Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216-224. ACM PRESS, August 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... is a list of d integers for the d dimensional case) The labels assigned to the links of the network are also multi dimensional (a list of d 1 dimensional intervals) The class of networks supporting linear IRS (in which the intervals are not cyclic) is already known for the 1 dimensional case [FG94]. In this paper, we generalize this result and completely characterize the class of networks supporting linear MIRS (or MLIRS) for a given number of dimensions d. We show that by increasing d, the class of networks supporting MLIRS is strictly expanded. We also give a characterization of the ....

.... has done a survey of results concerning this method [Gav00] It has been proved that any network supports an SIRS and therefore an IRS [SK85, vLT87] The class of networks which support LIRS and SLIRS have also been characterized by Fraigniaud and Gavoille which excludes a large class of networks [FG94]. They de ne a class of graphs called lithium graphs and show that a network supports an LIRS if and only if its underlying graph is not a lithium graph. They also show that a network supports an SLIRS if and only if its underlying graph is not a weak lithium graph. A very interesting extension ....

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Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216-224. ACM PRESS, August 1994.

.... is a list of d integers for the d dimensional case) The labels assigned to the links of the network are also multi dimensional (a list of d 1 dimensional intervals) The class of networks supporting linear IRS (in which the intervals are not cyclic) is already known for the 1 dimensional case [FG94]. In this paper, we generalize this result and completely characterize the class of networks supporting linear MIRS (or MLIRS) for a given number of dimensions d.Weshow that by increasing d, the class of networks supporting MLIRS is strictly expanded. We also give a characterization of the ....

.... has done a survey of results concerning this method [Gav00] It has been proved that any network supports an SIRS and therefore an IRS [SK85, vLT87] The class of networks which support LIRS and SLIRS have also been characterized by Fraigniaud and Gavoille which excludes a large class of networks [FG94]. They define a class of graphs called lithium graphs and show that a network supports an LIRS if and only if its underlying graph is not a lithium graph. They also show that a network supports an SLIRS if and only if its underlying graph is not a weak lithium graph. A very interesting extension ....

[Article contains additional citation context not shown here]

Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216--224. ACM PRESS, August 1994.

....r partite graphs K n 1 ;n 2 ; n r with r 2, n i 1, the product n i=1 G i if the graph G i has an optimum LIRS for each i [KKR94] unit interval graphs [FG98] CHAPTER 2. PRELIMINARIES 20 (a) b) Kernel Electrons Figure 2.9: a) A lithium graph b) A weak lithium graph De nition 9. [FG94] A lithium graph is a connected graph with four connected subgraphs E 1 ; E 2 ; E 3 and K such that (i) each component E i , i = 1; 2; 3 has at least 2 vertices; ii) there is no edge connecting E i with E j for i; j = 1; 2; 3 and i 6= j; iii) each component E i , i = 1; 2; 3 is connected ....

....has at least 2 vertices; ii) there is no edge connecting E i with E j for i; j = 1; 2; 3 and i 6= j; iii) each component E i , i = 1; 2; 3 is connected with K by exactly one bridge. Fraigniaud and Gavoille have completely characterized the class of networks which support an LIRS. Theorem 5. [FG94] A graph G supports an LIRS if and only if it is not a lithium graph. We can verify that an interval graph cannot be a lithium graph. Therefore, based on the previous theorem, every interval graph supports an LIRS. The class of networks supporting an SLIRS has also been characterized by ....

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Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216-224. ACM PRESS, August 1994.

....address. If the routing technique is correctly defined, at least one arc with such a property exists. The arc label is then cut off from the destination address memorized in the message. Interval Routing Schemes (IRS) where first introduced by [22] and have then been widely studied (see [2, 5, 6, 7, 8, 9, 10, 11, 15, 17, 18, 19, 20, 21]) They are based on the definition of Interval Labeling Schemes (ILS) A k Gamma ILS assigns at most k intervals to each arc of the graph. Each node is labeled with a unique integer, and each interval on an arc contains the subset of nodes that are on a shortest path through this arc. In Boolean ....

....complexity for Distance Routing is O(D) steps. Proof. In the first step the sender has to compare the pair of the partial order, i.e. two numbers of O(log n) bits. This can be done in constant time. If x y or x y the path is already known. If x y 1 2 3 4 5 6 7 8 9 10 11 [2,1] 7,2] [4,6] Figure 8: 1 IRS on a DSPG. then other steps are required. First a coded path of length at most D has to be compared, i.e. D steps are sufficient. If one of the path is included in the other we are done, otherwise part of the coded path is cut off, and the levels of nodes s 0 , t 0 , I x , ....

P. Fraigniaud and C. Gavoille. A characterization of networks supporting linear interval routing. In Proceedings of ACM Conference on Principles of Distributed Computing, pages 216--224, 1994.

....(called Compact Routing techniques) whose principal aim is to optimize the trade off between the length of the paths and the space required for routing. Later we will discuss some of these techniques that search for shortest paths and try to optimize the space. These are Interval Routing Schemes ([BLT91, FG94, FJ88, FGS94, KKR93, SK85, LT87]) Prefix Routing Schemes ( BLT90] Compact Routing ( CL92] and Boolean Routing ( FGS94] We study Boolean Routing in detail, analyze the time space required for routing, and we will also show how this method can apply to a particular topology of interest: a chordal ring. We will also extend ....

....in the graph, and the intersection is disjoint. This means that a unique shortest path is defined from each source to each destination node. A 1 ILS can easily be defined for simple topologies such as trees, rings, hypercubes, meshes, some chordal rings, every arbitrary acyclic network, etc. [FGS94, FG94, SK85, LT87]) A 1 LILS is defined for a more restricted set of topologies [BLT91, FG94] A limit of this method is that in presence of node edge failures the routing information has to be redefined. Traffic congestion is also hard to avoid. A solution to the last problem could be the definition of ....

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P. Fraigniaud, C. Gavoille, "A Characterization of Networks Supporting Linear Interval Routing", ACM Conference on Principles of Distributed Computing, Los Angeles, California, 216-224, (1994).

....is correctly defined, at least one edge with such a property exists. The edge whose label has the longest prefix is chosen, the message is routed through this edge [1] and the prefix is cut from the destination address. Interval Routing Schemes (IRS) have been widely studied in literature (see [1, 6, 4, 7, 11, 16, 13]) They are based on the definition of Interval Labeling Schemes (ILS) A k Gamma ILS assigns at most k intervals to each edge of the graph. Each node is labeled with an integer, and each interval on an edge contains the subset of nodes that are on a shortest path through this edge. In Boolean ....

P. Fraigniaud and C. Gavoille. A characterization of networks supporting linear interval routing. In Proc. of ACM Conference on Principles of Distributed Computing, pages 216--224, 1994.

....that admit optimal k linear or circular interval labelings Even for the case k = 1, the answer is not well understood. It is known that all graphs have non optimal interval labelings [16] More recently, it was shown that all graphs have 1 circular labelings with exactly one interval per edge [5]; again, this result does not apply to optimal schemes. There are many familiar kinds of networks that are known to be in 1 LIRS, such as complete graphs, meshes, hypercubes, complete bipartite graphs [13] and proper interval graphs [6] Other networks such as trees, tori, and unit circular arc ....

....that as we proceed from proper interval graphs to the realm of all interval graphs, it is no longer possible to guarantee strictness and linearity simultaneously in interval labelings. For instance, the graph K 1;3 a non proper interval graph, does not admit any strict, 1 linear interval labeling [5]. Furthermore, the 3 fan graph, another non proper interval graph, does not admit any 1 linear interval labeling, whether strict or non strict. In general, it is interesting to ask whether every interval graph can be circularly labeled in a strict manner. We establish the answer in the ....

P. Fragniaud and C. Gavoille. A characterization of networks supporting linear interval routing. In Principles of Distributed Computing (1994), 216--224.

....that define minimum hop paths in an unweighted graph (in some cases these techniques can be extended to weighted graphs) and that minimize the time (to find the outgoing edge) and space (i.e. the number of bits) required for the routing. Examples of these techniques are Interval Routing Schemes [BLT91, FG94, FJ88, FGS94, KKR93, SK85, LT87], Prefix Routing Schemes [BLT90] Compact Routing [CL92] and Boolean Routing [FGS94] We restrict our attention to two of these techniques: Interval Routing Schemes and Boolean Routing. Interval Routing Schemes (IRS) are based on an Interval Labeling Scheme (ILS) An ILS consists of giving a ....

....the intersection is disjoint. This means that a unique path is defined from each source to each destination node. An optimal (i.e. shortest path) 1 ILS can easily be defined for simple topologies such as trees, rings, hypercubes, meshes, some chordal rings, every arbitrary acyclic network, etc. [FGS94, FG94, SK85, LT87]) In terms of memory this technique is better then the complete routing tables since a generic k ILS or k LILS require O(nkd log n) bits. Since k is often a constant, this techniques has better space complexity. Boolean Routing Schemes (BRS) define all the shortest paths among all pairs of nodes. ....

P. Fraigniaud, C. Gavoille, "A Characterization of Networks Supporting Linear Interval Routing", ACM Conference on Principles of Distributed Computing, Los Angeles, California, 216-224, (1994).

.... graph families, and it has also been extended later to a wider variety of networks, cf. vLT87, FJ89, Fre93, FGS96, FG98] When considering interval routing schemes for classes of graphs other than trees, a natural question is to identify which graphs admit interval routing along shortest paths [FG94, NS96, Fla97, EMZ97]. Another natural extension is to allow using more than one interval on each edge, raising the question of how many intervals are necessary to ensure shortest path routing, and how such a scheme can be implemented [FvLS98, GG98, GP98, GP99] For surveys of the many recent developments in this area ....

P. Fraigniaud, and C. Gavoille. A characterization of networks supporting linear interval routing. In 13 Annual ACM Symp. on Principles of Distributed Computing, pages 216-224, August 1994.

....approaches: ffl nd the best routing scheme for a given (small) class of networks, ffl nd all networks requiring small memory requirements for a given routing scheme. The second approach consists in graph characterization. For the interval routing scheme, this approach has been investigated in [1, 6, 9]. The rst approach consists in nding lower and upper bounds for given classes of graphs. The state of the art in the compact routing problem is presented in [8] for the general case, and in [12, page 88] for graphs of maximum degree d. In this paper we study the case of bounded degree networks for ....

....most two intervals of consecutive labels. Therefore, Irs(G 0 ) 2. However, another vertex labeling and another choice of shortest paths might decrease this number. For example, there exists two shortest paths from 5 to 1, but one would force to add one more interval on the arc (5; 6) I (5;6) [6] [ 1] We will see in the next paragraph a simple proof showing that the compactness of G 0 cannot be 1, and thus Irs(G 0 ) 2. 5 3 4 1 2 6 [2,3] 5,6] 3,4] 7,2] 6] 3] 7] 4,5] 1,2] 5,1] 2] 4] 3] 1] 4,7] 2,3] 5,6] 4] 7] 7,1] 5] 3] 4] 1] 2] 6] Figure ....

[Article contains additional citation context not shown here]

P. Fraigniaud and C. Gavoille, A characterization of networks supporting linear interval routing, in 13 Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM PRESS, ed., Aug. 1994, pp. 216224.

....find the best routing scheme for a given class and for a small class of networks, ffl find all networks requiring small memory requirements for a given routing scheme. The second approach consists in graph characterization. For the interval routing scheme, this approach has been investigated in [1, 3, 6]. The first approach consists in finding lower and upper bounds for given classes of graphs. The state of the art in the compact routing problem is presented in [5] for the general case, and in [7, page 88] for graphs of maximum degree d. In this paper we study the case of 3 regular networks for ....

P. Fraigniaud and C. Gavoille, A characterization of networks supporting linear interval routing, in 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM PRESS, ed., Aug. 1994, pp. 216--224.

....approaches: find the best routing scheme for a given (small) class of networks, find all networks requiring small memory requirements for a given routing scheme. The second approach consists in graph characterization. For the interval routing scheme, this approach has been investigated in [1, 5, 8]. The first approach consists in finding lower and upper bounds for given classes of graphs. The state of the art in the compact routing problem is presented in [7] for the general case, and in [11, page 88] for graphs of maximum degree d. In this paper we study the case of 3 regular networks for ....

....of a graph is NP complete [3] More precisely, Flammini in [2] showed that to determine if a graph has a compactness 2 is already NP complete. Note that for every graph G, Irs(G) n=2, since any destination set I e cannot contain more than n=2 non consecutive integers. 5 3 4 1 2 6 [2,3] [5,6] [3,4] 7,2] 6] 3] 7] 4,5] 1,2] 5,1] 2] 4] 3] 1] 4,7] 2,3] 5,6] 4] 7] 7,1] 5] 3] 4] 1] 2] 6] 7 M0 = 6,7) 2,7) 4,7) 1 1 1 7 1 0 0 3 0 1 0 5 0 0 1 1 Fig. 1. An example of interval routing with at most 2 intervals per link. On the right, a matrix of constraints of this graph. ....

[Article contains additional citation context not shown here]

P. Fraigniaud and C. Gavoille, A characterization of networks supporting linear interval routing, in 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM PRESS, ed., Aug. 1994, pp. 216--224.

....approaches: ffl nd the best routing scheme for a given (small) class of networks, ffl nd all networks requiring small memory requirements for a given routing scheme. The second approach consists in graph characterization. For the interval routing scheme, this approach has been investigated in [1, 6, 9]. The rst approach consists in nding lower and upper bounds for given classes of graphs. The state of the art in the compact routing problem is presented in [8] for the general case, and in [12, page 88] for graphs of maximum degree d. In this paper we study the case of bounded degree networks ....

....most two intervals of consecutive labels. Therefore, Irs(G 0 ) 2. However, another vertex labeling and another choice of shortest paths might decrease this number. For example, there exists two shortest paths from 5 to 1, but one would force to add one more interval on the arc (5; 6) I (5;6) [6] [ 1] We will see in the next paragraph a simple proof showing that the compactness of G 0 cannot be 1, and thus Irs(G 0 ) 2. 5 3 4 1 2 6 [2,3] 5,6] 3,4] 7,2] 6] 3] 7] 4,5] 1,2] 5,1] 2] 4] 3] 1] 4,7] 2,3] 5,6] 4] 7] 7,1] 5] 3] 4] 1] 2] 6] 7 Figure 1: An example of ....

[Article contains additional citation context not shown here]

P. Fraigniaud and C. Gavoille, A characterization of networks supporting linear interval routing, in 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM PRESS, ed., Aug. 1994, pp. 216224.

....contains the label of its own node is called strict. It may happens that the number of intervals increases by 1 if we decide to split the intervals containing the label of its own node in order to make the labeling strict. Routing strategies that do not require shortest paths have been studied in [BvLT91, FG94a, SK85], where the authors give a complete characterization of the graphs requiring a small number of intervals for different restricted versions of interval routing. A hardware solution to the routing problem based on intervals was proposed by INMOS with its C104 chip (see [MTW93] Given a graph G of n ....

....Irs(G Delta ) 2 Omega Gamma n= log n) 2 ) More precisely, if we let n(k) denote the number of nodes of the smallest network G for which Irs(G) k, we show that 2k 1 n(k) 12k Gamma 11 for every integer k 2, and thus that Irs(n) n=12. The lower bound of 2k 1 on n(k) was obtained in [FG94a] with the following simple argument: by the pigeon hole principle, any integer labeling on n Gamma 1 nodes can give at most b(n Gamma 1) 2c wrap around intervals of consecutive integers. This lower bound on n(k) proves that our bound on Irs(n) is asymptotically tight. Now let Irs(n; Delta) ....

Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In ACM PRESS, editor, 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216--224, August 1994.

....replicated n times in an n node network, such a large local memory requirement implies a global information of Theta(n 2 log d) bits for the whole network, just for routing. Many authors propose methods to compress the routing tables (see for instance [16, 17, 32] Among them, interval routing [3, 10, 11, 33, 32] was chosen by Inmos for its C104 routing chip [6] A router finds the direction to forward a message by determining the interval (a set of consecutive addresses) that contains the destination address of the message, each interval being associated to one particular direction; eventually, more than ....

P. Fraigniaud and C. Gavoille, A characterization of networks supporting linear interval routing, in 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM PRESS, ed., Aug. 1994, pp. 216--224.

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Pierre Fraigniaud and Cyril Gavoille. A characterization of networks supporting linear interval routing. In 13 th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 216-224. ACM PRESS, August 1994.

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