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Calculating A Path Algorithm
 Science of Computer Programming
, 1992
"... . A calculational derivation is given of an abstract path algorithm, one instance of the algorithm being Dijkstra's shortestpath algorithm, another being breadthfirst/depthfirst search of a directed graph. The basis for the derivation is the algebra of regular languages. 1 Problem Statement ..."
Abstract

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. A calculational derivation is given of an abstract path algorithm, one instance of the algorithm being Dijkstra's shortestpath algorithm, another being breadthfirst/depthfirst search of a directed graph. The basis for the derivation is the algebra of regular languages. 1 Problem Statement Given is a (nonempty) set N and a j N j \Theta j N j matrix A, the rows and columns of which are indexed by elements of N . It is assumed that the matrix elements are drawn from a regular algebra 1 (S; +; \Delta; ; 0; 1) having the two additional properties that the ordering induced 2 by + is a total ordering (1) on the elements of S; and 1 is the largest element in the ordering: (2) In addition to N and A, one is given a 1\Theta j N j matrix b. Hereafter 1\Theta j N j matrices will be called "vectors", 1 \Theta 1 matrices will be called "elements" and j N j \Theta j N j matrices will be called "matrices". The problem is to derive an algorithm to compute the vector b \Delta A wher...