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Optimal Distance Labeling Schemes for Trees [article]

Ofer Freedman, Paweł Gawrychowski, Patrick K. Nicholson, Oren Weimann
2017 arXiv   pre-print
Prior to our work, all distance labeling schemes for trees could be reinterpreted as universal trees. A tree T is said to be universal if any tree on n nodes can be found as a subtree of T.  ...  For the particular case of trees, optimal bounds (up to low order terms) were recently obtained for adjacency labeling [FOCS'15], nearest common ancestor labeling [SODA'14], and ancestry labeling [SICOMP  ...  In particular, it means that for level-ancestor queries, the scheme of Alstrup et al. [8] is optimal (after some modifications described in Section 3.6). Labeling schemes for bounded distances.  ... 
arXiv:1608.00212v2 fatcat:pziujxmqafd55dratk57wte57e

Compact and localized distributed data structures

Cyril Gavoille, David Peleg
2003 Distributed computing  
The survey concentrates mainly on combinatorial and algorithmic techniques, such as adjacency and distance labeling schemes and interval schemes for routing, and covers complexity results on various applications  ...  , focusing on compact localized schemes for message routing in communication networks.  ...  Hence Θ(n) bits is the optimal distance label length for general unweighted graphs.  ... 
doi:10.1007/s00446-002-0073-5 fatcat:bqydpls23ncnnm57nnhmctnz3m

Distance Labeling for Permutation Graphs

Fabrice Bazzaro, Cyril Gavoille
2005 Electronic Notes in Discrete Mathematics  
The datastructure is localized and given as a distance labeling, that is each vertex receives a label of O(log n) bits so that distance queries between any two vertices are answered by inspecting on their  ...  labels only.  ...  Trees, bounded tree-width and bounded clique-width graphs have O(log 2 n) bit distance labeling schemes.  ... 
doi:10.1016/j.endm.2005.06.098 fatcat:qh6ijtinbbc6dj556hqndgspdu

Compact Multicast Routing [chapter]

Ittai Abraham, Dahlia Malkhi, David Ratajczak
2009 Lecture Notes in Computer Science  
For trees, routing requires (log 2 / log log ) memory and header size, and this is known to be optimal [6].  ...  So a complete information scheme should maintain at each node the next step(s) of the shortest spanning tree, for all possible subsets.  ...  It is an interesting open question to find the optimal trade-offs between storage and space for the various problems considered in this paper.  ... 
doi:10.1007/978-3-642-04355-0_40 fatcat:kjm25bnqi5f5xpxrj72dteoiiu

Approximate Distance Labeling Schemes

Cyril Gavoille, Christophe Paul
2000 Electronic Notes in Discrete Mathematics  
We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance  ...  We also establish lower bounds, and prove that most of our schemes are optimal in terms of the length of the labels generated and the quality of the approximation.  ...  We would like to thank Stéphane Pérennes for his help in developing the upper bound for trees, and Douglas West for his bibliographic support.  ... 
doi:10.1016/s1571-0653(05)80145-3 fatcat:zpns4qhc55fmbo4hqu7jfgvjrq

Informative Labeling Schemes for Graphs [chapter]

David Peleg
2000 Lecture Notes in Computer Science  
Considering the class of n-vertex trees with M-bit edge weights, it is shown that for this class there exists a Steiner labeling scheme using O((M + log n) log n) bit labels, which is asymptotically optimal  ...  We begin by developing f labeling schemes for three functions f over the class of n-vertex trees.  ...  Acknowledgements I am grateful to Michal Katz and Nir Katz for their helpful comments and suggestions.  ... 
doi:10.1007/3-540-44612-5_53 fatcat:tr3wjlnap5azzmjazyv2x2dqpq

Informative labeling schemes for graphs

David Peleg
2005 Theoretical Computer Science  
Considering the class of n-vertex trees with M-bit edge weights, it is shown that for this class there exists a Steiner labeling scheme using O((M + log n) log n) bit labels, which is asymptotically optimal  ...  We begin by developing f labeling schemes for three functions f over the class of n-vertex trees.  ...  Acknowledgements I am grateful to Michal Katz and Nir Katz for their helpful comments and suggestions.  ... 
doi:10.1016/j.tcs.2005.03.015 fatcat:aigpwzyhorh3rpueyf4ita7iwu

Proximity-Preserving Labeling Schemes and Their Applications [chapter]

David Peleg
1999 Lecture Notes in Computer Science  
It is shown that for the class of n-vertex weighted trees with M -bit edge weights, there exists such a proximity-preserving labeling scheme using O(M log n + log 2 n) bit labels.  ...  This paper considers informative labeling schemes for graphs.  ...  Acknowledgements I am grateful to Cyril Gavoille, Yehuda Hassin and Richard Tan for helpful discussions.  ... 
doi:10.1007/3-540-46784-x_5 fatcat:foltucx5ejdodb4enkkgbyjsyq

Space-Optimal Time-Efficient Silent Self-Stabilizing Constructions of Constrained Spanning Trees

Lelia Blin, Pierre Fraigniaud
2015 2015 IEEE 35th International Conference on Distributed Computing Systems  
known algorithms designed and optimized specifically for solving each of these two tasks.  ...  In this paper, we make a first attempt for answering the question of whether such an efficient algorithm exists for every task, by focussing on constrained spanning tree construction tasks.  ...  As for the distance-based labeling scheme using the pairs (ID, d), the pairs (ID, s) alone provide a proof-labeling scheme for spanning trees.  ... 
doi:10.1109/icdcs.2015.66 dblp:conf/icdcs/BlinF15 fatcat:r4od2ao7gbdhpbub7tjoksvhfq

Average Case Analysis for Tree Labelling Schemes [chapter]

Ming-Yang Kao, Xiang-Yang Li, WeiZhao Wang
2005 Lecture Notes in Computer Science  
For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least 1 8 log 2 n − O(log n) bits.  ...  In this paper, we present two new distance labelling schemes that not only achieve the optimal label length Θ(log n · log(H n (T ))), but also have a much smaller expected label length under certain tree  ...  In this paper, we study distance labelling schemes for unweighted trees. For trees, Gavoille et al.  ... 
doi:10.1007/11602613_15 fatcat:lrksd45xivdyxny557mepw4jty

On rotation distance between binary coupling trees and applications for 3nj-coefficients

V. Fack, S. Lievens, J. Van der Jeugt
1999 Computer Physics Communications  
This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2n − 1)!!.  ...  The vertices of this graph G n consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3nj-coefficients.  ...  A binary coupling tree on n + 1 leaves is a rooted binary tree such that • the n + 1 leaves are labelled 1, 2, . . . , n + 1; • the internal vertices are not labelled; • for each internal vertex, one can  ... 
doi:10.1016/s0010-4655(99)00216-7 fatcat:wilbt4vdmbfezlmdziikmnov5i

A highway-centric labeling approach for answering distance queries on large sparse graphs

Ruoming Jin, Ning Ruan, Yang Xiang, Victor Lee
2012 Proceedings of the 2012 international conference on Management of Data - SIGMOD '12  
Here, we propose a novel labeling scheme, referred to as Highway-Centric Labeling, for answering distance queries in a large sparse graph.  ...  It empowers the distance labeling with a highway structure and leverages a novel bipartite set cover framework/algorithm.  ...  There are several interval-labeling schemes [25] for a rooted directed tree. A simple scheme is as follows.  ... 
doi:10.1145/2213836.2213887 dblp:conf/sigmod/JinRXL12 fatcat:dzuxwapr4zfbvmhjhnrwauifzy

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

Victor Chepoi, Feodor F. Dragan, Bertrand Estellon, Michel Habib, Yann Vaxès, Yang Xiang
2010 Algorithmica  
routing protocol, and a distance labeling scheme with O(log 2 n) bit labels, O(δ log n) additive error and constant time distance decoder.  ...  As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ ≥ 1/2 admits a routing labeling scheme with O(δ log 2 n) bit labels, O(δ log n) additive stretch and O(log 2 (4δ)) time  ...  We would like to thank the referees for many useful suggestions and remarks.  ... 
doi:10.1007/s00453-010-9478-x fatcat:6dtmgfdgjnh4zdnut7jddioppq

Distance labeling schemes for trees [article]

Stephen Alstrup and Inge Li Gørtz and Esben Bistrup Halvorsen and Ely Porat
2015 arXiv   pre-print
We consider distance labeling schemes for trees: given a tree with n nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels  ...  We also give (1+ϵ)-stretch labeling schemes using Θ( n) bits for constant ϵ>0. (1+ϵ)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs  ...  Fix an optimal leaf distance labeling scheme σ which produces exactly g(h, W, a) distinct labels for the family of (h, W, a)-trees.  ... 
arXiv:1507.04046v1 fatcat:ritscusvkbde7bhmbezfmj443i

Distance labeling scheme and split decomposition

Cyril Gavoille, Christophe Paul
2003 Discrete Mathematics  
This yields to optimal label length schemes for the family of distance-hereditary graphs and for other families of graphs, allowing distance estimation in constant time once the labels have been constructed  ...  A distance labeling scheme is a distributed data-structure designed to answer queries about distance between any two vertices of a graph G.  ...  Applying this technique, we show that the optimal schemes for trees obtained by [17, 19] can be generalized to distance-hereditary graphs.  ... 
doi:10.1016/s0012-365x(03)00232-2 fatcat:blqilrue4nfanms7lpnvxal5ey
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