We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n + kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery.

Finding the k Shortest Paths

We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m+n log n+k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m+n log n+kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, and maximum inscribed polygons. >

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Finding the k shortest paths

Bicriterion path problems in directed graphs are systematically studied. A series of ten problems are defined and their computational complexity examined. Algorithms are provided for some of them, including polynomial algorithms for the MAXMIN-MAXMIN problem and the MINSUM-MAXMIN problem, and a pseudo-polynomial exact algorithm as well as a fully polynomial approximation scheme for the MINSUM-MINSUM problem.

Bicriterion Path Problems

We have evolved a classification scheme to characterize algorithms for solving shortestpath problems. The algorithms are classified according to (A) the problem type, i.e., the question being asked about the given network; (B) the input type, i.e., the salient features of the given network which impact on the design of the algorithm and selection of data structures; and (C) the type of underlying technique employed to solve the problem. An annotated bibliography of 79 selected references on shortest-path algorithms is included. We have also provided a more complete listing of 222 references carefully culled out of a larger body of literature on shortest-path algorithms through the year 1979.

Shortest-path algorithms: Taxonomy and annotation

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

Semirings and path spaces

This paper presents an algebra which is appropriate for the formulation and solution of k-Shortest-paths problems. The algebra is a generalization of “Schedule Algebra” studied by Giffler [3,4] for computing path values in a network. It is shown formally that these path values can be calculated by using direct methods of numerical linear algebra for solving systems of linear equations and an algorithm similar to the long division procedure of ordinary arithmetic. Such a method is then modified to yield an algorithm for finding k shortest elementary paths in a network.

Ahnont Wongseelashote

Papers

Semirings and path spaces

An algebra for determining all path‐values in a network with application to K‐shortest‐paths problems