Universal graphs and induced-universal graphs
Journal of Graph Theory
Implicit representation of graphs
SIAM Journal on Discrete Mathematics
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Compact labeling schemes for ancestor queries
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Reachability and distance queries via 2-hop labels
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved labeling scheme for ancestor queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A comparison of labeling schemes for ancestor queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Labeling schemes for small distances in trees
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Small Induced-Universal Graphs and Compact Implicit Graph Representations
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Informative Labeling Schemes for Graphs
MFCS '00 Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science
Short and Simple Labels for Small Distances and Other Functions
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Distance Labeling Schemes for Well-Separated Graph Classes
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
Proximity-Preserving Labeling Schemes and Their Applications
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Approximate Distance Labeling Schemes
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Compact and localized distributed data structures
Distributed Computing - Papers in celebration of the 20th anniversary of PODC
Labeling Schemes for Flow and Connectivity
SIAM Journal on Computing
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Distributed verification of minimum spanning trees
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Labeling schemes for weighted dynamic trees
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
General compact labeling schemes for dynamic trees
DISC'05 Proceedings of the 19th international conference on Distributed Computing
Labeling schemes for weighted dynamic trees
Information and Computation
An Optimal Labeling for Node Connectivity
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Labeling schemes for vertex connectivity
ACM Transactions on Algorithms (TALG)
SIROCCO'07 Proceedings of the 14th international conference on Structural information and communication complexity
Distributed relationship schemes for trees
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Labeling schemes for vertex connectivity
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs ${\mathcal F}$ labels the vertices of all graphs in ${\mathcal F}$ such that for every graph $G\in{\mathcal F}$ 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 ${\mathcal F}$. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let ${\mathcal F}(n)$ 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 ${\mathcal F}(n)$. We first investigate methods for translating f-labeling schemes for ${\mathcal F}(n)$ 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 ${\mathcal F}(n)$, one can construct an f-labeling scheme of size g(n)+loglogn+O(1) for ${\mathcal C}({\mathcal F},n)$. We also show that in several cases, the above mentioned extra additive term of loglogn+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 logn+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 1.5logn+O(1) and a lower bound of 4/3logn–O(1) for the size of any such labeling scheme.