The grammar problem, a generalization of the single-source shortest-path problem introduced by Knuth, is to compute the minimum-cost derivation of a terminal string from each non-terminal 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 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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Abstract. A recent industrial challenge for traffic information providers is to be able to compute point-to-point shortest paths very efficiently on road networks involving millions of nodes and arcs, where the arc costs represent travelling times that are updated every few minutes when new traffic information is available. Such stringent constraints defy classic shortest path algorithms. In this paper we review some existing methods that address this scenario and propose a new Polynomial-Time Approximation Scheme heuristic. Mots-Clefs. Shortest Paths; PTAS; Dynamic Road Networks.

Among the variants of the well known shortest path problem, those that refer to a dynamically changing graphs are theoretically interesting, as well as computationally challenging. Applicationwise, there is an industrial need for computing point-to-point shortest paths on large-scale road networks whose arcs are weighted with a travelling time which depends on traffic conditions. We survey recent techniques for dynamic graph weights as well as dynamic graph topology. 1