65,081 research outputs found
Parallel Transitive Closure and Point Location in Planar Structures
AMS(MOS) subject classifications. 68E05, 68C05, 68C25Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs.
Most of these algorithms achieve optimal O(logn) running time using n/logn processors in the EREW PRAM model, n being the number of vertices
Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams
We study a general family of facility location problems defined on planar
graphs and on the 2-dimensional plane. In these problems, a subset of
objects has to be selected, satisfying certain packing (disjointness) and
covering constraints. Our main result is showing that, for each of these
problems, the time brute force algorithm of selecting objects
can be improved to time. The algorithm is based on an idea
that was introduced recently in the design of geometric QPTASs, but was not yet
used for exact algorithms and for planar graphs. We focus on the Voronoi
diagram of a hypothetical solution of objects, guess a balanced separator
cycle of this Voronoi diagram to obtain a set that separates the solution in a
balanced way, and then recurse on the resulting subproblems. We complement our
study by giving evidence that packing problems have time
algorithms for a much more general class of objects than covering problems
have.Comment: 64 pages, 16 figure
Localization game on geometric and planar graphs
The main topic of this paper is motivated by a localization problem in
cellular networks. Given a graph we want to localize a walking agent by
checking his distance to as few vertices as possible. The model we introduce is
based on a pursuit graph game that resembles the famous Cops and Robbers game.
It can be considered as a game theoretic variant of the \emph{metric dimension}
of a graph. We provide upper bounds on the related graph invariant ,
defined as the least number of cops needed to localize the robber on a graph
, for several classes of graphs (trees, bipartite graphs, etc). Our main
result is that, surprisingly, there exists planar graphs of treewidth and
unbounded . On a positive side, we prove that is bounded
by the pathwidth of . We then show that the algorithmic problem of
determining is NP-hard in graphs with diameter at most .
Finally, we show that at most one cop can approximate (arbitrary close) the
location of the robber in the Euclidean plane
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
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