DOI: 10.1007
/
s004540032949y
Discrete Comput Geom 31:61–81 (2004)
Discrete & Computational
Geometry
© 2003 SpringerVerlag New York Inc.
Testing Homotopy for Paths in the Plane
∗
Sergio Cabello,
1
Yuanxin Liu,
2
Andrea Mantler,
2
and Jack Snoeyink
2
1
Institute of Information and Computer Science, Utrecht University,
3508 TB Utrecht, The Netherlands
[email protected]
2
Department of Computer Science, University of North Carolina,
Chapel Hill, NC 275993175, USA
{
liuy,mantler,snoeyink
}
@cs.unc.edu
Abstract.
In this paper we present an efFcient algorithm to test if two given paths are
homotopic; that is, whether they wind around obstacles in the plane in the same way. ±or
paths speciFed by
n
line segments with obstacles described by
n
points, several standard
ways achieve quadratic running time. ±or simple paths, our algorithm runs in
O
(
n
log
n
)
time, which we show is tight. ±or selfintersecting paths the problem is related to Hopcroft’s
problem; our algorithm runs in
O
(
n
3
/
2
log
n
)
time.
1.
Introduction
A basic topological question is determining if two paths are homotopic, so that one can
be deformed into another without leaving the containing space. SpeciFcally, suppose
that the input consists of a set
P
of up to
n
points in the plane, and two paths,
α
and
β
,
that start and end at the same points and are represented as polygonal lines of at most
n
segments each. The goal is to determine whether
α
is deformable to
β
without passing
over any points of
P
(±ig. 1). Equivalently, we determine whether the closed loop
αβ
R
,
which concatenates
α
with the reverse of
β
, is contractible in the plane minus
P
.We
assume (or simulate) general position, so that no three points are collinear and no two
points are on the same vertical line. We are primarily concerned with paths that have no
selfintersections.
The path homotopy question arises in several application areas: In circuit board design,
river routing
, where the homotopy class of each wire is speciFed, is one of the few
∗
The Frst author was partially supported by the Cornenlis Lely Stichting, and the rest by NS± Grants
9988742 and 0076984.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document62
S. Cabello, Y. Liu, A. Mantler, and J. Snoeyink
α
β
Fig. 1.
Are
α
and
β
homotopic?
polynomialtime solvable variations of the wire routing problem [21], [17]. In motion
path planning, one might check to see that two ways of getting from point
A
to point
B
are equivalent [19]. In geographic information systems (GIS), one may wish to simplify a
linear feature (road or river) while respecting the way in which the feature winds around
points [5], [8]. Michael Goodchild, in an invited lecture at the 11th ACM Symposium
on Computational Geometry, pointed out that even on a road map that has features with
60 m accuracy, you will still Fnd all the houses on the proper side of the road. In such
a case the operators entering data have used topological constraints to make sure that
the road winds properly when creating the road layer or building layer. Efrat et al. [15]
recently looked at computing shortest paths among obstacles by identifying and bundling
homotopic fragments. They independently developed a sweep algorithm that uses similar
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Topology, Algebraic Topology, Fundamental group, universal cover, Canonical Path

Click to edit the document details