Simpler Adjacency Labeling for Planar Graphs with B-Trees

Abstract

A permutation graph is the intersection graph of a set of segments between two parallel lines. In other words, they are defined by a permutation π on n elements, such that u and v are adjacent if an only if u < v but π(u) > π(v). We consider the problem of computing the distances in such a graph in the setting of informative labeling schemes.The goal of such a scheme is to assign a short bitstring ℓ(u) to every vertex u, such that the distance between u and v can be computed using only ℓ(u) and ℓ(v), and no further knowledge about the whole graph (other than that it is a permutation graph). This elegantly captures the intuition that we would like our data structure to be distributed, and often leads to interesting combinatorial challenges while trying to obtain lower and upper bounds that match up to the lower-order terms.For distance labeling of permutation graphs on n vertices, Katz, Katz, and Peleg [STACS 2000] showed how to construct labels consisting of O(log 2 n) bits. Later, Bazzaro and Gavoille [Discret. Math. 309(11)] obtained an asymptotically optimal bounds by showing how to construct labels consisting of 9 log n+O(1) bits, and proving that 3 log n−O(log log n) bits are necessary. This however leaves a quite large gap between the known lower and upper bounds. We close this gap by showing how to construct labels consisting of 3 log n + O(log log n) bits.

Sparse universal graphs for planarity

“…use of the bulk tree sequences from [12] as a "black box" in Sections 2 and 3, they can also be replaced with the approach based on B-trees from [14] in these proofs. This reduces the factor 𝑛 ⋅ 2 𝑂( √ log 𝑛⋅log log 𝑛) in Theorems 1 and 2 to 𝑛 ⋅ 2 𝑂( √ log 𝑛) .…”

“…Very recently, Gawrychowski and Janczewski [14] showed that the use of bulk tree sequences in [12] could be replaced with a simpler approach based on B‐trees, while leaving the rest of the proof essentially unchanged. This simplifies the data‐structure part of the proof in [12] and also gives a slightly improved bound of $$n\cdot 2^{O(\sqrt {\log n})}$$ on the number of vertices in the resulting induced‐universal graph for $$n$$‐vertex planar graphs, compared to $$n\cdot 2^{O(\sqrt {\log n \cdot \log \log n})}$$ in [12].…”

“…This simplifies the data‐structure part of the proof in [12] and also gives a slightly improved bound of $$n\cdot 2^{O(\sqrt {\log n})}$$ on the number of vertices in the resulting induced‐universal graph for $$n$$‐vertex planar graphs, compared to $$n\cdot 2^{O(\sqrt {\log n \cdot \log \log n})}$$ in [12]. Given our use of the bulk tree sequences from [12] as a “black box” in Sections 2 and 3, they can also be replaced with the approach based on B‐trees from [14] in these proofs. This reduces the factor $$n\cdot 2^{O(\sqrt {\log n \cdot \log \log n})}$$ in Theorems 1 and 2 to $$n\cdot 2^{O(\sqrt {\log n})}$$.…”

“…On the other hand, the proofs in Section 4 do depend on the inner workings of bulk tree sequences, and as such it is not immediately clear whether they could be replaced with B‐trees. As in [14], we leave this as an open problem.…”