Implicit representation of graphs
SIAM Journal on Discrete Mathematics
Randomized algorithms
A small universal graph for bounded-degree planar graphs
Proceedings of the tenth 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
A comparison of labeling schemes for ancestor queries
SODA '02 Proceedings of the thirteenth 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
Labeling Schemes for Small Distances in Trees
SIAM Journal on Discrete Mathematics
Compact Labeling Scheme for Ancestor Queries
SIAM Journal on Computing
Informative labeling schemes for graphs
Theoretical Computer Science - Mathematical foundations of computer science 2000
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
A note on labeling schemes for graph connectivity
Information Processing Letters
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Informative labeling schemes consist in labeling the nodes of graphs so that queries regarding any two nodes (e.g., are the two nodes adjacent?) can be answered by inspecting merely the labels of the corresponding nodes. Typically, the main goal of such schemes is to minimize the label size, that is, the maximum number of bits stored in a label. This concept was introduced by Kannan et al. [STOC'88] and was illustrated by giving very simple and elegant labeling schemes, for supporting adjacency and ancestry queries in n-node trees; both these schemes have label size 2log n. Motivated by relations between such schemes and other important notions such as universal graphs, extensive research has been made by the community to further reduce the label sizes of such schemes as much as possible. The current state of the art adjacency labeling scheme for trees has label size log n+O(log*n) by Alstrup and Rauhe [FOCS'02], and the best known ancestry scheme for (rooted) trees has label size log n+O(√log n) by Abiteboul et al., [SICOMP 2006]. This paper aims at investigating the above notions from a probabilistic point of view. Informally, the goal is to investigate whether the label sizes can be improved if one allows for some probability of mistake when answering a query, and, if so, by how much. For that, we first present a model for probabilistic labeling schemes, and then construct various probabilistic one-sided error schemes for the adjacency and ancestry problems on trees. Some of our schemes significantly improve the bound on the label size of the corresponding deterministic schemes, while the others are matched with appropriate lower bounds showing that, for the resulting guarantees of success, one cannot expect to do much better in term of label size.