236 research outputs found
Shortest Path in a Polygon using Sublinear Space
\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}
\newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due
to Tetsuo Asano, showing how to compute the shortest path in a polygon, given
in a read only memory, using sublinear space and subquadratic time.
Specifically, given a simple polygon \Polygon with vertices in a read
only memory, and additional working memory of size \Space, the new algorithm
computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected
time. This requires several new tools, which we believe to be of independent
interest
Negative curvature in graphical small cancellation groups
We use the interplay between combinatorial and coarse geometric versions of
negative curvature to investigate the geometry of infinitely presented
graphical small cancellation groups. In particular, we characterize
their 'contracting geodesics', which should be thought of as the geodesics that
behave hyperbolically.
We show that every degree of contraction can be achieved by a geodesic in a
finitely generated group. We construct the first example of a finitely
generated group containing an element that is strongly contracting with
respect to one finite generating set of and not strongly contracting with
respect to another. In the case of classical small cancellation
groups we give complete characterizations of geodesics that are Morse and that
are strongly contracting.
We show that many graphical small cancellation groups contain
strongly contracting elements and, in particular, are growth tight. We
construct uncountably many quasi-isometry classes of finitely generated,
torsion-free groups in which every maximal cyclic subgroup is hyperbolically
embedded. These are the first examples of this kind that are not subgroups of
hyperbolic groups.
In the course of our analysis we show that if the defining graph of a
graphical small cancellation group has finite components, then the
elements of the group have translation lengths that are rational and bounded
away from zero.Comment: 40 pages, 14 figures, v2: improved introduction, updated statement of
Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely
presented graphical small cancellation groups"), minor changes, to appear in
Groups, Geometry, and Dynamic
Memory-Constrained Algorithms for Simple Polygons
A constant-workspace algorithm has read-only access to an input array and may
use only O(1) additional words of bits, where is the size of
the input. We assume that a simple -gon is given by the ordered sequence of
its vertices. We show that we can find a triangulation of a plane straight-line
graph in time. We also consider preprocessing a simple polygon for
shortest path queries when the space constraint is relaxed to allow words
of working space. After a preprocessing of time, we are able to solve
shortest path queries between any two points inside the polygon in
time.Comment: Preprint appeared in EuroCG 201
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
Trajectory Visibility
We study the problem of testing whether there exists a time at which two entities moving along different piece-wise linear trajectories among polygonal obstacles are mutually visible. We study several variants, depending on whether or not the obstacles form a simple polygon, trajectories may intersect the polygon edges, and both or only one of the entities are moving. For constant complexity trajectories contained in a simple polygon with n vertices, we provide an (n) time algorithm to test if there is a time at which the entities can see each other. If the polygon contains holes, we present an (n log n) algorithm. We show that this is tight. We then consider storing the obstacles in a data structure, such that queries consisting of two line segments can be efficiently answered. We show that for all variants it is possible to answer queries in sublinear time using polynomial space and preprocessing time. As a critical intermediate step, we provide an efficient solution to a problem of independent interest: preprocess a convex polygon such that we can efficiently test intersection with a quadratic curve segment. If the obstacles form a simple polygon, this allows us to answer visibility queries in (nĀ³/4logĀ³ n) time using (nlogāµ n) space. For more general obstacles the query time is (log^k n), for a constant but large value k, using (n^{3k}) space. We provide more efficient solutions when one of the entities remains stationary
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