Good Quality Virtual Realization of Unit Ball Graphs

The quality of an embedding Φ : V 7→ R of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E ||Φ(u) − Φ(v)||2 to min{u,v}6∈E ||Φ(u) − Φ(v)||2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R, we seek algorithms to compute an embedding Φ : V 7→ R of best (smallest) quality. Note that G comes with no associated geometric information and in this setting, related problems such as recognizing if G is a unit disk graph (UDG), are NP-hard. While any connected unit disk graph (UDG) has a 2-dimensional embedding with quality between 1/2 and 1, as far as we know, Vempala’s random projection approach (FOCS 1998) provides the best quality bound of O(log n · √ log log n) for this problem. This paper presents a simple, combinatorial algorithm for computing a O(log n)-quality 2-dimensional embedding of a given graph, that is known to be a UBG in fixed dimensional Euclidean space R. The embedding produced by this algorithm is also an O(log n)-factor approximation to the problem of minimizing the ratio of the total edge length to minimum distance between non-adjacent vertices. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in R log . The first step of our algorithm constructs a “growthrestricted approximation” of the given UBG. While such a construction is trivial if the UBG comes with a geometric representation, we are not aware of any other algorithm that can perform this step without geometric information. Construction of a growth-restricted approximation permits us to bypass the standard and costly technique of solving a linear program with exponentially many “spreading constraints.” As a side effect of our construction, we get a constantfactor approximation to the minimum clique cover problem for UBGs, described without geometry. To the best of our knowledge, this is the first constant factor clique partition of a UBG without geometry. Furthermore, we compute such a cover in O(log n) rounds in a distributed manner. The second step of our algorithm combines the probabilistic decomposition of growth-restricted graphs due to Lee and Krauthgamer (STOC 2003) with Rao’s embedding algorithm for planar graphs (SoCG 1999) to obtain a (k, O( √ log n))-volume respecting embedding of growth-restricted graphs. Our problem is a version of the well known localization problem in wireless sensor networks, in which network nodes are required to compute virtual 2-dimensional Euclidean coordinates given little or (as in our case) no geometric information.