Constructing Labeling Schemes through Universal Matrices

Abstract

Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let ℱ(n) be a family of connected graphs of size at most n and let $\mathcal{C}(\mathcal{F},n)$denote the collection of graphs of size at most n, such that each graph in $\mathcal{C}(\mathcal{F},n)$is composed of a disjoint union of some graphs in ℱ(n). We first investigate methods for translating f-labeling schemes for ℱ(n) to f-labeling schemes for $\mathcal{C}(\mathcal{F},n)$. In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family ℱ(n), one can construct an f-labeling scheme of size g(n)+log log n+O(1) for $\mathcal{C}(\mathcal{F},n)$. We also show that in several cases, the above mentioned extra additive term of log log n+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5log n+O(1) and a lower bound of 4/3log n−O(1) for the size of any such labeling scheme.