M. Thorup and U. Zwick, Compact routing schemes, in Proc. 13th Ann. ACM Symp. on Par. Alg. and Arch. (SPAA 2001), ACM 2001, pp. 1--10.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is the design of optimal labeling schemes when clues are provided as distribution functions. 7. RELATED WORK One can design labeling schemes in a static or dynamic setting. The static setting, where the full structure of the tree is known in advance, has been the focus of several recent works [2, 4, 17, 10, 8]. The best proposed schemes use labels of length log n o(log n) bits, and there is a matching lower bound of log n on the maximum label length of any such labeling scheme. None of these schemes, however, is suitable for a dynamic setting where the trees undergo changes through time. Ancestor ....

U. Zwick and M. Thorup. Compact routing schemes. In Proceedings of th Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA, July 2001.

...., where d(u,v) is the length of the shortest u v path) can routing tables be constructed that use sublinear space at every node Though the name independent compact routing problem was first introduced in 1989 by Awerbuch, Bar Noy, Linial and Peleg [1] progress has been slow. Much recent work [4, 7, 6, 17] has occurred on the easier, but related com pact routing problem, where the compact routing scheme designer may assign his her own polylogarithmic sized node labels (generally O(logn) or O(log 2 n) bit) dependent on network topology. That is, when a packet destined for i arrives, i has been ....

.... 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 that our algorithms can be easily modified to determine the results of a handshaking scheme in the sense of [17] For example, if there is a whole stream of packets fi om a single origin ....

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures, pages i 10, July 2001.

....with labels of length 2 log n O(log log n) Alstrup and Rauhe [3] subsequently showed how to improve the bound of Abiteboul et al., to log n O( log n) by incorporating alphabetic trees [7] into their algorithm. A similar scheme has also been discovered independently by Zwick and Thorup [12] in the context of a routing application. As already mentioned our result builds upon the tree decomposition technique of Abiteboul et al. as presented in [2] We can obtain a labeling scheme for parent queries by taking the ancestor labeling scheme of Alstrup and Rauhe [3] and add to the label of ....

....multicast of information. In such context a router may want to switch to unicast or TTL (Time To Live) based broadcast when all destinations reachable from it are relatively close. A close relation between ancestor labeling schemes and routing in trees is suggested by the work of Zwick and Thorup [12]. Note that when the label of the destination (or labels of the destinations) is transmitted with the packet, long labels will increase the bandwidth. Each node also stores the labels of all destinations it may need to send data to, so if the labels are long these tables may become too large. ....

M. Thorup and U. Zwick. Compact routing schemes. In To appear in the Proceedings of the Thirteenth ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

....and makes this term be the dominant when n is the current average size of an XML file. In this paper we suggest labeling schemes which are likely to be winners in practice. We compare our schemes experimentally, using real XML data, to the interval scheme as well as to the recursive schemes of [2, 4, 20]. We also analyze our new schemes theoretically and prove worst case guarantees on the maximum label length. Our Results. In contrast to the interval scheme we suggest to use a prefix based approach, where the labeling is such that an ancestor query roughly amounts to testing whether one label is ....

.... for a tree which is a long path the labels are of length #(n) Therefore, a single skewed XML file, may increase the size of the index considerably, in particular for fixed [4] build upon the work of [2] A related scheme producing labels of length log n O(log n log log n) is also suggested in [20]. length representation. To remedy this we introduce the compressed prefix scheme that first transform the tree into a balanced one. We perform this balancing by partitioning the tree into paths and contracting each path into a single virtual node. Then we assign a collection of prefix free ....

M. Thorup and U. Zwick. Compact routing schemes. In 13 ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

.... it is used by algorithms for nding a maximum weighted matching in a graph [31] a minimum spanning tree in a graph [38, 19] and a dominator tree in a directed owgraph [3] It is also proved useful in several string algorithms [33, 23] for dynamic planarity testing [67] in network routing [62], for solving various geometric problems [12] including range searching [30] for nding evolutionary trees [24] and in bounded treewidth algorithms [13] The IT University of Copenhagen, Denmark, E Mail: stephen it c.dk LaBRI Universit e de Bordeaux, France, E Mail: gavoille labri.fr ....

....the identi er of nca(x; y) from the identi ers of x and y may prove useful. Our particular nca labeling scheme also allows to identify the rst edge on the shortest path from x to y from the labels of x and y alone. In that sense it generalizes recent labeling schemes for routing on trees [25, 62] (see also Section 3) Another possible application of our algorithm arises in XML search engines. Such search engines typically maintain a reverse index. This index is a hash table mapping each word or a name of a tag to all XML documents containing it. The engine can exploit the fact that an ....

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M. Thorup and U. Zwick. Compact routing schemes. In 13 ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

.... framework [Pel00a] introduced informative labeling schemes for graphs and thereby captured a whole set of already known results such as implicit representation or adjacency labeling [AR02b, KNR92, Spi02] whose objective is only to decide adjacency between two vertex labels, compact routing [Gav01, TZ01b] whose objective is to provide the rst edge on a near shortest path between the source and the destination, nearest common ancestor and other related functions for trees [ABR03, AGKR02, AKM01, KMS02] ow and connectivity [KKP02] A rst set of results about distance labeling can be founded in ....

M. Thorup and U. Zwick. Compact routing schemes. In 13 Symposium on Parallel Algorithms and Architectures (SPAA), pages 1-10, Hersonissos, Crete, Greece, July 2001. ACM PRESS.

....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 ....

....the destination) they even reduced the stretch to 2k 1. Results and well established Erd os conjectures [5] from structural graph theory allow to believe that the space time tradeo is optimal. However, it is let open whether one could decrease the size of the addresses and headers. As shown in [6, 15], these complexity issues strongly depend on whether or not we have the ability to set the output port numbers during the pre processing phase. The xed port model speci es that the port numbers are xed (say by an adversary) as opposed to the designer port model which speci es that the port ....

[Article contains additional citation context not shown here]

Mikkel Thorup and Uri Zwick. Compact routing schemes. In 13 Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 1-10, Hersonissos, Crete, Greece, July 2001. ACM PRESS. 11

.... on implicit graph representation can be found in [49] and an extensive survey on labeling schemes in [30] To test various kinds of relationships, in addition to adjacency, between two nodes alone from the labels, has been studied in a number of papers: Labeling schemes are given for ancestor in [34, 1, 52, 7, 35, 3], sibling and parent in [35] and distances in [31, 36, 44] Efficient labeling schemes are essential to a number of distributed computation problems, as e.g. routing technology [47, 52] Recently, papers on labeling schemes with applications to XML Based Search Engines have appeared, see e.g. ....

.... from the labels, has been studied in a number of papers: Labeling schemes are given for ancestor in [34, 1, 52, 7, 35, 3] sibling and parent in [35] and distances in [31, 36, 44] Efficient labeling schemes are essential to a number of distributed computation problems, as e.g. routing technology [47, 52]. Recently, papers on labeling schemes with applications to XML Based Search Engines have appeared, see e.g. 1] In [35] an adjacency labeling scheme for trees is given, which assign labels of size log n O( log n) bits. The labels assigned in [35] can, beside adjacency, be used to test for ....

M. Thorup and U. Zwick. Compact routing schemes. In The Thirteen Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

....balancing, though it could perhaps be modified to do so. Unfortunately, both the PRR and Awerbuch schemes assume full knowledge of the participating nodes, or, equivalently, they assume that the network is static. There is also an abundance of theoretical work on finding compact routing tables [2, 7, 13, 20] whose techniques are closely related to those in this paper. See [8] for a survey. A recent and closely related paper is that of Thorup and Zwick, who showed that a sampling based scheme similar to that of PRR could be used to find small stretch routing tables and or answer approximate distance ....

THORUP, M., AND ZWICK, U. Compact routing schemes. In Proc. of the 13th Annual Symp. on Parallel Algorithms and Architectures (2001), pp. 1--10.

....balancing, though it could perhaps be modified to do so. Unfortunately, both the PRR and Awerbuch schemes assume full knowledge of the participating nodes, or, equivalently, they assume that the network is static. There is also an abundance of theoretical work on finding compact routing tables [2, 7, 13, 20] whose techniques are closely related to those in this paper. See [8] for a survey. A recent and closely related paper is that of Thorup and Zwick, who showed that a sampling based scheme similar to that of PRR could be used to find small stretch routing tables and or answer approximate distance ....

THORUP, M., AND ZWICK, U. Compact routing schemes. In Proc. of the 13th Annual Symp. on Parallel Algorithms and Architectures (2001), pp. 1--10.

....is based solely on local label information such schemes can avoid costly access to large tables that stores global information for the tree. For large global tables this may avoid expensive memory lookups. Furthermore, the local distributed information is also required in some routing schemes. In [11] Zwick and Thorup show how to assign short labels to a packet which should be routed, minimizing the overhead information associated to the packet such that the ancestor relation can be determined when the packet arrives to a node in the network. Abiteboul, Kaplan and Milo [1] also study the ....

....of graphs including trees. The main motivation for their study is the construction of small vertex induced universal graphs. For trees they consider the ancestor and parent adjacency relations. They show how to obtain labels of size 2 log n for the ancestor relation. We refer the reader to [11, 1, 7] for further details of appilcations of labeling schemes supporting ancestor queries. Related results: Tsakalidis [12] shows that if we assign the preorder and postorder number to each node in a tree, a node v is an ancestor to a node w i preorder(v) preorder(w) postorder(v) This leads to a ....

[Article contains additional citation context not shown here]

M. Thorup and U. Zwick. Compact routing schemes. In ACM Symposium on Parallel Algorithms and Architectures, volume 13, 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 ....

....the destination) they even reduced the stretch to 2k 1. Results and well established Erd os conjectures [5] from structural graph theory allow to believe that the space time tradeo is optimal. However, it is let open whether one could decrease the size of the addresses and headers. As shown in [6, 15], these complexity issues strongly depend on whether or not we have the ability to set the output port numbers during the pre processing phase. The xed port model speci es that the port numbers are xed (say by an adversary) as opposed to the designer port model which speci es that the port ....

[Article contains additional citation context not shown here]

Mikkel Thorup and Uri Zwick. Compact routing schemes. In 13 th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 1-10, Hersonissos, Crete, Greece, July 2001. ACM PRESS.

....compute the identi er of nca(x; y) from the identi ers of x and y may prove useful. Our particular nca labeling scheme also allows to identify the rst edge on the shortest path from x to y from the labels of x and y alone. In that sense it generalizes recent labeling schemes for routing on trees [17, 42] (see also Section 1.3) Another possible application of our algorithm arises in XML search engines. Such search engines typically maintain a reverse index. This index is a hash table mapping each word or a name of a tag to all XML documents containing it. The engine can exploit the fact that an ....

....and other functions have also been studied [27, 28, 37] Unfortunately these labeling schemes for ancestor queries do not allow to identify the nearest common ancestor of x and y when they are unrelated. The labeling scheme that we describe in this paper do allow this extra functionality. In [42] and [17] it is demonstrated how compact labelings of a tree, can improve routing in trees and graphs. In the routing problem one assigns two labels to every node of the graph. The rst label is is the address of the node whereas the second label is a data structure called local routing table. The ....

[Article contains additional citation context not shown here]

M. Thorup and U. Zwick. Compact routing schemes. In 13 th ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

....constant routing time and 7 log n bits of addresses, and (2) a shortest path memory free routing scheme for trees with a 4:8 log n bits of addresses and polynomial routing time. All these results are stated in the designer port model. A similar result was founded independently by Thorup and Zwick [12]. 1 All the time complexities are computed in a word RAM model where standard integer operations on words of O(log n) bits are performed in constant time, including logical bitwise operations, integer division, and integer multiplication. All the logarithms are in base 2. 2. For the xed port ....

M. Thorup and U. Zwick, Compact routing schemes, in 13 th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), ACM PRESS, July 2001. To appear.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the 13th annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 1--10, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Porceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures, Crete, Greece, pages 1--10, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Proc. 13th SPAA, 2001. To appear.

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M. Thorup and U. Zwick. Compact routing schemes. In Porceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures, Crete, Greece, pages 1-10, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the 13th annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 1--10, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. Submitted for publication, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Porceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures, Crete, Greece, pages 1-10, 2001.

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M. Thorup and U. Zwick, Compact routing schemes, in Proc. 13th Ann. ACM Symp. on Par. Alg. and Arch. (SPAA 2001), ACM 2001, pp. 1--10.

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M. Thorup, U. Zwick, Compact routing schemes, in: Proc. 13th Ann. ACM Symp. on Par. Alg. and Arch. (SPAA

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures (SPAA '01), pages 1--10. ACM Press, 2001.

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M. Thorup, U. Zwick, Compact routing schemes, in Proc. 13th Ann. ACM Symp. on Par. Alg. and Arch. (SPAA 2001.

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Thorup, M., and Zwick, U. Compact routing schemes. In Proceedings of the 13th Annual Symposium on Parallel Algorithms and Architectures (July 2001), pp. 1--10.

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M. Thorup and U. Zwick. Compact routing schemes. In Proc. ACM SPAA, pages 183--192, 2001.

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M. Thorup and U. Zwick, "Compact routing schemes," in Proc. of the 13 SPAA. ACM, 2001.

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THORUP, M., AND ZWICK, U. Compact routing schemes. In Proc. of the 13th Annual Symp. on Parallel Algorithms and Architectures (July 2001), pp. 1--10.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures (SPAA '01), pages 1--10. ACM Press, 2001.

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M. Thorup and U. Zwick, "Compact routing schemes," in Proc. of the 13 SPAA. ACM, 2001.

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M. Thorup and U. Zwick. Compact routing schemes.

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Mikkel Thorup and Uri Zwick. Compact routing schemes. SPAA, 2001.

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Mikkel Thorup and Uri Zwick. Compact routing schemes. In Proceedings of the 13th annual ACM symposium on Parallel algorithms and architectures, pages 110. ACM Press, July 2001.

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Mikkel Thorup and Uri Zwick. Compact routing schemes. SPAA, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings pages 1--10, 2001.

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THORUP, M., AND ZWICK, U. Compact routing schemes. In Proc. of the 13th Annual Symp. on Parallel Algorithms and Arch.s (2001), pp. 1--10.

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THORUP, M., AND ZWICK, U. Compact routing schemes. In Proc. of the 13th Annual Symp. on Parallel Algorithms and Architectures (July 2001), pp. 1--10.

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M. Thorup and U. Zwick, "Compact routing schemes," in Proc. of the 13 SPAA. ACM, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, pages 1--10. ACM Press, 2001.

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M. Thorup and U. Zwick. Compact routing schemes. In Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, pages 1--10. ACM Press, 2001. 14

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Mikkel Thorup and Uri Zwick. Compact routing schemes. In 13 Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 1-10, Hersonissos, Crete, Greece, July 2001. ACM PRESS.

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U. Zwick and M. Thorup. Compact routing schemes. manuscript, February 2001. 12

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M. Thorup and U. Zwick. Compact routing schemes. In 13 ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2001.

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