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An Incremental Algorithm for a Generalization of the ShortestPath Problem
, 1992
"... The grammar problem, a generalization of the singlesource shortestpath problem introduced by Knuth, is to compute the minimumcost derivation of a terminal string from each nonterminal of a given contextfree grammar, with the cost of a derivation being suitably defined. This problem also subsume ..."
Abstract

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The grammar problem, a generalization of the singlesource shortestpath problem introduced by Knuth, is to compute the minimumcost derivation of a terminal string from each nonterminal of a given contextfree 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 singlesource shortestpath problem with positive edge lengths. The aspect of our work that distinguishes it from other work on the dynamic shortestpath 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 edgelength changes.
Fast Shortest Paths Algorithms in the Presence of Few Negative Arcs
"... The shortest paths problem on weighted directed graphs is one of the basic network optimization problems. Its importance is mainly due to its applications in various areas, such as communication and transportation. Given a source node s in a weighted directed graph G, with n nodes and m arcs, the si ..."
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The shortest paths problem on weighted directed graphs is one of the basic network optimization problems. Its importance is mainly due to its applications in various areas, such as communication and transportation. Given a source node s in a weighted directed graph G, with n nodes and m arcs, the singlesource shortest path problem (SSSP, for short) from s is the problem of finding the minimum weight paths from s to all other nodes of G. The allpairs shortest paths problem (APSP, for short) consists in finding the minimum weight paths for each pair of nodes in G. In this paper we present hybrid algorithms for the SSSP and the APSP problems which are asymptotically fast when run on graphs with few negative weight arcs. We begin by reviewing the relevant notations and terminology. A directed graph is represented as a pair G = (V,E), where V is a finite set of nodes and E ⊆ V × V is a set of arcs such that E does not contain any selfloop of the form (v, v). In this context, we usually put n = V  and m = E. A weight function ω on G = (V,E) is any real function ω: E → R. A path in G = (V,E) from u to v is any finite sequence (v0, v1,..., vk) of nodes such that v0 = u, vk = v, and (vi, vi+1) is an arc of G, for