102 research outputs found
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
An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism
We present the first optimal randomized algorithm for constructing the
order- Voronoi diagram of points in two dimensions. The expected running
time is , which improves the previous, two-decades-old result
of Ramos (SoCG'99) by a factor. To obtain our result, we (i)
use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination
with Ramos's cutting construction, to reduce the problem to verifying an
order- Voronoi diagram, and (ii) solve the verification problem by a new
divide-and-conquer algorithm using planar-graph separators.
We also describe a deterministic algorithm for constructing the -level of
lines in two dimensions in time, and constructing
the -level of planes in three dimensions in
time. These time bounds (ignoring the term) match the current best
upper bounds on the combinatorial complexity of the -level. Previously, the
same time bound in two dimensions was obtained by Chan (1999) but with
randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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
Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs
We consider the massively parallel computation (MPC) model, which is a
theoretical abstraction of large-scale parallel processing models such as
MapReduce. In this model, assuming the widely believed 1-vs-2-cycles
conjecture, solving many basic graph problems in rounds with a strongly
sublinear memory size per machine is impossible. We improve on the recent work
of Holm and T\v{e}tek [SODA 2023] that bypass this barrier for problems when a
planar embedding of the graph is given. In the previous work, on graphs of size
with machines, the memory size per machine needs to be
at least , whereas we extend their work to the
fully scalable regime, where the memory size per machine can be for any constant . We give the first constant round
fully scalable algorithms for embedded planar graphs for the problems of (i)
connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the
-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be
incorporated into our recursive framework to obtain constant-round
-approximation algorithms for the problems of computing (iii)
single source shortest path (SSSP), (iv) global min-cut, and (v) -max flow.
All previous results on cuts and flows required linear memory in the MPC model.
Furthermore, our results give new algorithms for problems that implicitly
involve embedded planar graphs. We give as corollaries constant round fully
scalable algorithms for (vi) 2D Euclidean MST using total memory and
(vii) -approximate weighted edit distance using
memory.
Our main technique is a recursive framework combined with novel graph drawing
algorithms to compute smaller embedded planar graphs in constant rounds in the
fully scalable setting.Comment: To appear in SODA24. 55 pages, 9 figures, 1 table. Added section on
weighted edit distance and shortened abstrac
Guard placement for efficient pointin-polygon proofs
{eppstein, goodrich, nodari} (at) ics.uci.edu We consider the problem of placing a small number of angle guards inside a simple polygon P so as to provide efficient proofs that any given point is inside P. Each angle guard views an infinite wedge of the plane, and a point can prove membership in P if it is inside the wedges for a set of guards whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, which we call “sculpture garden ” problems and for which we provide upper and lower bounds. In particular, we show there is a polygon P such that a “natural” angle-guard vertex placement cannot fully distinguish between points on the inside and outside of P (even if we place a guard at every vertex of P), which implies that Steinerpoint guards are sometimes necessary. More generally, we show that, for any polygon P, there is a set of n + 2(h − 1) angle guards that solve the sculpture garden problem for P, where h is the number of holes in P (so a simple polygon can be defined with n − 2 guards). In addition, we show that, for any orthogonal polygon P, the sculpture garden problem can be solved using n angle guards. We also give an 2 example of a class of simple (non-general-position) polygons that have sculpture garden solutions using O ( √ n) guards, and we show this bound is optimal to within a constant factor. Finally, while optimizing the number of guards solving a sculpture garden problem for a particular P is of unknown complexity, we show how to find in polynomial time a guard placement whose size is within a factor of 2 of the optimal number for any particular polygon
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