201 research outputs found
Time-Space Trade-offs for Triangulating a Simple Polygon
An s-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses O(s) additional words of space. We give a randomized s-workspace algorithm for triangulating a simple polygon P of n vertices, for any s up to n. The algorithm runs in O(n^2/s+n(log s)log^5(n/s)) expected time using O(s) variables, for any s up to n. In particular, the algorithm runs in O(n^2/s) expected time for most values of s
A Time-Space Tradeoff for Triangulations of Points in the Plane
In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm
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
Space-Time Trade-offs for Stack-Based Algorithms
In memory-constrained algorithms we have read-only access to the input, and
the number of additional variables is limited. In this paper we introduce the
compressed stack technique, a method that allows to transform algorithms whose
space bottleneck is a stack into memory-constrained algorithms. Given an
algorithm \alg\ that runs in O(n) time using variables, we can
modify it so that it runs in time using a workspace of O(s)
variables (for any ) or time using variables (for any ). We also show how the technique
can be applied to solve various geometric problems, namely computing the convex
hull of a simple polygon, a triangulation of a monotone polygon, the shortest
path between two points inside a monotone polygon, 1-dimensional pyramid
approximation of a 1-dimensional vector, and the visibility profile of a point
inside a simple polygon. Our approach exceeds or matches the best-known results
for these problems in constant-workspace models (when they exist), and gives
the first trade-off between the size of the workspace and running time. To the
best of our knowledge, this is the first general framework for obtaining
memory-constrained algorithms
Approximate Euclidean shortest paths in polygonal domains
Given a set of pairwise disjoint simple polygonal obstacles
in defined with vertices, we compute a sketch of
whose size is independent of , depending only on and the
input parameter . We utilize to compute a
-approximate geodesic shortest path between the two given points
in time. Here, is a user
parameter, and is a small positive constant (resulting from the time
for triangulating the free space of using the algorithm in
\cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a
-approximation algorithm to answer two-point Euclidean distance
queries for the case of convex polygonal obstacles.Comment: a few updates; accepted to ISAAC 201
A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-off Algorithms
We are given a read-only memory for input and a write-only stream for output. For a positive integer parameter s, an s-workspace algorithm is an algorithm using only O(s) words of workspace in addition to the memory for input. In this paper, we present an O(n^2/s)-time s-workspace algorithm for subdividing a simple polygon into O(min{n/s,s}) subpolygons of complexity O(max{n/s,s}).
As applications of the subdivision, the previously best known time-space trade-offs for the following three geometric problems are improved immediately: (1) computing the shortest path between two points inside a simple n-gon, (2) computing the shortest path tree from a point inside a simple n-gon, (3) computing a triangulation of a simple n-gon. In addition, we improve the algorithm for the second problem even further
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