Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

Topological Phases: An Expedition off Lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of `baby universe', Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger's theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.Comment: 38 pages, 22 figure

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc