J. A. Azevedo, M. E. O. S. Costa, J. J. E. R. S. Madeira, and E. Q. V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm 5 The main MS algorithm idea is still the copy of the successive p k paths nodes, for a later computation of paths p j , with j k. In what follows it will be tried to roughly describe the several improvements to Martins algorithm. For more complete details the reader is addressed to [1, 2, 3, 16]. As presented in Algorithm 1, for each one of the determined p k paths, each of its nodes is analysed and copied. The copy of node i 2 N k corresponds to a j th shortest path from s to i, for some 1 j k. This copy consists in a new node, somehow related with the corresponding node in the ....

J. A. Azevedo, M. E. O. S. Costa, J. J. E. R. S. Madeira, and E. Q. V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.

....term above can be omitted. The related problem of finding the k longest paths in a DAG [4] can be transformed to a shortest path problem simply by negating all edge lengths; we can therefore also solve it in the same time bounds. 1. 3 Related Work Many papers study algorithms for k shortest paths [3, 5,7,9,13,14,17,24,31,32,34,35,37 41,43 45, 47, 50, 51, 56 60, 63, 65 69]. Dreyfus [17] and Yen [69] cite several additional papers on the subject going back as far as 1957. One must distinguish several common variations of the problem. In many of the papers cited above, the paths are restricted to be simple, i.e. no vertex can be repeated. This has advantages in some ....

....several common variations of the problem. In many of the papers cited above, the paths are restricted to be simple, i.e. no vertex can be repeated. This has advantages in some applications, but as our results show this restriction seems to make the problem significantly harder. Several papers [3, 13, 17, 24, 41, 42, 58, 59] consider the version of the k shortest paths problem in which repeated vertices are allowed, and it is this version that we also study. Of course, for the DAGs that arise in many of the applications described above including scheduling and dynamic programming, no path can have a repeated vertex ....

J. A. Azevedo, M. E. O. Santos Costa, J. J. E. R. Silvestre Madeira, and E. Q. V. Martins. An algorithm for the ranking of shortest paths. Eur. J. Operational Research 69:97--106, 1993.

....the time bounds above should be modified to include the time to compute a single source shortest path tree in such networks. Similar results also hold for finding the k longest paths in acyclic networks [4] we omit the details. 1. 3 Related Work The k shortest paths problem has been well studied [3, 5, 7, 9, 12, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42] and many algorithms are known. Dreyfus [9] and Yen [42] cite several additional papers on the subject going back as far as 1957. One must distinguish several common variations of the problem. In many of the papers cited above, the paths are restricted to be simple, i.e. no vertex can be ....

....several common variations of the problem. In many of the papers cited above, the paths are restricted to be simple, i.e. no vertex can be repeated. This has advantages in some applications, but as our results show this restriction seems to make the problem significantly harder. Several papers [3, 9, 12, 25, 26, 34, 35] explicitly consider the version 2 of the k shortest paths problem in which repeated vertices are allowed, and it is this version that we also study. Of course, for acyclic digraphs (as used in many of the applications described above including scheduling and dynamic programming) no path can have ....

J. A. Azevedo, M. E. O. Santos Costa, J. J. E. R. Silvestre Madeira, and E. Q. V. Martins. An algorithm for the ranking of shortest paths. European J. Operational Research, 69:97--106, 1993.

....a lot of references can be found on the specialized literature. See [7, 8, 9, 10, 12, 15, 17, 18, 26] The problem of determining not only the shortest but also the second, the third, the K th shortest path between a given pair of nodes it is also a well known and important problem, [3, 2, 4, 11, 16, 17, 20, 21, 22, 23, 24, 25]) As an example we may refer the constrained shortest path problem, 1] where a set of some more or less complicated constraints has to be satisfied by the optimal path. The first feasible path in the ranking of the shortest paths is that one we are looking for. The shortest path ranking problem ....

....problem, 24] paths can have no common arcs (arc disjoint paths) or no common intermediate nodes (node disjoint paths) In this paper we are concerned with the unconstrained problem. A new algorithm will be proposed which significantly improves the memory space requirement of a former version, [3, 2, 4], allowing that really large problems can be solved. For the unconstrained ranking problem, the theoretically best algorithm appears to be that one due to Eppstein which runs in time O(m n log n K ) in a directed network with n nodes and m arcs, 11] Computational experiments are reported ....

J.A. Azevedo, M.E.O.S. Costa, J.J.E.R.S. Madeira and E.Q.V. Martins, An algorithm for the ranking of shortest paths, European Journal of Operational Research 69, (1993), 97--106.

....natural labeling algorithm generalization for the shortest path problem. Shier was the first one to propose forms of this algorithm in this subclass [21, 22, 23] Yet in the class of algorithms supported by the Optimality Principle, are included Martins algorithm, 14] and all its improvements, [2, 3, 4, 18]. Labeling algorithm will be presented in the next subsection. 3.1 Labeling Algorithm As all the forms of the shortest path labeling algorithm, their generalizations for the K shortest paths problem can be divided into two sets: the label correcting algorithms and the label setting algorithms, ....

....00 and 1 000 will be picked out from X. That is, P 2 = fp 1 ; p 2 g is determined, where p 1 = h5; 3; 4; 1i and p 2 = h5; 3; 2; 4; 1i; moreover, count t=1 Gamma 2 and the algorithm finishes. We must notice that either Dreyfus algorithm, 11] or MS algorithm, 18] and its previous versions, [2, 3, 4, 14], are also label setting algorithms for the shortest paths ranking problem. 10 . 5 ....

J.A. Azevedo, M.E.O.S. Costa, J.J.E.R.S. Madeira, and E.Q.V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.

....brute force methods with no practical utility. From theorem 1 it results also that all the algorithms supported by the Optimality Principle, such as Dreyfus algorithm, 4] all labeling algorithms due to Shier, 12, 13, 14] and all the versions of the path deletion algorithm of Martins et al. [1, 2, 3, 7, 10], can not be adapted for determining only loopless paths. 3 The tree of the K shortest loopless paths Such as in the unconstrained ranking path problem, the K shortest loopless paths form a pseudotree the tree of the K shortest loopless paths. In this pseudo tree all nodes can be repeated, ....

J.A. Azevedo, M.E.O.S. Costa, J.J.E.R.S. Madeira, and E.Q.V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.

....perspective the set of the K shortest paths forms a tree in the sense of Theory of Graphs, being new algorithms proposed which compute a tree containing at least all the K shortest paths. The reader is assumed to be familiarized with the problem. Notation is the one used in previous papers, [1, 2, 3, 15, 16, 18]. Despite, some definitions and notation are introduced in the next section. 2 Definitions and Notation Let (N ; A) denote a given network, where N = fv 1 ; v n g (or N = f1; ng to simplify) is a finite set with n elements called nodes or vertices and A = fa 1 ; am g is a ....

.... using a suitable data structure to represent X , the time complexity is also O(Km) To exemplify the algorithm, the problem depicted in figure 1 will be solved for K = 2 (see figure 4) We must notice that either Dreyfus algorithm, 10] or MS algorithm, 16] and its previous versions, [1, 2, 3, 14], are also label setting algorithms for the shortest paths ranking problem. ....

J.A. Azevedo, M.E.O.S. Costa, J.J.E.R.S. Madeira, and E.Q.V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.

....paths. Since routers in the link state algorithm have access to the entire network topology, any centralized graph algorithm for computing multiple paths can be used. We are particularly interested in algorithms that calculate suffix matched path sets. As we will show, the k ranked paths algorithm [6], and the initial link disjoint k paths algorithm [23] are in this class. For such algorithms, our multipath forwarding algorithm can be directly applied. As in MPDV, let OE k r;d be the identifier that router r assigns its k th path to destination d. The path ID used by router r is of the ....

....Matched Multipath Routing Algorithms We presented an efficient forwarding method for link state based multipath calculation algorithms that calculate suffix matching path sets. Several existing multipath algorithms generate suffix matched paths. Here we show that the k ranked path algorithm [6] produces suffix matched paths. Proposition: The unconstrained k shortest (ranked) simple paths algorithm [6, 7] produces suffix matched path sets. Proof: The unconstrained k shortest path algorithm uses the following invariant for computing multiple paths between all pairs of nodes in the ....

[Article contains additional citation context not shown here]

J. A. de Azevedo, M. E. O. S. Costa, J. J. E. R. S. Madeira, and E. Q. V. Martins. An algorithm for the ranking of shortest paths. European Journal of Operational Research, 69:97--106, 1993.

.... tree, 12] As an example we point out the classical Dreyfus algorithm, 8] Another class comprises the generalizations of labeling shortest path algorithms of Shier, 16] Finally, the last class comprises the algorithms based on the path deletion concept due to Martins, 14] A new algorithm, [2] and [3] for the general shortest path ranking problem that uses the path deletion concept is presented. Its theoretical computational complexity is studied and computational experiments showing its clearly better performance are also presented. As an example, we 3 report tests on random networks ....

Azevedo J., Costa M., Madeira J., Martins E., An Algorithm for the Ranking of Shortest Paths, October 1989, submitted for publication.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC