Tse, S. S. H. and Lau, F. C. M. (1997) A lower bound for interval routing in general networks. Networks, 29, 49--53.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... to a wider variety of networks, cf. 124, 53,50,39,45,33] 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 [45,93,38,32] or to minimize the length of the longest route [34,121, 82]. 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 [80,42,91,57,63] It is worth noting that an interval routing scheme, once computed ....

S.S.H. Tse and F.C.M. Lau. A lower bound for interval routing in general networks. Networks, 29(1):49--53, Jan. 1997.

.... tables and note that any assignment of more than n destinations to an edge can be represented by at most n intervals on the edge) But it appears to be much harder to decide whether a network admits an optimal k IRS for some smaller value of k (see [16] Some results exist that show that 1 IRS [34, 35, 37] and 2 IRS schemes [38] cannot even come close to optimality for some classes of net works. Indeed, when no specific assumption about the topology of a network is made, interval routing does not seem to reduce the space requirement for the routing information in the nodes significantly. In this ....

S.S.H. Tse, F.C.M. Lau. A lower bound for interval routing in general networks. Techn. Report, Dept. of Computer Science, The University of Hong Kong, Hong Kong, 1994.

....and strict LILSs, of G, respectively. For Interval Routing, upper and lower bounds for OPT ILS (G) are known. An upper bound of 2D was presented in [SK85] where D is the diameter of the network, and a lower bound of 1:5D 0:5 was presented in [R91] and was later improved to 1:75D Gamma 1 in [TL94]. In our proof we use the following characterization of graphs that admit a valid LILS. Definition6. FG94] A lithium graph is a connected graph G = V; E) V = V 0 [V 1 [V 2 [V 3 , the sets V i are disjoint, jV 1 j; jV 2 j; jV 3 j 2 and there is a unique edge connecting V 0 with each of the ....

Tse, S. S.H., Lau, F. C.M.: A Lower Bound for Interval Routing in General Networks. Tech. Rep. TR-94-09, Department of Computer Science, University of Hong Kong (1994). This article was processed using the L a T E X macro package with LLNCS style

....1 vertices. Corollary 4 implies that the 1 time of a graph of diameter D is at most 2D. However, we do not know if this upper bound is tight. For k = 1 and k = 2, we summarize below the most recent results about the k time as a function of the diameter. Theorem 9 (Tse, Lau) For every integer D, [29] there exists a graph of diameter D of 1 time at least 7D=4 Gamma 1; 28] there exists a graph of diameter D of 2 time at least 5D=4 Gamma 1. Many experimental results can be found in [17] where the author studies the 1 time of some graphs of compactness at least 2 like the torus. Still in ....

Svio S. H. Tse and Francis C. M. Lau. A lower bound for interval routing in general networks. To appear in Journal Network, June 1996.

....graph admits a valid ILS, under which every message will traverse a path of length at most 2D, where D is the diameter of the graph. In [R91] a lower bound of 1:5D 0:5 was proved for the maximal length of a path a message traverses under Interval Routing, and this bound was recently improved ([TL94]) to 1:75D Gamma 1. Linear Interval Routing ( BLT91] is a restricted variant of Interval Routing which uses only linear intervals (i.e. wraps around are not allowed) Its advantage over Interval Routing is that it can be used for routing in dynamic networks, where insertion and deletion of ....

....and strict LILSs, of G, respectively. For Interval Routing, upper and lower bounds for OPT ILS (G) are known. An upper bound of 2D was presented in [SK85] where D is the diameter of the network, and a lower bound of 1:5D 0:5 was presented in [R91] and was later improved to 1:75D Gamma 1 in [TL94]. In our proof we use the following characterization of graphs that admit a valid LILS. Definition 6 ( FG94] A lithium graph is a connected graph G = V; E) V = V 0 [ V 1 [ V 2 [ V 3 , the sets V i are disjoint, jV 1 j; jV 2 j; jV 3 j 2 and there is a unique edge connecting V 0 with each of ....

Tse, S. S.H., Lau, F. C.M.: A Lower Bound for Interval Routing in General Networks. Tech. Rep. TR-94-09, Department of Computer Science, University of Hong Kong (1994).

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Tse, S. S. H. and Lau, F. C. M. (1997) A lower bound for interval routing in general networks. Networks, 29, 49--53.

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S.S.H. Tse and F.C.M. Lau, "A Lower Bound for Interval Routing in General Networks", Networks, 29:49-53, 1997.

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Tse, S.S.H. and Lau, F.C.M. (1997) A Lower Bound for Interval Routing in General Networks. Networks, 29, 49--53.

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S.S.H. Tse and F.C.M. Lau, A lower bound for interval routing in general networks. Networks, to appear.

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S.S.H. Tse and F.C.M. Lau, "A Lower Bound for Interval Routing in General Networks", Networks, 29:49--53, 1997.

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S.S.H. Tse and F.C.M. Lau, A LowerBound for Interval Routing in General Networks, Networks, to appear.

....of this paper is to present a lower bound on the longest path in 1 label interval routing. Interestingly, only one upper bound exists, which is the 2D upper bound for 1IRS according to Santoro and Khatib [6] In this paper, the lower bounds are 2D 3 and 2D o(D) improving the result 7 1 in [8]. Since there is no any better algorithm yielding an upper bound less than 2D, and no any lower bound higher 3 there are still rooms for improvement on both sides. 2 Properties The network in question is a connected graphs, G = V,E) where V is the set of nodes, and E the set of the edges. ....

....if and only if A is a proper subset of every interval set containing it. If an interval B contains an interval B # , B # is called a subinterval of B. We use the notation u# v# w, to denote the cyclic ordering of node numbers, for u, v, w # 0, n 1 . Naturally, 0# 1# . # n 0. Asin[8],the expression u v, w # x means that v and w are contained in some interval and that they are ordered after u and before x, but the order of v and w is not shown. Property 1 (Completeness) The set of interval labels for edges directed from a node u is complete. That is, #u # V , V ....

S.S.H. Tse and F.C.M. Lau, A lower bound for interval routing in general networks, Networks, 29(1): 49--53, 1997.

....log n) labels [2] One could hope for a narrower gap in the future. Since many graph algorithms perform better in planar graphs than in non planar graphs, we would like to know how interval routing would perform in planar graphs. Several lower bounds have been proposed for non planar graphs (e.g. [10, 11]) but for planar graphs, there exists only one lower bound result 3 2 D 1 which is due to Ru zicka [7] 2 His proof is based on a simple planar graph, which he later referred to as a globe graph [4] see Figure 2 for an example) In [12] the authors improved the bound to 3 2 D 1 2 . ....

S.S.H. Tse and F.C.M. Lau, "A Lower Bound for Interval Routing in General Networks", Networks, 29:49--53, 1997.

....this paper is d 3 2 De Gamma 1. Let L(u; v) denote the interval label for the edge that goes from u to v. A node u is said to be contained in [p; q] if (1) p u q for p q, or (2) p u n Gamma 1 or 0 u q, otherwise. The following are some essential properties of a valid labeling scheme [6]. Property 1 (Completeness) The set of interval labels for edges directed from a node u is complete. That is, every other node (6= u) in the graph must be contained in some interval at u. Property 2 (No ambiguity) The interval labels for edges directed from a node u are disjoint. That is, every ....

.... [62,65] 66,61] 63,65] 0,62] 64,71] 61,63] 65,60] 61,64] 66,60] 61,65] 61,65] 66,60] 66,71] 0,65] 67,71] 0,66] 68,71] 0,67] 69,71] 66,68] 70,65] 66,69] 71,65] 66,70] 36,40] 41,35] 37,40] 41,36] 38,40] 0,37] 39,71] 36,38] 40,35] 36,39] 41,35] 36,40] 1,5] [6,0] [2,5] 6,1] 3,5] 0,2] 4,71] 0,3] 5,71] 0,4] 6,71] 0,5] 6,10] 11,5] 7,10] 11,6] 8,10] 0,7] 9,71] 6,8] 10,5] 6,9] 11,5] 6,10] A C A k k Figure 2: A labeled G (k = 3; s = 14) It is easy to check that the labeling satisfies the necessary conditions for a valid IRS. ....

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Tse, S.S.H. and Lau, F.C.M. (1997) A Lower Bound for Interval Routing in General Networks. Networks, 29, 49--53.

....L 2 (u; v) LM (u; v) A nodeu is said to be contained in hp; qi if (1) p u q for p q, or (2) p u n Gamma 1 or 0 u q, otherwise. We use the notation u OE v OE w, to denote the cyclic ordering of node numbers, for u; v; w 2 L. Naturally, 0 OE 1 OE : OE n Gamma 1 OE 0. As in [8], the expression u OE fv; wg OE x means that v and w are contained in some interval and that they are ordered after u and before x, but the order of v and w is not shown. Property 1 (Completeness) The set of interval labels for edges directed from a node u is complete. That is, every node in V 6= ....

S.S.H. Tse and F.C.M. Lau, A LowerBound for Interval Routing in General Networks, Networks, to appear.

....better in planar graphs than in non planar graphs, it is intuitive to think that interval routing would perform likewise. It is reasonable therefore to divide the class into planar and non planar graphs, as shown in Figure 2. A number of lower bounds have been proposed for non planar graphs (e.g. [8, 9]) but for planar graphs, there exists only one lower bound result 3 2 D Gamma 1 which is due to Ru zicka [4] 1 His proof is based on a simple planar graph, which he later called a globe graph [2] see Figure 3 for an example) In [10] the authors proved that the lower bound is ....

S.S.H. Tse and F.C.M. Lau, "A Lower Bound for Interval Routing in General Networks", Networks, 29:49-53, 1997.

....for point to point networks. The method has been incorporated in the design of a commercially available routing chip [7] and is a basic element in some compact routing methods (e.g. 4] With up to one interval label per edge, the method has been shown to be nonoptimal for arbitrary graphs [9, 13], where optimality is measured in terms of the longest (routing) path in a graph. It is intuitive to think that the longest path would depend on the number of interval labels used. In this paper, using some non planar graphs, we prove that even with a relatively large number of labels, interval ....

....path in multi label interval routing in general networks. For 1 IRS, Ru zicka, using some planar graphs, proved a lower bound of 3 2 D Gamma 1 on the longest path in general networks [9] This bound was improved to 7 4 D Gamma 1 by Tse and Lau who used some similar, but non planar graphs [13]. For M IRS, M 1, Tse and Lau proposed in [12] two lower bounds using planar graphs on the longest path 2M 1 2M D Gamma 1 and 6M 1 6M D Gamma 1 for M label IRS, where M is from 2 to Theta( 3 p n) and from 2 to Theta( p n) respectively. In this paper, we improve on these results ....

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S.S.H. Tse and F.C.M. Lau, A lower bound for interval routing in general networks, Networks, to appear. (Preliminary version: Technical Report TR-94-09, Department of Computer Science, The University of Hong Kong, 1994.)

....all the routing paths are shortest paths [5] For arbitrary graphs, Ru zicka proved that it is not possible to have an optimal 1 IRS [1] He gave a lower bound of 3 2 D 1 2 for the longest path in a family of graphs. This bound was recently improved to 7 4 D Gamma 1 by the authors [4]. The natural question then arises: Using more labels per edge, is it possible to have an optimal IRS for arbitrary graphs In this paper, we consider the case of using two labels per outgoing edge. We derive a lower bound of 3 2 D for the longest path, which again is greater than D. The answer ....

....presented in [3] In the following, we assume that interval labels are cyclic. In addition to interval labels, there could be null labels and complement labels [1] However, it can be easily seen that null labels and complements labels will not affect the result (see for example the arguments in [4]) 2 Preliminaries The network in question is a connected graph, G = V; E) where E is the set of edges, andV the set of the nodes. Everyedge in E is bidirectional. There are n nodes in V . To implement interval routing, each node is labeled with a unique integer, called node number, from the ....

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S.S.H. Tse and F.C.M. Lau, A lower bound for interval routing in general networks. Networks, to appear.

....on the longest path for arbitrary networks using any labeling. Interestingly, only one upper bound exists, which is the 2D upper bound according to Santoro and Khatib [2] In this paper, we give the lower bound 2D Gamma 3 which represents an improvement over the previous bound of 7 4 D Gamma 1 [4]. By allowing a larger diameter, this new bound can be modified to become 2D Gamma o(D) Because of space limitation, we omit many of the proofs. The full version is available as [5] 2 Properties The network in question is a connected graphs, G = V; E) where V is the set of nodes, and E the ....

....subset of every interval set that contains A. If an interval B contains an interval B 0 , B 0 is a subinterval of B. We use the notation u OE v OE w, to denote the cyclic ordering of node numbers, for u; v; w 2 f0; n Gamma 1g. Naturally, 0 OE 1 OE : OE n Gamma 1 OE 0. As in [4], the expression u OE fv; wg OE x means that v and w are contained in some interval and that they are ordered after u and before x, but the order of v and w is not shown. Property 1 (Completeness) The set of interval labels for edges directed from a node u is complete. That is, 8u 2 V , V Gamma ....

S.S.H. Tse and F.C.M. Lau, A Lower Bound for Interval Routing in General Networks, Networks, 29(1): 49--53, 1997.

....is still needed to narrow the gap. Let L(u; v) denote the interval label for the edge that goes from u to v. A node u is said to be contained in [p; q] if (1) p u q for p q, or (2) p u n Gamma 1 or 0 u q, otherwise. The following are some essential properties of a valid labeling scheme [6]. Property 1 (Completeness) The set of interval labels for edges directed from a node u is complete. That is, every other node (6= u) in the graph must be contained in some interval at u. Property 2 (No ambiguity) The interval labels for edges directed from a node u are disjoint. That is, every ....

.... [62,65] 66,61] 63,65] 0,62] 64,71] 61,63] 65,60] 61,64] 66,60] 61,65] 61,65] 66,60] 66,71] 0,65] 67,71] 0,66] 68,71] 0,67] 69,71] 66,68] 70,65] 66,69] 71,65] 66,70] 36,40] 41,35] 37,40] 41,36] 38,40] 0,37] 39,71] 36,38] 40,35] 36,39] 41,35] 36,40] 1,5] [6,0] [2,5] 6,1] 3,5] 0,2] 4,71] 0,3] 5,71] 0,4] 6,71] 0,5] 6,10] 11,5] 7,10] 11,6] 8,10] 0,7] 9,71] 6,8] 10,5] 6,9] 11,5] 6,10] Figure 2: A labeled G (k = 3; s = 14) It is easy to check that the labeling satisfies the necessary conditions for a valid IRS. Proposition 1 All ....

[Article contains additional citation context not shown here]

Tse, S.S.H. and Lau, F.C.M. (1997) A Lower Bound for Interval Routing in General Networks. Networks, 29, 49--53.

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