Labeling Schemes with Queries.

Recently, quite a few papers studied methods for representing network properties by assigning informative labels to the vertices of a network. Consulting the labels given to any two vertices u and v for some function f (e.g. "distance(u,v)") one can compute the function (e.g. the graph distance between u and v). Some very involved lower bounds for the sizes of the labels were proven. In this paper, we demonstrate that such lower bounds are very sensitive to the number of vertices consulted. That is, we show several almost trivial constructions of such labeling schemes that beat the lower bounds by large margins. The catch is that one needs to consult the labels of three vertices instead of two. We term our generalized model labeling schemes with queries. Additional contributions are several extensions. In particular, we show that it is easy to extend our schemes for tree to work also in the dynamic scenario. We also demonstrate that the study of the queries model can help in designing a scheme for the traditional model too. Finally, we demonstrate extensions to the non-distributed environment. In particular, we show that one can preprocess a general weighted graph using almost linear space so that flow queries can be answered in almost constant time. [ABSTRACT FROM AUTHOR]