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... hence at least (n) bits at some node. (See Fraigniaud and Gavoille [14], Gavoille and Pérennès [20], and Gavoille and Gengler [19]. The claim also follows easily from the lower bounds given by us in =-=[25]-=-.) Furthermore, any routing scheme of stretch < 5 must use a total space of (n 3=2 ) bits on some n-node network, and hence at least (n 1=2 ) bits at some node. (This follows again from [25].) The sch...

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...to those of Awerbuch and Peleg [1]. Table 1 gives a summary of some of the previous results along with ours. Note that our result for general metrics can be improved using results of Thorup and Zwick =-=[32]-=- to use only O(n log n) space. Techniques: The crux of our method for inserting nodes into the network lies in an algorithm for maintaining nearest neighbors in a restricted metric space. Our approach...

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...nce (or reachability relation) between two vertices using only the labels of these two vertices. Distance labels were considered, among others, by Peleg [10], Gavoille et al. [4] and Thorup and Zwick =-=[12]-=-. All these papers, however, consider only undirected graphs , and only worst case results. We are interested mainly in directed graphs, and in the performance of the proposed labeling scheme on netwo...

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... log 2 n + ε −O(dim) n,S = ε −O(dim) n,Q = 2 O(dim) ,κ = 1 + ε)-CRS. (b) (P = 2 O(dim) · poly(n) + ε −O(dim) n,S = ε −O(dim) n,Q = O(dim),κ = 1 + ε)-CRS. For general n-point metrics, Thorup and Zwick =-=[45]-=- obtained a (kn 1+1/k ,kn 1+1/k ,O(k), 2k −1)- CRS, where k ∈ N is a prescribed parameter. The trade-off between the approximation and the space is essentially tight for general metrics. Closer in spi...

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...anisms. On the other hand, distance labels fall into the general class of compact graph representation problems, and approximate distance oracles have been studied, amongst others by Thorup and Zwick =-=[43]-=-. Such oracles for the geometric setting have been explored in [23]. Compact routing schemes have been widely studied, e.g. [39, 7, 8, 16, 17, 44]. For a general overview of efficient routing schemes,...

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...hen route to it. Indeed, optimal stretch 3 compact schemes for labeled routing are already known. The first stretch 3 scheme was given by Cowen [6] with � O(n 2/3 ) memory 1 . Later, Thorup and Zwic=-=k [17, 18] improved -=-the memory bound to only �O( √ n ) bits. They also gave an elegant generalization of their scheme, achieving stretch 4k − 5 (and even 2k − 1 with handshaking) using only � O(n 1/k ) bits. Ad...

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...ws that any weighted graph on n vertices has a (2k 1)-spanner with O(n 1+1=k ) edges. This result is believed to be tight for any k 1. It is proved, however, only for k = 1; 2; 3 and 5. (See, e.g., [=-=73-=-].) The fastest known implementation of the algorithm of Althofer et al. [7] runs in O(mn 1+1=k ) time. If the graph is unweighted, then a (2k 1)-spanner of size O(n 1+1=k ) can be easily found in O(m...