R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with interval routing, at most O(d) space is needed at a node, where d is the node s degree. In general, d is smaller than n, the size of the network, and we say that the routing information stored at a node as required by interval routing is compact . See the survey by Tan and van Leeuwen [7] for an overview of the field of compact routing. One of the main questions in interval routing research is that given G, how to label its nodes and edges so that all the routing paths are shortest paths, where G represents either a specific kind of graphs or arbitrary graphs (general networks) ....

.... the routing paths are shortest paths, where G represents either a specific kind of graphs or arbitrary graphs (general networks) A successful labeling satisfying the condition constitutes an optimum interval routing scheme (IRS) For a number of specific graphs, optimum IRSs are known to exist [7]. What about arbitrary graphs Ru zicka answered this in the negative way by constructing a graph that has no optimum IRS [4] In practice, it might not always be necessary to insist on shortest path routing, as long as the paths are not too far from the optimal. Santoro and Khatib have proposed ....

R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.

....Each routing table requires O(n) space in an n node network, which is not ecient (and even feasible) for large networks of computers. The methods to reduce the amount of space needed at each node have been intensively studied and there are many techniques to compress the size of routing tables [FJ88, FJ89, ABNLP90, TvL95]. The general idea is to group the destination addresses that correspond to the same outgoing link (at a node) and to encode the group so that it is easy to verify if a given destination address is in the group or not. A well known solution is to use intervals as groups of destination addresses. ....

Richard B. Tan and Jan van Leeuwen. Compact routing methods: A survey. In Proceedings of Colloquium on Structural Information and Communication Complexity (SICC'94), SCS, Carleton University, Ottawa, pages 99-109, 1995. 16

....Each routing table requires O(n)spaceinann node network, which is not e#cient (and even feasible) for large networks of computers. The methods to reduce the amount of space needed at each node have been intensively studied and there are many techniques to compress the size of routing tables [FJ88, FJ89, ABNLP90, TvL95]. The general idea is to group the destination addresses that correspond to the same outgoing link (at a node) and to encode the group so that it is easy to verify if a given destination address is in the group or not. A well known solution is to use intervals as groups of destination addresses. ....

Richard B. Tan and Jan van Leeuwen. Compact routing methods: A survey. In Proceedings of Col loquium on Structural Information and Communication Complexity (SICC'94), SCS, Carleton University, Ottawa, pages 99--109, 1995. 16

....routing if the routing tables are associated with the edges leaving from each vertex (in fact, the degree of any vertex is usually much smaller than N ) and such tables are made more compact by establishing some relations among the vertices associated with an edge. See the survey contained in [1]. An approach to obtain such a reduction is interval routing, introduced by Santoro and Khatib [2] that has found industrial applications in the C104 router chip used in the INMOS T9000 transputer design [3, 4] A rather complete review is contained in [5] This method assigns a distinct label ....

....can be realized, in IRS, with a simple variation of a binary search on the sequence of interval extremes. In MIRS, this strategy is no longer possible because the pairs I,F related to different intervals may overlap. For instance, we can have the following intervals on the same edge: 8, 5] [7, 15, 1], 5, 2] The search for a destination x can then be organized as follows: i) preprocessing) in each vertex, build an ordered list L of the values of I and F of all the intervals; ii) perform a binary search of x into L, to reduce the candidate intervals to the ones for which I F ; ....

Tan, R. and van Leeuwen, J. (1997) Compact Routing Methods: a Survey. Research Report (Proc. Research Meeting on Structural Information and Communication Complexity), UniversitadiSiena.

....= 0 and l j mod l j Gamma1 = 0, 1 j h; 4. k = 1 and c = n for any n Theta n grid Gn Thetan ; 5. k = 2 and c = 2n for any n Theta n torus Tn Thetan ; 6. k = 1 and c = n=2 for any hypercube H d of d = log n dimensions. Proof. All such networks admit a shortest path k IRS with the claimed k [36, 17]. Let us then consider each network separately. 1. Trees: As observed above, any tree admits a shortest path 1 IRS and, as the tree has a unique simple (shortest) path connecting any source destination pair, the path system induced by any k IRS is the same. 2. Rings: Consider a ring Rn of n ....

J. van Leeuwen and R.B. Tan. Compact routing methods: A survey. In 1st Colloquium on Structural Information and Communication Complexity (SICC), pages 71--93. Carleton University Press, 1994. 24

....to each link; in particular, a 2 IRS, i.e. a scheme associating at most 2 intervals for each edge, is proposed for 2 dimensional doubly wrapped grids. Other characterization and computational complexity results related to k IRS and compact routing schemes can be found in [9,13,14,15,16,17,18] see [19] for a survey) Multi label k IRS have been implemented on the latest generation of INMOS Transputer C104 Router chips [2] The time complexity of devising minimal space k IRS has been first investigated in [18] where it is shown that given an integer k and a weighted network G, the problem of ....

J. van Leeuwen and R.B. Tan. Compact routing methods: a survey. In B. Mans P. Flocchini and N. Santoro, editors, Colloquium on Structural Information and Communication Complexity (SICC'94), pages 71--93. Carleton University Press, 1994.

....2 See also [68] A good overview of many of the issues involved here has been made by Mohring [64] 15.2 Interval Routing Schemes Consider a distributed processor network, in which processors want to send messages to each other. Research has been done on so called compact routing methods (see [102] for an overview) methods in which processors decide over what link to forward messages that take relatively little space for storing such routing information. One type of these methods is interval routing. In the case of k interval routing, each processor is numbered with a unique integer, and ....

J. van Leeuwen and R. B. Tan, Compact routing methods: A survey, Technical Report UU-CS-1995-05, Department of Computer Science, Utrecht University, Utrecht, 1995.

.... Robertson and Seymour, in their graph minors series (for a survey, see [12] A simpler proof of Theorem 1, with a much better bound for c H , was given by Robertson, Seymour, and Thomas in [14] where c H 20 2(2jV (H)j 4jE(H)j) 5 (additional results on general bounds for c H can be found in [11]) Much research has been done towards proving tighter bounds for c H when H is restricted to certain families of planar graphs. Such kind of bounds have been found in [1] trees) 9] cycles and subgraphs of cycles) 5] disjoint copies of K 3 ) and [3] graphs that are minors of a circus ....

J. van Leeuwen and R. B. Tan, Compact routing methods: A survey, Technical Report UU-CS-1995-05, Department of Computer Science, Utrecht University, Utrecht, 1995.

....with interval routing, at most O(d) space is needed at a node, where d is the node s degree. In general, d is smaller than n, the size of the network, and we say that the routing information stored at a node as required by interval routing is compact . See the survey by Tan and van Leeuwen [3] for an overview of the field of compact routing. One of the main questions in interval routing research is that given G, how to label its nodes and edges so that all the routing paths are shortest paths, where G represents either a specific kind of graphs or arbitrary graphs (general networks) A ....

.... the routing paths are shortest paths, where G represents either a specific kind of graphs or arbitrary graphs (general networks) A successful labeling satisfying the condition constitutes an optimum interval routing scheme (IRS) For a number of specific graphs, optimum IRSs are known to exist [3]. What about arbitrary graphs Ruzicka answered this in the negative way by constructing a graph that has no optimum IRS [1] In practice, it might not always be necessary to insist on shortest path routing, as long as the paths are not too far from the optimal. Santoro and Khatib have proposed an ....

R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.

....over the link over which they are sent. We also concentrate mainly on results pertaining to networks with shortest path or minimum cost routing. A preliminary version of this paper was presented in the Colloquium on Structural Information and Communication Complexity at Carleton University ([LT94]) 2 Interval Routing The idea behind an Interval Labeling Scheme (ILS) is to group the destination nodes belonging to the same outgoing link in a cyclic interval (modulo n) This is done by first labeling all the nodes in the graph with some unique integer in [0. n 1] Each link is then ....

J. van Leeuwen and R. Tan, Compact Routing Methods: A Survey, Proceedings Colloquium on Structural Information and Communication Complexity (SICC '9d), Carleton University Press, 1994.

....identified useful classes of network topologies for which the shortest path information can indeed be coded more succinctly at the nodes, usually with a different regime for the routing decisions as well. Tables typically use in the order of dlogn bits per node, hence O (elogn) bits total. See [41, 42] for an overview. We will consider the most popular of these techniques called interval routing and will demonstrate that for general networks it is not likely to work as well. Interval routing is based on a suitable naming scheme for the nodes and edges in a network. A node label is any ....

J. van Leeuwen, R..B. Tan. Compact routing methods: a survey, Techn. Report R. UU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.

....down to hopefully O(d) items, where d is the degree of the node. Many common networks such as trees, rings and hypercubes have efficient Interval Routing Schemes. This compact routing method has also found practical applications, as it has been implemented by INMOS on its Transputer chips (see [14] and [10] for surveys) This work has been supported in part by the EU ESPRIT Long Term Research Project ALCOMIT under contract N. 20244 and IASI CNR. y Dipartimento di Matematica Pura ed Applicata, University of L Aquila, via Vetoio loc. Coppito, I 67010 L Aquila, Italy. Email: ....

J. van Leeuwen and R. B. Tan. Compact Routing Methods: A Survey. In Proc. Colloquium on Structural Information and Communication Complexity (SICC'94), Carleton University Press (1994).

....open problems. We assume that the nodes and links do not fail and that messages eventually arrive over the link over which they are sent. We also concentrate mainly on results pertaining to networks with optimal shortest path or minimum cost routing. A preliminary version of this paper appeared as [TvL94] at the Colloquim on Structural Information and Communication Complexity. 2 Interval Routing The idea behind an Interval Labeling Scheme (ILS) is to group the destination nodes belonging to the same outgoing link in a cyclic interval (modulo n) This is done by first labeling all the nodes in the ....

R. B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Proceedings of the Colloquim on Structural Information and Communic ation Complexity (SICC 94), Carleton University Press (1994).

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R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.

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R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995.

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Tan, R. B.---van Leeuwen, J.: Compact Routing Methods: A Survey. Proceedings of Colloquium on Structural Information and Communication Complexity (SICC'94), SCS, Carleton University, Ottawa, 1995, pp. 99--109.

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