Better Distance Labeling for Unweighted Planar Graphs

Abstract

A distance labeling scheme is an assignment of labels, that is, binary strings, to all nodes of a graph, so that the distance between any two nodes can be computed from their labels without any additional information about the graph. The goal is to minimize the maximum length of a label as a function of the number of nodes. A major open problem in this area is to determine the complexity of distance labeling in unweighted planar (undirected) graphs. It is known that, in such a graph on n nodes, some labels must consist of $\varOmega (n^{1/3})$ bits, but the best known labeling scheme constructs labels of length $\mathcal {O}(\sqrt{n}\log n)$ [Gavoille, Peleg, Pérennes, and Raz, J. Algorithms, 2004]. For weighted planar graphs with edges of length polynomial in n, we know that labels of length $\varOmega (\sqrt{n}\log n)$ are necessary [Abboud and Dahlgaard, FOCS 2016]. Surprisingly, we do not know if distance labeling for weighted planar graphs with edges of length polynomial in n is harder than distance labeling for unweighted planar graphs. We prove that this is indeed the case by designing a distance labeling scheme for unweighted planar graphs on n nodes with labels consisting of $\mathcal {O}(\sqrt{n})$ bits with a simple and (in our opinion) elegant method. We augment the construction with a mechanism that allows us to compute the distance between two nodes in only polylogarithmic time while increasing the length by $\mathcal {O}(\sqrt{n\log n})$. The previous scheme required $\varOmega (\sqrt{n})$ time to answer a query in this model.