Results 1  10
of
16
Greedy Drawings of Triangulations
, 2007
"... Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the fol ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the following conjecture: Any 3connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s = v1,v2,...,vk = t in a drawing is said to be distance decreasing if �vi − t � < �vi−1 − t�, 2 ≤ i ≤ k where �... � denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Walter Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the KnasterKuratowskiMazurkiewicz Theorem, that some drawing of G belonging to this class is greedy. 1 1
Convex drawings of graphs with nonconvex boundary
 Proc. of WG 2006
, 2006
"... Abstract. In this paper, we study a new problem of finding a convex drawing of graphs with a nonconvex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a starshaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Abstract. In this paper, we study a new problem of finding a convex drawing of graphs with a nonconvex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a starshaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such a drawing, called an innerconvex drawing, can be obtained in linear time. 1
Schnyder woods and orthogonal surfaces
 In Proceedings of Graph Drawing
, 2006
"... In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the BrightwellTrotter Theorem which says, that the face lattice of a 3polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time. 1
Bijections for Baxter Families and Related Objects
, 2008
"... The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2) ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)
Drawing graphs in the plane with a prescribed outer face and polynomial area
 Proc. 18th Int. Symp. on Graph Drawing (GD 2010
"... We study the classic graph drawing problem of drawing a planar graph using straightline edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addi ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We study the classic graph drawing problem of drawing a planar graph using straightline edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straightline drawing of a combinatoriallyembedded genusg graph with the graph’s canonical polygonal schema drawn as a convex polygonal external face. Submitted:
Embedding Stacked Polytopes on a PolynomialSize Grid
, 2011
"... We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) × O(n4) × O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts t ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) × O(n4) × O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by Günter M. Ziegler, and is the first nontrivial subexponential upper bound on the longstanding open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about efficient versions of Steinitz’s 1916 theorem. An immediate consequence of our result is that O(log n)bit coordinates suffice for a greedy routing strategy in planar 3trees.
Orthogonal Surfaces and their CPorders
, 2007
"... Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with c ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which nongeneric orthogonal surfaces have a polytopal structure. We review the state of knowledge of the 3dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cporders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cporders can lack key properties of face lattices. We investigate extra requirements which may help to have cporders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.
Stateless and Delivery Guaranteed Geometric Routing on Virtual Coordinate System
, 2008
"... Stateless geographic routing provides relatively good performance at a fixed overhead, which is typically much lower than conventional routing protocols such as AODV. However, the performance of geographic routing is impacted by physical voids, and localization errors. Accordingly, virtual coordinat ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Stateless geographic routing provides relatively good performance at a fixed overhead, which is typically much lower than conventional routing protocols such as AODV. However, the performance of geographic routing is impacted by physical voids, and localization errors. Accordingly, virtual coordinate systems (VCS) were proposed as an alternative approach that is resilient to localization errors and that naturally routes around physical voids. However, VCS also faces virtual anomalies, causing their performance to trail geographic routing. In existing VCS routing protocols, there is a lack of an effective stateless and delivery guaranteed complementary routing algorithm that can be used to traverse voids. Most proposed solutions use variants of flooding or blind searching when a void is encountered. In this paper, we propose a spanningpath virtual coordinate system which can be used as a complete routing algorithm or as the complementary algorithm to greedy forwarding that is invoked when voids are encountered. With this approach, and for the first time, we demonstrate a stateless and delivery guaranteed geometric routing algorithm on VCS. When used in conjunction with our previously proposed aligned virtual coordinate system (AVCS), it outperforms not only all geometric routing protocols on VCS, but also geographic routing with accurate location information.
Natural Wireless Localization is NPhard
"... We consider a special class of art gallery problems inspired by wireless localization. Given a polygonal region P, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the boundary ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We consider a special class of art gallery problems inspired by wireless localization. Given a polygonal region P, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the boundary of P. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. We prove NPhardness of one variant of the problem, namely the natural setting where guards may be placed aligned to a boundary edge or two consecutive boundary edges of P only. Introduction. Art gallery problems are a classic
Pointed Drawings of Planar Graphs
, 2008
"... We study the problem how to draw a planar graph such that every vertex is incident to an angle greater than π. In general a straightline embedding cannot guarantee this property. We present algorithms which construct such drawings with either tangentcontinuous biarcs or quadratic Bézier curves (par ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We study the problem how to draw a planar graph such that every vertex is incident to an angle greater than π. In general a straightline embedding cannot guarantee this property. We present algorithms which construct such drawings with either tangentcontinuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straightline embedding of the graph. Moreover, the graph can be embedded with circular arcs if the vertices can be placed arbitrarily. The topic is related to noncrossing drawings of multigraphs and vertex labeling.