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A walk over the shortest path Dijkstras Algorithm viewed as fixed-point computation.pdf
Information Processing Letters 77 (2001) 197–200
A walk over the shortest path: Dijkstra’s Algorithm viewed as ?xed-point computation
Jayadev Misra ?
Department of Computer Sciences, University of Texas at Austin, Austin, TX 78712-1188, USA
Abstract
We present a derivation of Dijkstra’s shortest path algorithm [Numer. Math. 1 (1959) 83]. We view the problem as computation of a “greatest solution” of a set of equations. A UNITY-style computation [Chandy and Misra, Parallel Program Design: A Foundation, 1988] is then prescribed whose implementation results in Dijkstra’s algorithm. ? 2001 Elsevier Science B.V. All rights reserved.
Keywords: Design of algorithms; Graph algorithms; Combinatorial problems; Program derivation
0. Introduction
1. The shortest path problem
Dijkstra’s shortest path algorithm [1] has, by now, become a classic (the cited paper has received such an of?cial designation from the Citation Index Service). Typical descriptions (and derivations) of this algorithm start by postulating that the shortest paths be enumerated in the order of increasing distances from the source. In this note, we present a derivation that is quite different in character. We view the problem as computation of a “greatest solution” of a set of equations. A UNITY-style computation [0] is then prescribed whose implementation results in Dijkstra’s algorithm.
The bulk of the work in our derivation is in designing the appropriate heuristics that guarantee termination; this is in contrast to traditional derivations where most of the effort is directed toward postulating and maintaining the appropriate invariant.
Given is a ?nite directed graph that has (i) a source node, henceforth designated by s, and (ii) for each edge (i, j ) a non-negative real number
wij , called its length. The length of a path is the sum of edge lengths along the path. It is required to compute the shortest path, i.e., a path of minimum length, from s to every node. Henceforth, “shortest path to a node” means
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