1,320 research outputs found
Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of
the same plane graph. The morph preserves the convexity of the drawing at any
time instant and moves each vertex along a piecewise linear curve with linear
complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip
Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs
A greedy embedding of a graph into a metric space is a
function such that in the embedding for every pair of
non-adjacent vertices there exists another vertex adjacent
to which is closer to than . This notion of greedy
embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci.
2005), where authors conjectured that every 3-connected planar graph has a
greedy embedding (possibly planar and convex) in the Euclidean plane. Recently,
greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008).
However, their algorithm do not result in a drawing that is planar and convex
for all 3-connected planar graph in the Euclidean plane. In this work we
consider the planar convex greedy embedding conjecture and make some progress.
We derive a new characterization of planar convex greedy embedding that given a
3-connected planar graph , an embedding x: V \to \bbbr^2 of is
a planar convex greedy embedding if and only if, in the embedding , weight
of the maximum weight spanning tree () and weight of the minimum weight
spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq
(\card{V}-1)^{1 - \delta}, for some .Comment: 19 pages, A short version of this paper has been accepted for
presentation in FCT 2009 - 17th International Symposium on Fundamentals of
Computation Theor
Orderly Spanning Trees with Applications
We introduce and study the {\em orderly spanning trees} of plane graphs. This
algorithmic tool generalizes {\em canonical orderings}, which exist only for
triconnected plane graphs. Although not every plane graph admits an orderly
spanning tree, we provide an algorithm to compute an {\em orderly pair} for any
connected planar graph , consisting of a plane graph of , and an
orderly spanning tree of . We also present several applications of orderly
spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem,
(2) the first area-optimal 2-visibility drawing of , and (3) the best known
encodings of with O(1)-time query support. All algorithms in this paper run
in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of
the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001),
Washington D.C., USA, January 7-9, 2001, pp. 506-51
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Minimizing the stabbing number of matchings, trees, and triangulations
The (axis-parallel) stabbing number of a given set of line segments is the
maximum number of segments that can be intersected by any one (axis-parallel)
line. This paper deals with finding perfect matchings, spanning trees, or
triangulations of minimum stabbing number for a given set of points. The
complexity of these problems has been a long-standing open question; in fact,
it is one of the original 30 outstanding open problems in computational
geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide
is negative for a number of minimum stabbing problems by showing them NP-hard
by means of a general proof technique. It implies non-trivial lower bounds on
the approximability. On the positive side we propose a cut-based integer
programming formulation for minimizing the stabbing number of matchings and
spanning trees. We obtain lower bounds (in polynomial time) from the
corresponding linear programming relaxations, and show that an optimal
fractional solution always contains an edge of at least constant weight. This
result constitutes a crucial step towards a constant-factor approximation via
an iterated rounding scheme. In computational experiments we demonstrate that
our approach allows for actually solving problems with up to several hundred
points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational
Geometry". Previous version (extended abstract) appears in SODA 2004, pp.
430-43
Locally optimal unstructured finite element meshes in 3 dimensions
This paper investigates the adaptive finite element solution of a general class of variational problems in three dimensions using a combination of node movement, edge swapping, face swapping and node insertion. The adaptive strategy proposed is a generalization of previous work in two dimensions and is based upon the construction of a hierarchy of locally optimal meshes.
Results presented, both for a single equation and a system of coupled equations, suggest that this approach is able to produce better meshes of tetrahedra than those obtained by more conventional adaptive strategies and in a relatively efficient manner
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