75 research outputs found
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems
Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n^{ (11+ o(1)) sqrt{n} }, significantly improving the previous best running time of O(2^n n^2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O(sqrt{n}) of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in n^{O(sqrt{n})} time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, 3-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, 3-regular graphs, and more
Peeling and nibbling the cactus: Subexponential-time algorithms for counting triangulations and related problems
Given a set of points in the plane, a triangulation of is a
maximal set of non-crossing segments with endpoints in . We present an
algorithm that computes the number of triangulations on a given set of
points in time , significantly improving the previous
best running time of by Alvarez and Seidel [SoCG 2013]. Our main
tool is identifying separators of size of a triangulation in a
canonical way. The definition of the separators are based on the decomposition
of the triangulation into nested layers ("cactus graphs"). Based on the above
algorithm, we develop a simple and formal framework to count other non-crossing
straight-line graphs in time. We demonstrate the usefulness
of the framework by applying it to counting non-crossing Hamilton cycles,
spanning trees, perfect matchings, -colorable triangulations, connected
graphs, cycle decompositions, quadrangulations, -regular graphs, and more.Comment: 47 pages, 23 Figures, to appear in SoCG 201
Shortest Reconfiguration of Perfect Matchings via Alternating Cycles
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Reconstructing Geometric Structures from Combinatorial and Metric Information
In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable.
Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets.
Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding
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