On randomized representations of graphs using short labels

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 2 log 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.