We address the problem for finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k−1) shortest path trees. Consequently, we design an O(+Km) time and O(K+m) space algorithm to determine the K shortes...

The technique of dynamic time warping (DTW) is relied on heavily in isolated word recognition systems. The advantage of using DTW is that reliable time alignment between reference and test patterns is obtained. The disadvantage of using DTW is the heavy computational burden required to find the optimal time alignment path. Several alternative procedures have been proposed for reducing the compu...

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Recently, we have shown that every arc of a strongly connected tournament of order n is contained in a directed path of order r(n + 3)/21. This is no longer valid for strongly connected in-tournaments, because there exist examples containing an arc with the property that the longest directe...

We define general algebraic frameworks for shortest-distance problems based on the structure of semirings. We give a generic algorithm for finding single-source shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently classical shortest paths problems or to find the k-shortest d...

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by CO(D) (CC(D)) and its size by co(D) (cc(D)). Gutin, Johnstone...

The Path Partition Conjecture for digraphs states that for every digraph D, and every choice of positive integers λ1, λ2 such that λ1 + λ2 equals the order of a longest directed path in D, there exists a partition of D in two subdigraphs D1,D2 such that the order of the longest path in Di is at most λi for i = 1, 2. We present sufficient conditions for a digraph to satisfy the Path Partition Co...

We show two improvements on time complexities of the all pairs shortest path (APSP) problem for directed graphs that satisfy certain properties. The idea for speed-up is information sharing by n single source shortest path (SSSP) problems that are solved for APSP. We consider two parameters, in addition to the numbers of vertices, n, and edges, m. First we improve the time complexity of O(mn+n ...

The Split algorithm is an essential building block for route-first cluster-second heuristics and modern genetic algorithms for vehicle routing problems. The recent survey of [Prins, Lacomme and Prodhon, Transport Res. C (40), 179–200] lists more than 70 articles that use this technique. Split is classically assimilated to the search for a shortest path in an acyclic directed graph and performed...

Let G be a non-trivial, simple, finite, connected and undirected graph of order n and size m. An induced acyclic graphoidal decomposition (IAGD) of G is a collection ψ of non-trivial and internally disjoint induced paths in G such that each edge of G lies in exactly one path of ψ. For a labeling f : V → {1, 2, 3, . . . , n}, let ↑ Gf be the directed graph obtained by orienting the edges uv of G...

The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a chordal graph which does not contain a hole...