4 research outputs found
Distance labeling schemes for trees
We consider distance labeling schemes for trees: given a tree with nodes,
label the nodes with binary strings such that, given the labels of any two
nodes, one can determine, by looking only at the labels, the distance in the
tree between the two nodes.
A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg
(J. Graph Theory 2000) establish that labels must use
bits\footnote{Throughout this paper we use for .}. Gavoille et.
al. (ESA 2001) show that for very small approximate stretch, labels use
bits. Several other papers investigate various
variants such as, for example, small distances in trees (Alstrup et. al.,
SODA'03).
We improve the known upper and lower bounds of exact distance labeling by
showing that bits are needed and that bits are sufficient. We also give ()-stretch labeling
schemes using bits for constant .
()-stretch labeling schemes with polylogarithmic label size have
previously been established for doubling dimension graphs by Talwar (STOC
2004).
In addition, we present matching upper and lower bounds for distance labeling
for caterpillars, showing that labels must have size . For simple paths with nodes and edge weights in , we show that
labels must have size
Distributed Relationship Schemes for Trees
We consider a distributed representation scheme for trees, supporting some special relationships between nodes at small distance. For instance, we show that for a tree T and an integer k we can assign local information on nodes such that we can decide for any two nodes u and v if the distance between u and v is at most k and if so, compute it only using the local information assigned. For trees with n nodes, the local information assigned by our scheme are binary labels of log n + O(k log(k log(n/k))) bits, improving the results of Alstrup, Bille, and Rauhe (SODA ’03)