Labeling Schemes for Small Distances in Trees

Abstract

We consider labeling schemes for trees, supporting various relationships between nodes at small distance. For instance, we show that given a tree T and an integer k we can assign labels to each node of T such that given the label of two nodes we can decide, from these two labels alone, if the distance between v and w is at most k and, if so, compute it. For trees with n nodes and k ≥ 2, we give a lower bound on the maximum label length of log n + Ω(log log n) bits, and for constant k, we give an upper bound of log n+O(log log n). Bounds for ancestor, sibling, connectivity, and bi-and triconnectivity labeling schemes are also presented.

On the Succinct Representation of Equivalence Classes

“…In the direct equivalence query problem, we give each element a unique label such that we can answer the equivalence query by computing directly from the two labels without using any auxiliary data except for the value of n. This problem was studied by Alstrup et al [2], where they showed that lg n + Θ(lg lg n) bits of space are necessary and sufficient to represent the labels. This bound was further strengthened by Lewenstein et al [16] to lg n + lg lg n + Θ(1)).…”

“…This problem has been studied by Katz et al [15], Alstrup et al [2], and Lewenstein et al [16]. Lewenstein et al [16] showed that with no auxiliary data structure, a label space of size n i=1 n i is necessary and sufficient to represent the equivalence relation.…”

“…They tightened the lower bound in [2] to a label space of n i=1 n i , which can be represented in lg n + lg lg n − Ω(1) bits, and proved that it is sufficient. Moreover, they showed that one can solve the problem using lg n + lg lg n + 2 bits such that equivalence queries can be answered in Θ(1) time.…”

“…As there is a huge body of research in 'labeling schemes' (see [2]), investigation into such a tradeoff for other labeling schemes maybe interesting.…”

Labeling Schemes for Vertex Connectivity

“…Our focus is thus on informative labeling schemes using relatively short labels (say, of length polylogarithmic in n). Labeling schemes of this type were recently developed for different graph families and for a variety information types, including vertex adjacency [Alstrup and Rauhe 2002;Breuer 1966;Breuer and Folkman 1967;Kannan et al 1992;Korman et al 2006], distance [Alstrup et al 2005;Katz et al 2000;Kaplan and Milo 2001;Korman et al 2006;Peleg 1999;Thorup 2004], tree routing [Fraigniaud and Gavoille 2001;Fraigniaud and Gavoille 2002;Thorup and Zwick 2001], vertex-connectivity [Alstrup and Rauhe 2002;Katz et al 2004], flow Katz et al 2004], tree ancestry [Abiteboul at al. 2001;Abiteboul at al.…”