D. Peleg and A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the algorithm for computing all pairs shortest paths. Spanners are also used in computational biology [4] in the process of reconstructing phylogeny trees from matrices whose entries represent genetic distances among contemporary living species. For other applications, please refer to the papers [2, 3, 11, 13]. 1.1 Lower bound on the size of All the applications of spanners require a t spanner of smallest possible size. Therefore, from a graph theoretic perspective, the following question arises : How sparse can a t spanner be In this regard, a lot of work [11, 15] has been done to establish a lower ....

....please refer to the papers [2, 3, 11, 13] 1.1 Lower bound on the size of All the applications of spanners require a t spanner of smallest possible size. Therefore, from a graph theoretic perspective, the following question arises : How sparse can a t spanner be In this regard, a lot of work [11, 15] has been done to establish a lower bound on the size of spanner in terms of its stretch factor. These results use the following simple relationship between the stretch of a spanner and the girth (length of the smallest cycle) of a graph. A graph has girth at least t 2 if and only if it does ....

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D. Peleg and A. A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

....is that it requires precomputing and storing a matrix with all the O(n ) distances among the objects of U. This high space requirement has prevented it from being seriously considered except in very small domains. On the other hand, the concept of a t spanner is well known in graph theory [9]. Let G be a connected graph G(V; E) with a nonnegative cost function d(e) assigned to its edges e 2 E, and dG (u; v) be the cost of the cheapest path between u; v 2 V . Then, a t spanner of G is a subgraph G (V; E ) where E E and 8u; v 2 V; dG (u; v) t dG (u; v) It should be ....

D. Peleg and A. Schaer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989.

....Graphs, Probabilistic Approximations 1 Introduction 1.1 Metric approximations. Approximating a given metric by structurally simpler metrics has been an area of much research motivated by several different perspectives such as functional analysis [9] graph theory [7] distributed computation [14], approximation algorithms [12] and computational biology [1] The general notion of approximation is to sandwich a simpler metric between the given metric and some multiple of the given metric. A popular simple metric used to approximate a given metric is an additive or tree metric, represented ....

D. Peleg and A. A. Schaeffer. Graph spanners. Journal of Graph Theory, 13(1):99-106, 1989.

.... the border nodes of the original graph (as explained later, this losslessness does not hold under multiple QoS parameters) The second step involves mapping the full mesh into a more compact topology, such as symmetric node (simple node) star [20] minimum spanning tree [19] and t spanner [2, 15]. Graph reduction is performed by pruning several links of the full mesh. The compact topology is 3 then represented as a complex node, which is broadcasted to the rest of the network. One problem in the above approach is that the amount of lossyness that results from graph reduction is not ....

D. Peleg and A. A. Schaffer. Graph spanners. Journal of Graph Theory, 13(1):99--116, 1989.

.... geometry and biology [PU89, ADJS93] Peleg and Ullman [PU89] introduced the concept of graph spanners as a means of constructing synchronisers for Hypercubic networks (their results have been recently improved in [DZ] The problem of finding a 2 spanner with the minimum number of edges is NPhard [Pm89]. In this Section we present a reduction from the problem of finding a sparsest 2 spanner in a graph to that of finding a largest induced matching in a bipartite graph without small cycles. For every G on n vertices and m edges, let B(G) be a bipartite graph with vertex sets U = fu e : e 2 E(G)g ....

D. Peleg and A. A. Schaffer. Graph Spanners. Journal of Graph Theory, 13(1):99--116, 1989.

....G) t. Indeed, for the identity map on the vertices I : V 7 V , kIk t and kI Gamma1 k = 1. Thus spanners with few edges provide some answers to our questions. The first explicit construction of sparse spanners for arbitrary unweighted graphs was accomplished in Awerbuch [2] Peleg and Shaffer [11]. A different construction appeared more recently in Althofer et al. 1] it improves the constants, and works for weighted graphs as well. Theorem 1.1: Althofer et al. 1] Let H be an arbitrary (weighted) graph with n vertices. Then, for all integer t 1, H has a t spanner with at most n ....

D. Peleg, A. Schaffer, Graph spanners, Journal of Graph Theory, 13 (1989), 99-116.

....pair of vertices in G, the distance between them in S is at most t times the distance between them in G. We are interested in finding a sparsest t spanner. This problem has many applications in areas as far afield as distributed computing, networks, computational geometry, robotics and biology [1, 12, 13]. We refer to the quantity t as the dilation of the spanner. The cardinality of the edge set of a spanner denotes its size. Peleg and Ullman [13] introduced the concept of graph spanners as a means of constructing synchronisers of hypercubic networks. They showed that the d dimensional hypercube ....

....0000 c Hermes Science Publications 2 J. of Discrete Algorithms, Vol. 0 No. 0, 0000 bound results for finding sparse hypercube 3 spanners. In recent years, there has been a great deal of research in this area and many complexity results are now known. For general graphs, Peleg and Schaffer [12] showed that the problem of finding a sparsest 2 spanner (S2S) is NP hard. Since then, Cai [4] extended this result to include all dilations greater than 2. Cai and Keil [5] gave a linear time algorithm for S2S in graphs with maximum degree 4. They also showed that finding a sparsest t spanner ....

D. Peleg and A.A. Schaffer. Graph Spanners. Journal of Graph Theory, 13(1):99--116, 1989.

....information exchange, the main reason why hierarchical routing is scalable. Topology aggregation essentially derives a simpler connectivity graph out of a much more complicated real connectivity graph. Various topology aggregation methods exist, among them are tree scheme [6] and spanner scheme [2, 14]. 2.2 Effect of the hierarchy structures We study the effect of network hierarchy structures on the hierarchical routing performance where various aggregation methods are used. That is, we consider how various hierarchies imposed on the same topology would affect the performance of hierarchical ....

D. Peleg and A. A. Schaffer. Graph spanners. Journal of Graph Theory, 13(1):99 -- 116, 1989.

.... in the Proceedings of the 24th International Annual Workshop on Graph Theoretic Concepts in Computer Science (WG 98) This work originates from the second author s thesis [17] Supowit [12] and Arikati et al. 2] Surveys of results on the existence and e cient constructibility can be found in [20] and [24] Depending on the objective for choosing a subnetwork, various kinds of spanners have been considered see [1, 3, 16, 19, 21, 23] for a selection of variants. Since the main motivation is to obtain a network of small total weight, particular attention has focused on tree spanners, ....

D. Peleg and A. A. Schaer. Graph spanners. Journal of Graph Theory, 13 (1989), pp. 99-116.

.... [20] who introduced spanners to synchronize asynchronous networks) They have also been used for simplifying geometric data structures see Chew [11] Dobkin, Friedman, and Supowit [12] and Arikati et al. 2] Surveys of results on the existence and efficient constructibility can be found in [19] and [23] Depending on the objective for choosing a subnetwork, various kinds of spanners have been considered see the list of references for a selection of variants. Since the main motivation is to obtain a network of small total weight, particular attention has focused on tree spanners, ....

D. Peleg and A. A. Schaffer. Graph spanners. Journal of Graph Theory, 13 (1989), pp. 99--116.

....the algorithm and the obtained stretch factor. For any even t, stretch t ffl paths between all pairs of vertices can be found in O(n 2 2=t ) time. The works of Awerbuch et al. ABCP93] and Cohen [Coh93] are based on the construction of sparse spanners (Awerbuch [Awe85] Peleg and Schaffer [PS89] A t spanner of a graph G = V; E) is a subgraph G 0 = V; E 0 ) of G such that for every u; v 2 V we have ffi G 0 (u; v) t Deltaffi G (u; v) where ffi G (u; v) is the distance between the vertices u and v in the (possibly weighted) graph G. A different approach all together was ....

....graph. Here we want a single graph 13 that will supply good approximations of all distances. Constructing a sparse k emulator is therefore harder than computing surplus k distances. The definition of k emulators is related to the definition of k spanners (Awerbuch [Awe85] Peleg and Schaffer [PS89] Let G = V; E) be a weighted undirected graph. A subgraph G 0 = V; E 0 ) of G is said to be a k spanner of G if and only if for every u; v 2 V we have ffi G 0 (u; v) k Delta ffi G (u; v) As G 0 is a subgraph of G, we always have ffi G (u; v) ffi G 0 (u; v) This definition ....

D. Peleg and A.A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

....that edges residing on the same page do not intersect; cf. 14] The memory bound is 8n o(n) bits per vertex. The schemes can be extended to g genus graphs with n log g O(n) memory bits per vertex. As another example, the construction of 3spanners for the family of chordal graphs described in [101] can be used to construct routing schemes for these graphs with stretch factor 3 and O(n log n) bits of memory in total. For Euclidean networks, namely, networks whose sites are embedded in the 2dimensional plane with Euclidean distances, recent papers have dealt with proposing efficient ....

D. Peleg and A.A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. A. Scha#er. Graph Spanners. Journal of Graph Theory, 13(1):99--116, 1989.

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D. Peleg and A.A. Scha#er. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A.A. Scha#er. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A.A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. A. Schaffer, Graph Spanners, Journal of Graph Theory, 13 (1989), pp 99-116.

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D. Peleg and A.A. Schaer. Graph spanners. Journal of Graph Theory, 13:99-116, 1989.

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D. Peleg and A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A.A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. A. Scha#er. Graph Spanners. Journal of Graph Theory, 13(1):99--116, 1989.

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D. Peleg and A. Schaffer. Graph spanners. Journal of Graph Theory, 13:99--116, 1989.

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D. Peleg and A. Schaer. Graph spanners. Journal of Graph Theory, 13:99{ 116, 1989.

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