J. W. Suurballe, R. E. Tarjan. A Quick Method for Finding ShortestPairs of Disjoint Paths. Networks, 14:325336, 1984  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of minimum energy disjoint path routing has not been looked at before. However, when taken as separate problems, considerable work has been done on energy e#cient routing in wireless networks [2, 3, 4, 5, 6, 17, 18, 22, 28, 29] as well as disjoint path routing in both wired and wireless networks [10, 11, 12, 13, 14, 15, 23, 24, 25, 27]. The energy e#ciency aspect of our work builds upon that of Wieselthier et al. 2] on energy e#cient broadcasting and multicasting in wireless networks. Although they present only heuristic solutions to the problem (the problem was subsequently proven to be NP Hard [3, 4, 5, 6] their work ....

....paths problem. The problem of finding k node (link) disjoint source destination paths in a network, is a well studied problem in graph theory. Polynomial O(kN ) running time algorithms that find minimum weight k node (link) disjoint source destination paths have existed for decades [10, 11, 12]. While these algorithms do not address the minimum energy disjoint paths problem, they serve as basic building blocks for the algorithms developed in this paper. The remainder of the paper focuses on developing optimal polynomial running time algorithms for finding minimum energy disjoint paths. ....

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J. W. Suurballe and R.E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths," Networks, 14 (1984) pp. 325-336.

....the connectivity information, a set of transmission paths are generated for each demand pair by the k shortest path method [Dre69] and then loop variables are generated by identifying disjoint paths from this set of paths. We note that there are better algorithms to generate node disjoint paths [Sur74,SuT84] which we have not implemented in our present study) An approach to solve the nonsmooth dual problem is to use the subgradient methods discussed in [Po169, HWC74] However, in general, the subgradient algorithms described there are not ascent algorithms. An alternative procedure to solve the ....

J. W. Suurballe & R. E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths," Networks, vol. 14, pp. 325-336, 1984. 0676 22.4.5.

....between sources and destinations in an offline pre computation phase. While this simplifies the online process of comparing relative costs of different primary backup route candidates, it is still an inelegant bruteforce technique that could potentially be optimized. Efficient solutions exist [42, 43, 39] for the problem of finding the minimum cost disjoint route pair (MinCostPP) where the cost metric is simply the sum of weights for each link in the disjoint route pair. The problem handled by LCBR has two important differences with the above problem. First, our cost metric cost(G) does not lend ....

J.W. Suurballe and R.E. Tarjan. A quick method for finding shortest pairs of disjoint paths. Networks, 14:325--336, 1984.

.... number of paths whenever the k successively shortest link disjoint path algorithm selects a path which blocks other potential paths (well illustrated in [3] using the generalised trap topology) or even worse they may overestimate the number of link disjoint paths (e.g. 2] On the other hand, [5, 6, 7] concentrate on finding only a pair of disjoint paths between a given pair of nodes, by optimizing the physical length of paths: 5] finds the shortest pair of node disjoint paths but cannot be applied at the span (physical) level (e.g. physical links sharing a common conduit) 6] finds a pair of ....

.... illustrated in [3] using the generalised trap topology) or even worse they may overestimate the number of link disjoint paths (e.g. 2] On the other hand, 5, 6, 7] concentrate on finding only a pair of disjoint paths between a given pair of nodes, by optimizing the physical length of paths: [5] finds the shortest pair of node disjoint paths but cannot be applied at the span (physical) level (e.g. physical links sharing a common conduit) 6] finds a pair of disjoint paths between a given pair of nodes taking into consideration any span sharing by links, however the solution for networks ....

J.W. Suurballe, R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths", Networks 14, pp. 325-336, 1984.

....in [6] of different topologies for the complete and partially linkdisjoint case yields that the commonly used paths have to be equal to paths of the original single shortest path to provide feasible shared hops. The algorithmic approach in this paper is based on the Suurballe algorithm ( 7] [8]) and uses an efficient Dijkstra implementation [9] for the shortest path algorithms. The partially disjoint paths are collected in a specially designed data structure during the execution of the modified algorithm. It is called partially disjoint path label (PDPL) Section II introduces the PDPL ....

....A. Correctness For the correctness discussion, two questions are addressed: Do the changes applied to the Suurballe algorithm influence the correctness of finding two completely disjoint paths And secondly: Are all partially disjoint paths synthesised during the algorithm execution In [8] the correctness of the Suurballe algorithm is proven. The changes made are: Additional Label: The PDPL results in no changes of the general procedure of the original algorithm. It remains passive during the execution of the algorithm and is described in detail in Section II. It has no influence ....

J.W. Suurballe and R. E Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336, 1984.

....the link is 2 critical. We approximate the problem of determining the path set that minimizes interference to the problem of determining the path set (two disjoint paths) that minimizes the sum of the criticality indices of the links in the path set. We use the algorithm of Suurballe and Tarjan [14] to solve this disjoint path problem as outlined in Section IV. III. PROBLEM STATEMENT Let # # # # # # # # # describe the given network, where # is the set of routers (nodes) and # the set of links (edges) and # the bandwidth of the links. Let # denote the number of nodes and the number of ....

....link criticality index as just computed. Also, there is a supply of 2 units at node and a demand of 2 units at node . Any standard min cost flow algorithm can be used to solve this problem. A very fast and simple algorithm for this min weight (shortest) disjoint path problem is presented in [14] and is given here for the sake of completeness. The algorithm works as described below. In the algorithm, it is assumed that the length of any link is set to its criticality index, Algorithm shortest disjoint path . Step 1. Determine the shortest path tree from node .Let ....

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J. W. Suurballe, R. E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths", Networks, Vol. 14, 1984.

....ABW (say w units) is to be allocated (or deallocated) the amount of BBW to be allocated (or deallocated) in any shared path protection schemes depends on many factors including which links are used by the corresponding AP. This is why shortest pair of path (or SPP) algorithms such as the one in [2] can no longer guarantee minimum TBW allocation for a given connection establishment request, let al..one those based on the so called active path first or APF heuristic [3] In [1] an approach which uses Integer Linear Programming (ILP) to determine a pair of AP and BP, called Sharing with ....

....for use as the BP. This ultra fast algorithm takes less than 0. 05 seconds for each request in a 70 node network (approximately 20 times faster than the quick method [1] One alternative to DPIM M A is to remove all links with R e w, and find a shortest pair of paths using the SPP algorithm in [2]. Then, the shorter of the two can be used as the AP and the other as the BP. Note that, once the BP is chosen, however, minimal BBW (# w) will be allocated on each link along the BP in the proposed DPIM schemes (as to be described next) This implies that some of the links removed earlier ....

J.W. Suurballe and R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336, 1984.

....problem (P) or exploration of any other better algorithm is beyond the scope of the present paper. A component that feeds into the model is the generation of candidate cycle paths. It may be noted that Suurballe and Tarjan have developed an algorithm for generating shortest pair of disjoint paths [11, 12]; this, however, helps in generating only the shortest cycle, not a set of candidate cycles. In our case, we have implemented a simple procedure by extending the K shortest path algorithm where the K paths generated by the K shortest path algorithm are compared to each other to filter out common ....

J. W. Suurballe, R. E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths", Networks, Vol. 14, pp. 325-336, 1986.

....along a BP depends on many factors including which links are used by the corresponding AP. This is why APF is not ideal as it does not consider (nor cares about) the potential cost along the BP yet to be chosen when selecting the AP. The shortest pair of path (or SPP) algorithm such as the one in [7] does not take possible BBW sharing into consideration either in that it IEEE INFOCOM 2002 4 essentially assumes that the cost of each link on a BP is also equal to #. Therefore, such an SPP algorithm may be useful only when each edge node cannot obtain a better (i.e. more accurate) estimation of ....

J.W. Suurballe and R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336., 1984.

....each node into two halves with a virtual directed link in between. IEEE INFOCOM 2002 2 (or deallocated) in any shared path protection schemes depends on many factors including which links are used by the corresponding AP. This is why shortest pair of path (or SPP) algorithms such as the one in [2] can no longer guarantee minimum TBW allocation for a given connection establishment request, let al..one those based on the so called active path first or APF heuristic [3] In [1] an approach which uses Integer Linear Programming (ILP) to determine a pair of AP and BP, called Sharing with ....

....for use as the BP. This ultra fast algorithm takes less than 0. 05 seconds for each request in a 70 node network (approximately 20 times faster than the quick method [1] One alternative to DPIM M A is to remove all links with # # ##, and find a shortest pair of paths using the SPP algorithm in [2]. Then, the shorter of the two can be used as the AP and the other as the BP. Note that, once the BP is chosen, however, minimal BBW (# #) will be allocated on each link along the BP in the proposed DPIM schemes (as to be described next) This implies that some of the links removed earlier ....

J.W. Suurballe and R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336., 1984.

....the cost of dual hub architecture which will be discussed in section 4. In section 2, we give some problem formulations. Section 5 summarizes the work. 2 Problem Formulations Here we give the mathematical programming formulations and definitions of some problems. These are as given in [CDSW] and [ST]. 2.1 Ring Routing Problem Given a network G with node set N , an arbitrary subset M of N , and directed links with cost distances weights c ij (some c ij are infinite, indicating that the directed link does not exist) the ring routing problem can be formulated as follows: RRP(G;M ) ....

....graph. 3 if (u; v) is an edge then (v; u) is not an edge 10 hub 1 hub 2 v v v 1 2 n 2 Figure 3: Dual Hub Architecture 4. 1 An Algorithm for Shortest Pairs Problem Given a directed asymmetric graph G = V; E) with a special node s and cost c(u; v) on each edge, we describe an algorithm([ST]) to find two link disjoint paths from s to all other vertices. We use following steps, consisting of two passes of Dijkstra s algorithm [D 59] Step 1: Find a shortest path tree T rooted at s. Such a tree contains, for every vertex v, a shortest path from s to v. Compute d(s; v) the shortest ....

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J. Suurballe, R. Tarjan,"A Quick Method for Finding Shortest Pairs of Disjoint Paths", Networks, Vol. 14,1984. 14

....the link is 2 critical. We approximate the problem of determining the path set that minimizes interference to the problem of determining the path set (two disjoint paths) that minimizes the sum of the criticality indices of the links in the path set. We use the algorithm of Suurballe and Tarjan [13] to solve this disjoint path problem as outlined in Section IV. III. PROBLEM STATEMENT Let G = N , L, B) describe the given network, where N is the set of routers (nodes) and L the set of links (edges) and B the bandwidth of the links. Let n denote the number of nodes and m the number of ....

....link criticality index as just computed. Also, there is a supply of 2 units at node s and a demand of 2 units at node d. Any standard min cost flow algorithm can be used to solve this problem. A very fast and simple algorithm for this min weight (shortest) disjoint path problem is presented in [13] and is given here for the sake of completeness. The algorithm works as described below. In the algorithm, it is assumed that the length of any link l is set to its criticality index, w(l) Algorithm shortest disjoint path(s, d) Step 1. Determine the shortest path tree from node s. Let d i ....

[Article contains additional citation context not shown here]

J. W. Suurballe, R. E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths", Networks, Vol. 14, 1984, pp. 325-336. 11

.... two different costs, one for the first path, one for the second one (see Li et al. LMSL92] The problem in which a maximum number of disjoint paths must be found between two nodes with a total length smaller than a given value M is also NP complete (see Itai et al. IPS82] Suurballe and Tarjan [ST84] propose an algorithm to solve the problem of finding two minimum cost paths between nodes o k and d k . They show that, if all arc costs are positive, these two paths can be obtained in O(mlog (1 m n ) n) time using two iterations of the algorithm of Dijkstra. We outline this algorithm here ....

....46 CHAPTER 1: CLASSIFICATION costs in the transformed network are all positive, and the algorithm of Dijkstra can be executed a second time. To solve the very close problem in which two arc disjoint paths are required from one source node o k to all other nodes of the graph, Suurballe and Tarjan [ST84] propose a method whose running time is the same as that of the above algorithm. This method is a variant of the shortest path algorithm of Dijkstra, since a set of node labels is updated at each step. At the end of the algorithm, the labels give the total cost of the two paths from o k to each ....

J.W. Suurballe and R.E. Tarjan. A quick method for finding shortest pairs of disjoint paths. Networks, 14:325--336, 1984.

....fact that, for any vertex (edge) redundant graph, there exists a pair of vertex (edge) disjoint paths between any two vertices is a consequence of Menger s theorem [57] 43] Fig. 4 shows edge disjoint and vertex disjoint paths. Such approaches are presented in [59] for edge disjoint paths and in [56], 72] for vertex disjoint paths, usually associated with some sort of shortest path selection. Applications of these techniques to networks are presented in [4] 51] However, none of these schemes guarantees trees in the case of a single Fig. 4. Pair of link disjoint paths and pair of ....

J. W. Suurballe and R. E. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336, 1984.

....the procedure to produce HB . If the branchwidth of HA and HB are at most the branchwidth of G then (A; B) can be used for an optimal branch decomposition. An example of an initial split is illustrated by Figure 5. We used an augmenting path algorithm by Suurballe [5] and Suurballe and Tarjan [6] to find cycles in M (G) Since there may exist separations of G with order equal to the branchwidth of G but are not found in any optimal branch decomposition of G and there is no algorithm guaranteed to find all cycles of a graph with a particular length, we rely on the edge contraction method ....

J. W. Suurballe and R. E. Tarjan. A quick method for finding shortest pairs of disjoint paths. Networks, 14:325--336, 1984. 13

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J. W. Suurballe, R. E. Tarjan. A Quick Method for Finding ShortestPairs of Disjoint Paths. Networks, 14:325336, 1984

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J. W. Suurballe, R. E. Tarjan, A Quick Method for Finding Shortest Pairs of Disjoint Paths, Networks, Vol. 14, pp. 325-336, 1986.

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J.W. Suurballe and R.E. Tarjan, A quick method for finding shortest pairs of disjoint paths, Networks, Vol. 14(1984), pp. 325--336.

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J. Suurballe and R. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths," Networks, vol. 14, pp. 325--336, 1984.

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J. Suurballe and R. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, pp. 325--336, 1984.

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J.W. Suurballe and R.E. Tarjan. A quick method for finding shortest pairs of disjoint paths. Networks, 14:325--336, 1984.

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J.W. Suurballe, R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths", Networks 14, pp. 325-336, 1984.

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J. W. Surrballe and R. E. Tarjan, "A Quick Method for Finding Shortest Pairs of Disjoint Paths," Networks, 14(2):325-336, 1984.

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J. Suurballe and R. Tarjan, "A quick method for finding shortest pairs of disjoint paths," Networks, vol. 14, 1984, pp.325-336.

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J.W. Suurballe, R.E. Tarjan, "A quick method for finding shortest pairs of disjoint paths", Networks 14, pp. 325-336, 1984.

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