This problem arises commonly in speech recognition applications when a ranked list of unique recognizer hypotheses is desired. We believe this is the first n-best algorithm to remove redundant hypotheses before rather than after the n-best determination. We give a detailed description of the algorithm and demonstrate its correctness. We report experimental results showing its efficiency and practicality even for large n in a 40; 000-word vocabulary North American Business News (NAB) task. In particular, we show that 1000-best generation in this task requires negligible added time over recognizer lattice generation.