S. Dreyfus. An appraisal of some shortest path algorithms. Operations Research, 17:395--412, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for trip planning. A more interesting application of travel times is in route guidance systems that compare travel times on alternative paths to find the path that meets the demand of travelers. The literature has seen a wide variety of algorithms applicable to finding desired paths for travelers [1, 5, 7]. Many of these algorithms assume constant cost, such as geometric distance, for traveling between connected locations. In contrast, link travel times are rarely constant, and should be treated as random variables in actual transportation networks. Moreover, in these so called stochastic ....

S.E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395412, 1969.

....the Bellman equations for the case K = 2 [BK60] The algorithm proposed by these authors consisted of first finding the shortest path from s to all other nodes and, then, of computing their respective second shortest paths by an iterative procedure similar to the Bellman Ford algorithm. Dreyfus [Drey69] extended the Bellman Kalaba equations to the general case and proposed important improvements to the method of resolution. The time complexity of the Dreyfus algorithm is O(A K Delta jV j Delta log d) where d is the maximum input degree [Fox73, Fox78] This algorithm computes the K shortest ....

.... nodes, s, t 2 V , and a positive integer K, find 1 (t) 2 (t) K (t) 3 An Algorithm for the K Shortest Paths Problem In this section, the K shortest paths problem is stated as the resolution of a set of recursive equations which were first presented rather informally by Dreyfus [Drey69]. First, a simple formal derivation of these equations is given based on the recursive problem formulation of Section 2. Later on, a new algorithm is proposed consisting of direct recursive resolution of these equations. Finally, the correctness and efficiency of the algorithm are proved. 3.1 ....

S. E. Dreyfus: "An Appraisal of some Shortest-Path Algorithms", Operations Research, vol. 17, pp. 395--412. (1969)

....problem where it is intended to rank the K shortest paths between an initial and a terminal node in a network. The first algorithm for solving this problem was proposed by Hoffman and Pavley in 1959, 9] and ever since other papers presenting algorithms have been proposed among which we refer [6, 7, 10, 11, 13, 16, 17]. A very complete bibliography on the problem, due to Eppstein, can be found in the URL address http: liinwww.ira.uka.de bibliography Theory k path.html. These algorithms can be divided into two classes: one based on the Optimality Principle generalization and another based on the determination ....

S. E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395--412, 1969.

....choose among them by considering the other criteria. We recently implemented a similar technique as a heuristic for the NP hard problem of, given a graph with colored edges, finding a shortest path using each color at most once [20] This type of application is the main motivation cited by Dreyfus [17] and Lawler [39] for k shortest path computations. Model evaluation. Paths may be used to model problems that have known solutions, independent of the path formulation; for instance, in a k shortest path model of automatic translation between natural languages [30] a correct translation can ....

....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 ....

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S. E. Dreyfus. An appraisal of some shortest path algorithms. Operations Research 17:395--412, 1969.

....were the source or destination, our encoding structure would result in considerably tighter bounding and consequently in much less effort. The Dijkstra algorithm has been used as the benchmark, since it is generally regarded to be the best algorithm for finding shortest path between two points [14]. One would like to make the effort ratio as close to zero as possible. Following [5] synthetic graphs were used as data sets. The distance between two nodes was assumed to be a uniform random variable over a specified positive interval. The number of nodes were varied to obtain databases of ....

S. E. Dreyfus, "An Appraisal of Some Shortest Path Algorithms," Operations Res., 17(3), 1969, pp. 395-412.

....departure times A related problem is the Minimum Time Dynamic Path Problem for all departure times, which looks for minimum arrival time paths from any node i d to a given destination node d for all the possible departure times. This problem was first addressed by Cooke and Halsey (1966) and by Dreyfus (1969). Recently, it was solved by Ziliaskopoulos and Mahmassani (1993) by extending a classical shortest path approach, in O(n 3 q 2 ) time. Observe that, if Chrono SPT is called q times, i.e. once for each t T, then an algorithm is obtained which solves the problem Q( E q) time, i.e. in O(mq 2 ....

S. E. Dreyfus (1969), "An appraisal of some shortest-path algorithms", Operations Research 17, 395-412.

....hand, some ecient algorithms for certain classes of SP problems can be exploited to provide some classes of SCLP programs with an ecient way to compute their semantics. Keywords: Shortest path problems, soft constraints, constraint logic programming. 1. Introduction Shortest Path (SP) problems [4, 11] are among the most studied network optimization problems. They are mainly used to represent and solve transportation problems, where the optimization may involve di erent criteria, says cost, time, resources, etc. Most interesting is the multi criteria case, where the optimization involves a set ....

S. E. Dreyfus. An appraisal of some shortest-paths algorithms. Operation Research, 17:395{ 412, 1969.

....the computation of fastest paths from one origin node, departing at time interval 0, to all other nodes. We analyse the problem for the two types of waiting at node policies defined above. 3. 1 Waiting at nodes is not allowed This is the most studied variation of the problem (Cook and Halsey, 1966; Dreyfus, 1969; Kaufman and Smith, 1993) The most celebrated result for this variation of the problem is the following: when the FIFO condition is valid, Dijkstra s algorithm can be generalized to solve the time dependent fastest paths problem with the same time complexity as the static shortest paths problem. ....

....Kaufman and Smith, 1993) The most celebrated result for this variation of the problem is the following: when the FIFO condition is valid, Dijkstra s algorithm can be generalized to solve the time dependent fastest paths problem with the same time complexity as the static shortest paths problem. Dreyfus (1969) is the first to mention this generalization. Later Kaufman and Smith (1993) formally proved that this generalization is valid only if the FIFO condition is satisfied. During our literature review, we found an even earlier proof due to Ahn and Shin (1991) Below we present yet another proof which ....

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Dreyfus, S. E. (1969). An Appraisal of Some Shortest-Path Algorithms. Operations Research, 17, 395-412.

....i ; v i 1 ) v i 1 ; v i 2 ) Delta Delta Delta ; v k Gamma1 ; v k ) Furthermore, the cost of a path, Pi, between MLC cells i and k, is derived from the cost of its edges so that the least costly path represents the most likely path to be followed by the mobile. A k shortest paths algorithm [5] is then used to obtain the set, K, of k most likely paths to be followed by the mobile unit. For each path, Pi 2 K, between MLC cell i and j, we define the path residence time as the sum of the residence time of each cell in the path. Let Pi s and Pi l in K, represent the paths with the ....

Sturat E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395--412, 1969.

....should connect its endpoints reasonably directly, but there may be more or less community support for one option or another. A typical solution is to compute several short paths and then choose among them by considering the other criteria. This type of application is the reason cited by Dreyfus [9] and Lawler [23] for k shortest path computations. Sensitivity analysis. By computing more than one shortest path, one can determine how sensitive the optimal solution is to variation of the problem s parameters. For instance, in biological sequence alignment, one typically wishes to see ....

....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 ....

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S. E. Dreyfus. An appraisal of some shortest path algorithms. Operations Research, 17:395--412, 1969. Romanovsky cites this as being pp. 345-412. I haven't doublechecked which is correct.

....C in (N ; A) and P si 6= P it 6= for any i 2 N . Then the k th shortest path p k , from s to t, is formed by j th shortest paths, from s to every intermediate node j of p k , and j k. Proof See [16] A class of ranking shortest paths algorithms is supported by theorem 3. Dreyfus, [11], was the first one to propose an algorithm in this class. His algorithm determines not only the K shortest paths between a given pair of nodes but also the K shortest paths from all nodes to t. Also in this class of algorithms we can consider the natural labeling algorithm generalization for the ....

....variable count) Notice that, after step 7 nodes 1 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 . ....

S.E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395--412, 1969.

....to determine the entire set of loopless paths in order to determine P K . That is, label correcting algorithms would be 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, ....

S.E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395--412, 1969.

.... has been extensively studied (see [6, 7, 8, 9, 12, 22] among hundreds of others possible references) The problem of determining not only the shortest path, but also the second, the third, the K th shortest path (for a given integer K 1) is also a classical one (see the classical papers, [10, 13, 23]) but has not been studied so intensively, despite its obvious practical interest; however more than one hundred of papers can be found in the literature (see the very complete online bibliography due to Eppstein, whose WWW adress is http: liinwww.ira.uka.de bibliography Theory k path.html ) ....

....i k j; so, as c(p j h ) is finite (theorem 2) we conclude that c(p s ) c(p i h ) c(p ht ) c(p k ) for any i k j. That is, p k is not the k th shortest path which contradicts the assumption made. A class of ranking shortest paths algorithms is supported by theorem 3. Dreyfus, [10], was the first one to propose an algorithm in this class. This algorithm determines not only the K shortest paths between a given pair of nodes but also the K shortest paths from all nodes to t. Yet in this class of algorithms we can consider the generalization of the labeling algorithms for the ....

[Article contains additional citation context not shown here]

S.E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17:395--412, 1969.

....any k th shortest path is made up of j th shortest sub paths, where j k. Algorithms in this class have to solve a functional equation, that is explicit only when nodes are processed in order of their level in the shortest 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 ....

Dreyfus S., An appraisal of some shortest paths algorithms, Operations Research 17, (1969), 395--412.

....the bounding property anymore, and we therefore refer to this approach as a heuristic. However, for a sufficiently large number of paths it is very likely that we still obtain a stochastic upper bound. In order to find the K longest paths we use a variant of the algorithm proposed by Dreyfus [6] which additionally is able to handle parallel arcs and is tailored for acyclic networks. It could also be valuable to choose the paths on the basis of other criteria cf. e.g. Dodin [4] In the following, we make use of this heuristic approach to cope with incomplete information. In practice, it ....

S. E. Dreyfus. An appraisal of some shortest-path algorithms. Operations research, 17:395--412, 1969.

....in a depth first manner. This difference may be pronounced if the source and destination are relatively close to each other. 3.2.4 Space Analysis Memory required is easily seen to be the same as for S . 3. 3 Comparison with other Methods Other methods for the general path case are described in [Dreyfus 69] and [Shier 79] In particular Shier compares the methods empirically, and finds S to be significantly superior to other methods when graphs are sparse (E = O(V ) which is the case we are interested in. For dense graphs other methods may be better. Shier uses a different implementation of the ....

....either the general or simple path version of S (or S) we can replace the edge weight function with a new function d(u; v; t) that not only depends on the edge (u; v) but also on the arrival time t at u. That this can be done with Dijkstra s algorithm seems to have been first noticed by Dreyfus ( Dreyfus 69] The only modification to the algorithm of Figure 1 needed is to replace d(u; v) in lines 2 and 7 of Relax with d(u; v; b k [u] For the algorithm to work, the function must satisfy the following property for all edges (u; v) t d(u; v; t) t 0 d(u; v; t 0 ) for all t and t 0 t ....

S. E. Dreyfus. An appraisal of some shortest path algorithms. Operations Research, Vol. 17, No. 3, 1969. pp. 395--412.

....protocols described in this paper. These results are mostly taken from Orda and Rom[6] Given a source node s and a starting time t S and allowing unrestricted waiting, we look for shortest paths and minimum delays between s and all other nodes for that starting time. As noted in Dreyfus[10] this is a straightforward extension to such algorithms as Dijkstra s[11] or Bellman Ford s[12] In other words, in the UW model, for any source destination pair and any starting time, there exists a simple and concatenated shortest path. Therefore, there exist efficient polynomial algorithms that ....

S.E. Dreyfus, "An Appraisal of some Shortest Path Algorithms," Operations Research 17 pp. 395412 (1969).

....departure times A related problem is the Minimum Time Dynamic Path Problem for all departure times, which looks for minimum arrival time paths from any node i 6= d to a given destination d for all the possible departure times. This problem was first addressed by Cooke and Halsey [1966] and by Dreyfus [1969]. Recently, it was solved by Ziliaskopoulos and Mahmassani [1993] by extending a classical shortest path approach, in O(n 3 q 2 ) time. A more efficient approach can be devised, which is based on a reverse chronological visit of the Reverse Space Time Network R 0 , where R 0 differs from ....

S. E. Dreyfus (1969). "An appraisal of some shortest-path algorithms". Operations Research 17, 395-412.

....we are concerned with. So far, the most 1 In contrast with Deo [29] a path is by default oriented. 2 contrarily to Deo [29] the shortest path problem 9 in depth classification of shortest path problems is agreed to be the Deo and Pang s taxonomy [30] Other general surveys can be found in [37, 95]. The next three sections aim at locating our shortest path problem within the Deo and Pang s classification. 2.2.1 The problem type We recall that, in the context of our inverse problem, shortest paths occur in constraint descriptions (see (1.5) and (1.6) in Chapter 1) These involve shortest ....

S.E. Dreyfus, "An appraisal of some shortest path algorithms", Operations Res., vol. 17, pp. 393--412, 1969.

....the predicted or the observed manner in which the delays change. Another approach is to consider a deterministic, time dependent network, in which the exact behavior of link delays is described by a given set of delay functions. Shortest path problems for such a model were studied in several works[6,7,8,9] and recently in the context of computer networks [10,11] The timedependent approach is beneficial when the dynamics of the network is deterministic or can be predicted fairly well. However, in many computer networks (e.g. multisatellite networks) such knowledge is not readily available. More ....

S.E. Dreyfus, "An Appraisal of some Shortest Path Algorithms," Operations Research 17 pp. 395412 (1969).

....is discrete. The first algorithms of that kind were considered yet by W. Hoffman and R. Pavley [1] and by R. Bellman and R. Kalaba at the very beginning of dynamic programming [2] The greatest interest was paid to this approach in connection with the problem of the shortest path in a graph [3, 4, 5, 6, 7], and this interest is yet al..ive as it follows from the recent discussion in electronic conferences. The discussion gave some new titles in this field, unfortunately I have no possibility to classify or even to list them now. We consider here a general approach to enumeration of solutions rather ....

Dreyfus S. E., An appraisal of some shortest path algorithms, Operations Research, 1969, 17:3, 345--412.

....problem of finding minimum delay paths in networks with time dependent link delays. Among the works addressing the problem Klafszky[2] and Cooke Halsey[3] dealt with discrete delay functions whose domain and range are integers. Arbitrary functions for link delays were briefly treated by Dreyfus[4] and Ling et al. 5] who addressed only limited cases. Halpern[6] dealt with a more general case, in which waiting at nodes may be useful but is allowed only at certain times. The general problem was addressed recently in [7] where algorithms for finding the shortest path under various waiting ....

S.E. Dreyfus, "An Appraisal of some Shortest Path Algorithms," Operations Research 17 pp. 395-412 (1969).

....directly [4,5] and indirectly in the context of maximal flow[6,7] In this paper we address the shortest path problem without these restrictions, i.e. we allow arbitrary functions for link delays. In this respect this is the broadest generalization. Such a problem was briefly treated by Dreyfus[8] and Ling et.al. 9] which address only limited cases. The most direct treatment to date was done by Halpern[10] where arbitrary waiting times are also considered. In this latter work an algorithm is proposed for various waiting constraints, but this algorithm cannot be bounded by network topology ....

....(t , t ) D ik (t , t) Note that in some cases such a value may not exist. Consider, for example, a link (i , k) for which d ik (t) 1 100 t 10 t10 Then, for t = 0, t 0 inf D ik (0 , t) 11 but this value cannot be achieved for any t 0. This point was overlooked in previous works[8,9,10]. In the following we shall implicitly assume that delay functions are such that for all (i , k) and every time instant t a proper optimal waiting time t does exist. This property clearly holds for continuous functions while for piecewise continuous functions a sufficient condition for this ....

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S.E. Dreyfus, "An Appraisal of some Shortest Path Algorithms," Operations Research 17 pp. 395412 (1969).

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S. Dreyfus. An appraisal of some shortest path algorithms. Operations Research, 17:395--412, 1969.

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S. E. Dreyfus. An appraisal of some shortest-path algorithms. The Journal of the Operations Research Society of America, 17, 1969, pages 266--276.

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D. Dreyfus. An Appraisal of Some Shortest Path Algorithms. Technical Report RM-5433, Rand Corporation, Santa Monica, CA, 1967.

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