An Incremental Algorithm for a Generalization of the Shortest-Path Problem

Abstract

The grammar problem, a generalization of the single-source shortest-path prob-Ž Ž . Ž . . lem introduced by D. E. Knuth Inform. Process. Lett. 6 1 1977 , 1᎐5 is to compute the minimum-cost derivation of a terminal string from each nonterminal of a given context-free grammar, with the cost of a derivation being suitably defined. This problem also subsumes the problem of finding optimal hyperpaths in Ž . directed hypergraphs under varying optimization criteria that has received attention recently. In this paper we present an incremental algorithm for a version of the grammar problem. As a special case of this algorithm we obtain an efficient incremental algorithm for the single-source shortest-path problem with positive edge lengths. The aspect of our work that distinguishes it from other work on the dynamic shortest-path problem is its ability to handle ''multiple heterogeneous modifications'': between updates, the input graph is allowed to be restructured by an arbitrary mixture of edge insertions, edge deletions, and edge-length changes.

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