M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, July 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

.... best known implementation of the algorithm of Althofer et al. 2] for (2k 1) spanner has a running time of O(mn ) Cohen [8] presented a randomized algorithm with O(kmn ) expected running time for computing a spanner of O(kn ) size and slightly larger stretch (2k ) Thorup and Zwick [15], improved the result of Cohen [8] and presented a randomized algorithm for computing a (2k 1) spanner of optimal size in O(kmn ) expected time. All these existing algorithms for computing spanners in undirected weighted graphs require computation of shortest distance information between many ....

[Article contains additional citation context not shown here]

M. Thorup and U. Zwick. Approximate distance oracle. In Proceedings of 33rd ACM Symposium on Theory of Computing (STOC), pages 183--192, 2001. 21

....Their resource require ments are listed in Figure 1, along with a comparison of our results with previous results in this model. The principal in gredients of our schemes include the following: the O(log rs) greedy approximation to dominating set, used in the same fashion as in [2, 7, 6, 16] for most of the constructions; the sparse neighborhood covers of [3] for the construction in Section 4; a distributed dictionary, as first defined by Pe leg [14] the schemes of [6] and [17, 8] for compact routing on trees; and a new randomized block assignment of ranges of addresses. We remark ....

....blocks to nodes. Lemma 2.5, used in Section 2, is a special case of the preceding lemma, given by k 2. A component of the algorithm is the name dependent routing algorithm of Thorup and Zwick which uses ( n ) space per node, O(log n) sized headers and which delivers messages with stretch 2k i [16]. We note that this is the version of their algorithm which requires handshaking, but our scheme stores the precomputed handshake information with the destination address. Let TZR(u, v) denote the address required for routing from u to v, including the extra O(log n) bits Thorup and Zwick ....

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. 33rd ACM Syrup. on Theory of Computing, pages 183 192, May 2001.

.... were given an (8; 0) approximate distance labeling scheme, for every integer 1, with O( Delta n log n Delta log D) bit labels [98] where D is the weighted diameter of the graph, and later an improved (2 Gamma1; 0) approximate scheme with O(n 1 Gamma1= n Delta log(nD) bit labels [120]. The time to decode the estimated distance is O( This implies a (2 log n; 0) approximate scheme n) bit labels for general unweighted graphs. These results are complemented by a lower bound in ) on the label size of ( Omega ( 0) approximate schemes, presented independently in [120] and ....

....[120] The time to decode the estimated distance is O( This implies a (2 log n; 0) approximate scheme n) bit labels for general unweighted graphs. These results are complemented by a lower bound in ) on the label size of ( Omega ( 0) approximate schemes, presented independently in [120] and in [60] It is interesting to notice that a small variation on the quality of the estimators, say, moving from (1; 0) approximate to (1 o(1) 0) approximate or to (1; O(1) approximate schemes, results in a significant impact on the label size. Trees, and more generally graphs with ....

[Article contains additional citation context not shown here]

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. 33 ACM Symp. on Theory of Computing, pages 183--192, Jul. 2001.

....upper and lower bounds for these two latter results. The upper bound for planar graphs is O( n log n) coming from a more general result about graphs having small separators. Related works concern distance labeling schemes in dynamic tree networks [22] and approximate distance labeling schemes [12,27,28]. Several ecient schemes have been designed for speci c graph families: interval and permutation graphs [21] distance hereditary graphs [13] bounded tree width graphs (or graphs with bounded vertex separator) and more generally bounded clique width graphs [9] All support an O(log ....

M. Thorup and U. Zwick, Approximate distance oracles, in 33 ACM Symp. on Theory of Computing (STOC), 2001, pp. 183-192.

....for these two latter results. The upper bound for planar graphs is O( n log n) coming from a more general result about graphs having small separators. Related works concern distance labeling schemes in dynamic tree networks [KPR02] and approximate distance labeling schemes [GKK 01, Tho01, TZ01a] Several ecient schemes have been designed for speci c graph families: interval and permutation graphs [KKP00] distance hereditary graphs [GP01a] bounded tree width and more generally bounded clique width graphs [CV01] all support an O(log labeling scheme. Excepted for the two rst ....

M. Thorup and U. Zwick. Approximate distance oracles. In 33 Symposium on Theory of Computing (STOC), pages 183-192, Hersonissos, Crete, Greece, July 2001.

....requiring less space at each node of the network The answer is yes , and the main tool to achieve that improvement is a combination of clustering and tree covers. In particular, we refer to the hierarchical routing schemes presented in [1, 2, 13] and to the more recent schemes presented in [3, 4, 14, 15]) Although the stretch factors (i.e. the maximum, over all pairs of nodes, of the ratio between the length of the route over the length of a shortest path between two nodes) resulting from these schemes might be not optimal, the improvement in term of memory space is often signi cant. All these ....

....scheme in [13] uses addresses on O(log n) bits, though headers on O(log n) bits only. 2) The size of the local data structure stored at each node. This size plus the size of the addresses is informally called memory requirement. And (3) The length of the routes, i.e. the stretch factor. In [14] it is proved that the previous results in the seminal All logarithms are in base two. paper [13] Namely, they show that any network support a routing scheme with stretch 4k 5 using local data structures of size O(kn 1=k ) and using addresses and headers of size n= log log n) for ....

[Article contains additional citation context not shown here]

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In 33 rd Annual ACM Symposium on Theory of Computing (STOC), pages 183-192, Hersonissos, Crete, Greece, July 2001.

....assumed by PRR) to get results similar to those of Awerbuch and Peleg [1] This is a qualitative statement at this time. Table 1 gives a summary of some of the previous results along with ours. Note that the result for general metrics can be improved using results of Thorup and Zwick [19] to use only ### ### ## 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 follows that of Karger and Ruhl [10] who give a sequential algorithm for answering nearest ....

....used by Bourgain [4] for metric embeddings. In particular, we show that this scheme leads to a covering of the graph by trees such that for any two nodes # and # at distance they are in a tree of diameter ### #. Indeed, by modifying the PRR scheme along the lines proposed by Thorup and Zwick [19] one can improve the space bounds by a logarithmic factor, but we do not address this issue here. The remainder of this paper is divided as follows: Section 2 describes the details of Tapestry, highlighting differences with the PRR scheme and introducing concepts and terminology for the remainder ....

[Article contains additional citation context not shown here]

THORUP, M., AND ZWICK, U. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp. (2001), pp. 183--192.

....assumed by PRR) to get results similar to those of Awerbuch and Peleg [1] This is a qualitative statement at this time. Table 1 gives a summary of some of the previous results along with ours. Note that the result for general metrics can be improved using results of Thorup and Zwick [19] 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 follows that of Karger and Ruhl [10] who give a sequential algorithm for answering nearest ....

....used by Bourgain [4] for metric embeddings. In particular, we show that this scheme leads to a covering of the graph by trees such that for any two nodes u and v at distance they are in a tree of diameter log n. Indeed, by modifying the PRR scheme along the lines proposed by Thorup and Zwick [19] one can improve the space bounds by a logarithmic factor, but we do not address this issue here. The remainder of this paper is divided as follows: Section 2 describes the details of Tapestry, highlighting differences with the PRR scheme and introducing concepts and terminology for the remainder ....

[Article contains additional citation context not shown here]

THORUP, M., AND ZWICK, U. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp. (2001), pp. 183--192.

....distance labelings. In the case of planar graphs, we can use our ideas to get a stretch 3 distance labelings of size O(log n) for planar graphs. Previously, no subpolynomial labeling schemes were known for planar graphs (even with constant distortion) 10] A recent paper of Thorup and Zwick [22] gives constructions of a slightly different variety of tree covers. Though their definitions differ from ours, they can also be used for MPLS routing. Their results imply that for general graphs, there exist tree covers of size ) with stretch O(k) This gives an MPLS routing scheme with ....

....path between u and v. Note that, since each tree is a subtree of G, dG (u; v) d T i (u; v) When D = 1, we often say that there is no stretch; furthermore, in this case, we will often omit mentioning the stretch. Note that this definition of tree covers is slightly different from that in [22], since it does not place a restriction on the number of trees in which a vertex appears, but instead places a uniform restriction on the number of trees in the family. Of course, it is easy to see that the size of a tree cover may be large: if we require a stretch 1 tree cover for the complete ....

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33nd Annual ACM Symposium on Theory of Computing, 2001. To appear.

....see [1, 11, 14] In numerous algorithms, query versions frequently appear as subroutines. In such a query, we are given two vertices and have to compute or approximate the shortest path between them. The latest in a series of results for undirected weighted graphs is by Thorup and Zwick [23]; their algorithm computes (2k 1) approximate solutions to the query version of the SP problem in O(k) time, using a data structure that takes (expected) time O(kmn 1=k ) to construct and utilizes O(kn 1 1=k ) space. It is not an approximation scheme in the true sense because the value k ....

M. Thorup and U. Zwick. Approximate distance oracles. In 33rd STOC, 2001.

....requiring less space at each node of the network The answer is yes , and the main tool to achieve that improvement is a combination of clustering and tree covers. In particular, we refer to the hierarchical routing schemes presented in [1, 2, 13] and to the more recent schemes presented in [3, 4, 14, 15]) Although the stretch factors (i.e. the maximum, over all pairs of nodes, of the ratio between the length of the route over the length of a shortest path between two nodes) resulting from these schemes might be not optimal, the improvement in term of memory space is often signi cant. All these ....

....scheme in [13] uses addresses on O(log 2 n) bits, though headers on O(log n) bits only. 2) The size of the local data structure stored at each node. This size plus the size of the addresses is informally called memory requirement. And (3) The length of the routes, i.e. the stretch factor. In [14] several trade o s between the memory requirement and the stretch factor have been given, improving the previous results of the seminal paper [13] Namely, they show that any network support a routing scheme with stretch 4k 5 using local data structures of size 2 O(kn 1=k ) and using ....

[Article contains additional citation context not shown here]

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In 33 rd Annual ACM Symposium on Theory of Computing (STOC), pages 183-192, Hersonissos, Crete, Greece, July 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33rd annual ACM Symposium on Theory of Computing (STOC), pages 183--192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183--192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of 33rd STOC, pages 183--192, 2001. Full version to appear in the Journal of the ACM.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183-192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33rd annual ACM Symposium on Theory of Computing (STOC), pages 183--192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, 2001. To appear.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183-192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183-192, 2001.

....vertex of the network a distance or reachability label such that we can later calculate the distance (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 networks that occur in practice. Our labels are based on the concept of 2 hop covers. Let G = V, E) be ....

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183--192, 2001.

....vertex of the network a distance or reachability label such that we can later calculate the distance (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 networks that occur in practice. Our labels are based on the concept of 2 hop covers. Let G = V; E) ....

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pages 183--192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, July 2001.

No context found.

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, 2001.

No context found.

M. Thorup, U. Zwick, Approximate distance oracles, Proc. 33 rd Ann. ACM Symp. on Theory of Computing (STOC

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the thirty-third annual ACM symposium on Theory of computing (STOC '01), pages 183--192. ACM Press, 2001.

No context found.

Thorup, M., and Zwick, U. Approximate distance oracles. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (July 2001), pp. 183--192.

No context found.

M. Thorup and U. Zwick, Approximate distance oracles, in 33 rd Annual ACM Symposium on Theory of Computing (STOC), July 2001, pp. 183--192.

No context found.

THORUP, M., AND ZWICK, U. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp. (July 2001), pp. 183--192.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the thirty-third annual ACM symposium on Theory of computing (STOC '01), pages 183--192. ACM Press, 2001.

No context found.

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. 33rd Annu. ACM Sympos. Theory Comput., pages 183-192, 2001.

No context found.

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, 2001.

No context found.

THORUP, M., AND ZWICK, U. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp. (2001), pp. 183--192.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. 33rd Annu. ACM Sympos. Theory Comput., pages 183-192, 2001.

No context found.

THORUP, M., AND ZWICK, U. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp. (July 2001), pp. 183--192.

No context found.

M. Thorup and U. Zwick, "Approximate distance oracles," in Proc. of the 33 rd STOC. ACM, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, July 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of the 33th Annual ACM Symp. on Theory of Comp., pages 183--192, July 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 183--192. ACM Press, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 183--192. ACM Press, 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In 33rd Annual ACM Symposium on Theory of Computing, pages 183-192, 2001.

No context found.

Mikkel Thorup and Uri Zwick. Approximate distance oracles. In 33 rd Annual ACM Symposium on Theory of Computing (STOC), pages 183-192, Hersonissos, Crete, Greece, July 2001.

No context found.

M. Thorup and U. Zwick. Approximate distance oracles. In Proc. ACM Symposium on Theory of Computing, pages 183--192, 2001.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC