On the Geometric Dilation of Closed Curves, Graphs and Point Sets

2005

Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound > 1. In this paper we provide the first upper and lower bounds for the embedding problem.

Siam Journal on Discrete Mathematics, 2009

Fomin and Thilikos in conjectured that there is a constant c such that, for every 2connected planar graph G, pw(G * ) ≤ 2pw(G)+c (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and actually is tight by Coudert, Huc and Sereni [4]. In , Fomin and Thilikos proved that there is a constant c such that the pathwidth of every 3-connected graph G satisfies: pw(G * ) ≤ 6pw(G) + c. In this paper we improve this result by showing that the dual a 3-connected planar graph has pathwidth at most 3 times the pathwidth of the primal plus two. We prove also that the question can be answered positively for 4-connected planar graphs.

2010

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the dilation in T of any pair p, p ∈ P , which is the ratio of the length of the shortest path from p to p in T over the Euclidean distance pp , can be at most π/2 ≈ 1.5708. In this paper, we show how to construct point sets in convex position with dilation > 1.5810 and in general position with dilation > 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case dilation for that distribution.

Journal of Combinatorial Theory, Series A, 1997

We prove that every nite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive.

International Journal of Computational Geometry and Applications, 2010

Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortestpath distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worst-case ratio of these two values, for all pairs of vertices, is called the vertex-to-vertex dilation of G.

Lecture Notes in Computer Science, 2009

Data access time becomes the main bottleneck in applications dealing with large-scale graphs. Cache-oblivious layouts, constructed to minimize the geometric mean of arc lengths of graphs, have been adapted to reduce data access time during random walks on graphs. In this paper, we present a constant factor approximation algorithm for the Minimum Geometric Mean Layout (MGML) problem for bounded-degree planar graphs. We also derive an upper bound for any layout of the MGML problem. To the best of our knowledge, these are the first results for the MGML problem with bounded-degree planar graphs.

Discrete Mathematics, 1990

We first show that the removal of 4fi vertices from an n-vertex planar graph with non-negative vertex weights summing to no more than 1 is sufficient to cleave or recursively separate it into components of weight no more than a given E, thus improving on the 2fia bound shown in . We then derive worst-case bounds on the number of vertices necessary to separate a planar graph of a given radius into components of weight no more than E.

Networks, 2007

The dilation-free graph of a planar point set S is a graph that spans S in such a way that the distance between two points in the graph is no longer than their planar distance. Metrically speaking, those graphs are equivalent to complete graphs; however they have far fewer edges when considering the Manhattan distance (we give here an upper bound on the number of saved edges). This article provides several theoretical, algorithmic, and complexity features of dilation-free graphs in the l 1-metric, giving several construction algorithms and proving some of their properties. Moreover, special attention is paid to the planar case due to its applications in the design of printed circuit boards.

Discrete & Computational Geometry, 1994

Given an undirected edge-weighted graph G = (V, E), a subgraph G' = (IT, E') is a t-spanner of G if, for every u, v ~ V, the weighted distance between u and v in G' is at most t times the weighted distance between u and v in G. We consider the problem of approximating the distances among points of a Euclidean metric space: given a finite set V of points in ~a, we want to construct a sparse t-spanner of the complete weighted graph induced by V. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for any t > 1 and any V c R a, a t-spanner G of V exists such that G has degree bounded by a function of d and r The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclidean t-spanners, even compared with spanners of bounded average degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimension d = 2. It is fairly easy to see that, for some t (t > 7.6), t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed) t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. * This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico, Proc 203039/87.4 (Brazil). 214 J. Soares shortest path between x and y. We say that a subgraph G' = (V, E') (with the same weights on E') is a t-spanner of G if, for every x, y 6 V, dG,(x, y) < t" da(x, y). The number t is a measure of how well G' approximates G with respect to the distances. The construction of t-spanners has received recent attention in several works: [2], [3], [5], [8], [9], [11], and [18], among others. Given a set V ~ •a the complete Euclidean graph on V is the complete graph on V where each edge weight is the Euclidean distance I[x-Y]I. In this paper we consider the problem of constructing bounded degree spanners of complete Euclidean graphs. For brevity we write t-spanner of V instead of t-spanner of the complete Euclidean graph on V. Let A(G) denote the maximum degree of a graph G. Dobkin et al. [5] mention that Feder and others had shown that, for some fixed t and for any set V of points in the Euclidean plane, a t-spanner G of V exists such that A(G) < 7. Then they ask what would be the minimum A for which such a result is possible? This paper has a partial answer to this question. Our main result (Section 4) is that, for some fixed t, t-spanners with A < 5 exist. Nisan [10] has proved the same for A < 6. Section 2 contains the basic algorithm used to construct bounded degree t-spanners. Although the algorithm has been used before by Althrfer et al. [1] and Soares [16] to construct t-spanners for arbitrary graphs, it was not known that the algorithm also constructs bounded degree spanners for complete Euclidean graphs. Section 3 contains a brief analysis of the problem when V is in d-dimensional Euclidean space. We show that, for any t > 1 and any V c ~d, a t-spanner G of V exists where A(G) is bounded by a function that depends only on d and t. This answers a question proposed by Keil and Gutwin in [8]. This bound on the maximum degree implies an improvement on the previously known upper bounds on the number of edges sufficient to build Euclidean spanners. Then we show that, for each dimension d, the least A(G) for which our algorithm constructs Od(1)spanners coincides with the kissing number in dimension d. (Od(1) denotes some function of d, i.e., a constant for each d.) Section 4 contains our main result, the construction of O(1)-spanners of degree 5 for any set of points in the Euclidean plane.

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2018

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k 2 log n) bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k = √ n, the upper bound is O(n) and their constructions imply an Ω(n 3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric intoÕ(min(k 2 , √ k • n)) bits, which is optimal up to log factors. Our data structure circumvents an Ω(k 2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs withÕ(√ k • n) space, andÕ(n 3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n 1/2) vs. Ω(n 1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.