Unweighted Planar Graphs Research Articles

Better Distance Labeling for Unweighted Planar Graphs

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 O(sqrt{n}log n) (Gavoille, Peleg, Pérennes, and Raz in J Algorithms 53:85–112, 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 in 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 O(sqrt{n}) bits with a simple and (in our opinion) elegant method. We also show how to extend this to graphs with small weight and (unweighted) graphs with bounded genus. We augment the construction for unweighted planar graphs with a mechanism (based on Voronoi diagrams) that allows us to compute the distance between two nodes in only polylogarithmic time while increasing the length to O(sqrt{nlog n}). The previous scheme required varOmega (sqrt{n}) time to answer a query in this model.

Non-Crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

Given a set of terminal pairs on the external face of an undirected unweighted planar graph, we give a linear-time algorithm for computing the union of non-crossing shortest paths joining each terminal pair, if such paths exist. This allows us to compute distances between each terminal pair, within the same time bound. We also give a novel concept of incremental shortest path subgraph of a planar graph, i.e., a partition of the planar embedding in subregions that preserve distances, that can be of interest itself.

Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs

In this article, we show how to compute for n -vertex planar graphs in O ( n 11/6 polylog( n )) expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In O ( n 15/8 polylog( n )) expected time, we can also compute the number of pairs of vertices at distances smaller than a given threshold. These are the first algorithms for these problems using time O ( n c ) for some constant c < 2, even when restricted to undirected, unweighted planar graphs.

Tree [formula omitted]-spanners in outerplanar graphs via supply demand partition

A treet-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the Treet-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from Treet-spanner in outerplanar graphs to the supply–demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve Treet-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(nlogn) time.

Counting and sampling minimum [formula omitted]-cuts in weighted planar graphs in polynomial time

We give an O ( n d + n log n ) algorithm computing the number of minimum ( s , t ) -cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest s – t path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomial-time algorithm for unweighted planar graphs with both s and t lying on the outer face. Our results hold for all locations of s and t and weighted graphs, and have direct applications in computer vision.

Spanners in sparse graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t ⩽ 3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t ⩾ 4 . In this work we resolve the open question of Fekete and Kremer by proving much more general results: • The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear. • Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus. • Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t ⩾ 4 , the problem of finding a tree t-spanner is NP-complete on K 6 -minor-free graphs.

Plane Embeddings of Planar Graph Metrics

Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires \(\Theta(\sqrt n)\) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matousek [M3] by proving that some planar graph metrics require \(\Omega(n^{2/3})\) distortion in any embedding into the plane, proving the first separation between these two types of graph metrics. We also prove that some planar graph metrics require \(\Omega(n)\) distortion in any crossing-free straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied notion of crossing-free straight-line planar drawings. Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with \(O(\sqrt n)\) distortion, generalizing the previous results on trees (both the worst-case bound and the approximation algorithm) and building techniques for handling cycles in plane embeddings of graph metrics.