Abstract
Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes.
Let ℱ(n) be a family of connected graphs of size at most n and let \(\mathcal{C}(\mathcal{F},n)\) denote the collection of graphs of size at most n, such that each graph in \(\mathcal{C}(\mathcal{F},n)\) is composed of a disjoint union of some graphs in ℱ(n). We first investigate methods for translating f-labeling schemes for ℱ(n) to f-labeling schemes for \(\mathcal{C}(\mathcal{F},n)\) . In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family ℱ(n), one can construct an f-labeling scheme of size g(n)+log log n+O(1) for \(\mathcal{C}(\mathcal{F},n)\) . We also show that in several cases, the above mentioned extra additive term of log log n+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5log n+O(1) and a lower bound of 4/3log n−O(1) for the size of any such labeling scheme.
Similar content being viewed by others
References
Alstrup, S., Bille, P., Rauhe, T.: Labeling schemes for small distances in trees. In: Proc. 14th ACM-SIAM Symp. on Discrete Algorithms (2003)
Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest common ancestors: A survey and a new distributed algorithm. Theory Comput. Syst. 37, 441–456 (2004)
Abiteboul, S., Kaplan, H., Milo, T.: Compact labeling schemes for ancestor queries. In: Proc. 12th ACM-SIAM Symp. on Discrete Algorithms (2001)
Alstrup, S., Rauhe, T.: Improved labeling scheme for ancestor queries. In: Proc. 19th ACM-SIAM Symp. on Discrete Algorithms (2002)
Alstrup, S., Rauhe, T.: Small induced-universal graphs and compact implicit graph representations. In: Proc. 43rd IEEE Symp. on Foundations of Computer Science (2002)
Chung, F.R.K.: Universal graphs and induced-universal graphs. J. Graph. Theory 14(4), 443–454 (1990)
Cohen, E., Halperin, E., Kaplan, H., Zwick, U.: Reachability and distance queries via 2-hop labels. In: Proc. 13th ACM-SIAM Symp. on Discrete Algorithms (2002)
Cohen, E., Kaplan, H., Milo, T.: Labeling dynamic XML trees. In: Proc. 21st ACM Symp. on Principles of Database Systems (2002)
Fraigniaud, P., Gavoille, C.: Routing in trees. In: Proc. 28th Int. Colloq. on Automata, Languages & Prog. LNCS, vol. 2076, pp. 757–772. Springer, Berlin (2001)
Gavoille, C., Paul, C.: Split decomposition and distance labelling: an optimal scheme for distance hereditary graphs. In: Proc. European Conf. on Combinatorics, Graph Theory and Applications (2001)
Gavoille, C., Peleg, D.: Compact and localized distributed data structures. Distrib. Comput. 16, 111–120 (2003)
Gavoille, C., Katz, M., Katz, N.A., Paul, C., Peleg, D.: Approximate distance labeling schemes. In: Proc. 9th European Symp. on Algorithms, pp. 476–488 (2001)
Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. In: Proc. 12th ACM-SIAM Symp. on Discrete Algorithms, pp. 210–219 (2001)
Korman, A.: General compact labeling schemes for dynamic trees. In: Proc. 19th Symp. on Distributed Computing (2005)
Korman, A., Kutten, S.: Distributed verification of minimum spanning trees. In: 25th ACM Symp. on Principles of Distributed Computing (2006)
Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5, 596–603 (1992)
Kaplan, H., Milo, T.: Short and simple labels for small distances and other functions. In: Workshop on Algorithms and Data Structures (2001)
Kaplan, H., Milo, T.: Parent and ancestor queries using a compact index. In: Proc. 20th ACM Symp. on Principles of Database Systems (2001)
Kaplan, H., Milo, T., Shabo, R.: A comparison of labeling schemes for ancestor queries. In: Proc. 19th ACM-SIAM Symp. on Discrete Algorithms (2002)
Katz, M., Katz, N.A., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. In: Proc. 19th ACM-SIAM Symp. on Discrete Algorithms (2002)
Katz, M., Katz, N.A., Peleg, D.: Distance labeling schemes for well-separated graph classes. In: Proc. 17th Symp. on Theoretical Aspects of Computer Science, pp. 516–528 (2000)
Korman, A., Peleg, D.: Labeling schemes for weighted dynamic trees. In: Proc. 30th Int. Colloq. on Automata, Languages & Prog. (2003)
Korman, A., Peleg, D.: Compact separator decompositions in dynamic trees and applications to labeling schemes. In: Proc. 21st Symp. on Distributed Computing. SV-LNCS, vol. 4731, pp. 313–327. Springer, Berlin (2007)
Korman, A., Peleg, D., Rodeh, Y.: Labeling schemes for dynamic tree networks. Theory Comput. Syst. 37, 49–75 (2004)
Lozin, V.V.: On minimal universal graphs for hereditary classes. J. Discrete Math. Appl. 7(3), 295–304 (1997)
Lozin, V.V., Rudolf, G.: Minimal universal bipartite graphs. Ars Comb. 84 (2007)
Moon, J.W.: On minimal n-universal graphs. Proc. Glasgow Math. Soc. 7, 32–33 (1965)
Peleg, D.: Proximity-preserving labeling schemes and their applications. In: Proc. 25th Int. Workshop on Graph-Theoretic Concepts in Computer Science, pp. 30–41, June 1999
Peleg, D.: Informative labeling schemes for graphs. Theor. Comput. Sci. 340, 577–593 (2005)
Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)
Rado, R.: Universal graphs and universal functions. Acta Arith. 331–340 (1964)
Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51, 993–1024 (2004)
Thorup, M., Zwick, U.: Compact routing schemes. In: Proc. 13th ACM Symp. on Parallel Algorithms and Architecture, Hersonissos, Crete, pp. 1–10 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
A. Korman is supported in part at the Technion by an Aly Kaufman fellowship.
D. Peleg is supported in part by grants from the Israel Science Foundation and the Israel Ministry of Science and Art.
Rights and permissions
About this article
Cite this article
Korman, A., Peleg, D. & Rodeh, Y. Constructing Labeling Schemes through Universal Matrices. Algorithmica 57, 641–652 (2010). https://doi.org/10.1007/s00453-008-9226-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-008-9226-7