Simpler, faster and shorter labels for distances in graphs

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label length and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different tradeoffs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with labels of length log3/2n + o(n) bits1 and constant decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(log log n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for the label size e: 1/2(n -- 1) log [W/2 + l] ≤ e ≤ 1/2n log (2W + 1) + O(log n · log(nW)). Furthermore, for r-additive approximation labeling schemes, where distances can be off by up to an additive constant r, we present both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency labeling scheme, namely n/2. We also give results for bipartite graphs as well as for exact and 1-additive distance oracles.

[1]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[2]  Peter Winkler,et al.  Proof of the squashed cube conjecture , 1983, Comb..

[3]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[4]  David Peleg,et al.  Proximity-preserving labeling schemes , 2000, J. Graph Theory.

[5]  Mikkel Thorup,et al.  Adjacency Labeling Schemes and Induced-Universal Graphs , 2014, STOC.

[6]  Shiri Chechik,et al.  Compact Routing Schemes , 2016, Encyclopedia of Algorithms.

[7]  David Peleg,et al.  Informative labeling schemes for graphs , 2000, Theor. Comput. Sci..

[8]  M. Chiani.  Error Detecting and Error Correcting Codes , 2012 .

[9]  Amos Korman,et al.  Labeling schemes for vertex connectivity , 2007, TALG.

[10]  Feodor F. Dragan,et al.  Distance and routing labeling schemes for non-positively curved plane graphs , 2006, J. Algorithms.

[11]  Melvin A. Breuer,et al.  An unexpected result in coding the vertices of a graph , 1967 .

[12]  F. Chung.  Universal graphs and induced-universal graphs , 1990 .

[13]  Cyril Gavoille,et al.  Optimal Distance Labeling for Interval Graphs and Related Graph Families , 2008, SIAM J. Discret. Math..

[14]  David Peleg,et al.  Compact routing schemes with low stretch factor , 2003, J. Algorithms.

[15]  Béla Bollobás,et al.  Sparse distance preservers and additive spanners , 2003, SODA '03.

[16]  Melvin A. Breuer,et al.  Coding the vertexes of a graph , 1966, IEEE Trans. Inf. Theory.

[17]  Cyril Gavoille,et al.  Localized and compact data-structure for comparability graphs , 2005, Discret. Math..

[18]  Bruno Courcelle,et al.  Query efficient implementation of graphs of bounded clique-width , 2003, Discret. Appl. Math..

[19]  Rephael Wenger,et al.  Extremal graphs with no C4's, C6's, or C10's , 1991, J. Comb. Theory, Ser. B.

[20]  Yang Xiang,et al.  A highway-centric labeling approach for answering distance queries on large sparse graphs , 2012, SIGMOD Conference.

[21]  Gary Chartrand,et al.  A new look at Hamiltonian walks , 2004 .

[22]  Rajeev Raman,et al.  Succinct representations of permutations and functions , 2011, Theor. Comput. Sci..

[23]  Edith Cohen,et al.  Labeling dynamic XML trees , 2002, PODS '02.

[24]  Pierre Fraigniaud,et al.  An optimal ancestry scheme and small universal posets , 2010, STOC '10.

[25]  Pierre Fraigniaud,et al.  Compact ancestry labeling schemes for XML trees , 2010, SODA '10.

[26]  Nicolas Bonichon,et al.  Short Labels by Traversal and Jumping , 2006, Electron. Notes Discret. Math..

[27]  S. Hedetniemi,et al.  On the hamiltonian completion problem , 1974 .

[28]  David Peleg,et al.  Labeling Schemes for Weighted Dynamic Trees , 2003, ICALP.

[29]  Robert Krauthgamer,et al.  Algorithms on negatively curved spaces , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[30]  J. Moon,et al.  On minimal n-universal graphs , 1965, Proceedings of the Glasgow Mathematical Association.

[31]  Pierre Fraigniaud,et al.  On randomized representations of graphs using short labels , 2009, SPAA '09.

[32]  Moni Naor,et al.  Implicit representation of graphs , 1992, STOC '88.

[33]  David Peleg,et al.  Labeling schemes for flow and connectivity , 2002, SODA '02.

[34]  Cyril Gavoille,et al.  Approximate Distance Labeling Schemes , 2000, Electron. Notes Discret. Math..

[35]  Paolo Ferragina,et al.  On compact representations of All-Pairs-Shortest-Path-Distance matrices , 2010, Theor. Comput. Sci..

[36]  James Cheng,et al.  TF-Label: a topological-folding labeling scheme for reachability querying in a large graph , 2013, SIGMOD '13.

[37]  Haim Kaplan,et al.  A comparison of labeling schemes for ancestor queries , 2002, SODA '02.

[38]  Cyril Gavoille,et al.  Distance labeling scheme and split decomposition , 2003, Discret. Math..

[39]  Mathias Bæk Tejs Knudsen,et al.  Optimal Induced Universal Graphs and Adjacency Labeling for Trees , 2015, FOCS.

[40]  J. Ian Munro,et al.  Succinct encoding of arbitrary graphs , 2013, Theor. Comput. Sci..

[41]  John Harold Muller,et al.  Local structure in graph classes , 1988 .

[42]  J. Ian Munro,et al.  A Uniform Paradigm to Succinctly Encode Various Families of Trees , 2012, Algorithmica.

[43]  Gonzalo Navarro,et al.  Rank and select revisited and extended , 2007, Theor. Comput. Sci..

[44]  Andrew V. Goldberg,et al.  Hub Labels: Theory and Practice , 2014, SEA.

[45]  Ran Raz,et al.  Distance labeling in graphs , 2001, SODA '01.

[46]  Robert Krauthgamer,et al.  Bounded geometries, fractals, and low-distortion embeddings , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[47]  Mikkel Thorup.  Compact oracles for reachability and approximate distances in planar digraphs , 2004, JACM.

[48]  Mikkel Thorup,et al.  Changing base without losing space , 2010, STOC '10.

[49]  Nicola Santoro,et al.  Labelling and Implicit Routing in Networks , 1985, Computer/law journal.

[50]  Takuya Akiba,et al.  Fast exact shortest-path distance queries on large networks by pruned landmark labeling , 2013, SIGMOD '13.

[51]  Feodor F. Dragan,et al.  Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs , 2008, SCG '08.

[52]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[53]  Haim Kaplan,et al.  Compact labeling schemes for ancestor queries , 2001, SODA '01.

[54]  Kunal Talwar,et al.  Bypassing the embedding: algorithms for low dimensional metrics , 2004, STOC '04.

[55]  Stephen Alstrup,et al.  Small induced-universal graphs and compact implicit graph representations , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[56]  David Peleg,et al.  A note on exact distance labeling , 2011, Inf. Process. Lett..

[57]  Stephen Alstrup,et al.  Improved labeling scheme for ancestor queries , 2002, SODA '02.

[58]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .

[59]  Philip Bille,et al.  Labeling schemes for small distances in trees , 2003, SODA '03.

[60]  Igor Nitto,et al.  On Compact Representations of All-Pairs-Shortest-Path-Distance Matrices , 2008, CPM.

[61]  Stephen Alstrup,et al.  Compact Labeling Scheme for Ancestor Queries , 2006, SIAM J. Comput..

[62]  R. Graham,et al.  On embedding graphs in squashed cubes , 1972 .

[63]  Stephen Alstrup,et al.  Near-optimal labeling schemes for nearest common ancestors , 2013, SODA.

[64]  Lenore Cowen,et al.  Compact routing with minimum stretch , 1999, SODA '99.

[65]  Olivier Ly,et al.  Distance Labeling in Hyperbolic Graphs , 2005, ISAAC.