On the complexity of minimum-link path problems
Citation for published version (APA):
Kostitsyna, I., Löffler, M., Polishchuk, V., & Staals, F. (2017). On the complexity of minimum-link path problems.
Journal of Computational Geometry
,
8
(2), 80-108. https://doi.org/10.20382/jocg.v8i2a5
DOI:
10.20382/jocg.v8i2a5
Document status and date:
Published: 01/01/2017
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be
important differences between the submitted version and the official published version of record. People
interested in the research are advised to contact the author for the final version of the publication, or visit the
DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page
numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please
follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
Download date: 25. Aug. 2022
JoCG 8(2), 80–108, 2017 80
Journal of Computational Geometry jocg.org
ON THE COMPLEXITY OF MINIMUM-LINK PATH PROBLEMS
∗
.
Irina Kostitsyna,
†
Maarten Löffler,
‡
Valentin Polishchuk,
§
and Frank Staals
¶
Abstract. We revisit the minimum-link path problem: Given a polyhedral domain and two
points in it, connect the points by a polygonal path with minimum number of edges. We
consider settings where the vertices and/or the edges of the path are restricted to lie on the
boundary of the domain, or can be in its interior. Our results include bit complexity bounds,
a novel general hardness construction, and a polynomial-time approximation scheme. We
fully characterize the situation in 2 dimensions, and provide first results in dimensions 3 and
higher for several variants of the problem.
Concretely, our results resolve several open problems. We prove that computing
the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is
NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open
problem [28] despite a large body of work on the topic. We also resolve the open problem
from [41] mentioned in the handb ook [29] (see Chapter 27.5, Open problem 3) and The Open
Problems Project [17] (see Problem 22): “What is the complexity of the minimum-link path
problem in 3-space?” Our results imply that the problem is NP-hard even on terrains (and
hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a
PTAS.
✶ ■♥tr♦❞✉❝t✐♦♥
The minimum-link path problem is fundamental in computational geometry [5, 27, 30, 33,
35, 38, 41, 49]. It concerns the following question: given a polyhedral domain
D
and two
points
s
and
t
in
D
, what is the polygonal path connecting
s
to
t
that lies in
D
and has as
few links as possible?
In this paper, we revisit the problem in a general setting which encompasses several
specific variants that have been considered in the literature. First, we nuance and tighten
results on the bit complexity involved in optimal minimum-link paths. Second, we present and
apply a novel generic NP-hardness construction. Third, we extend a simple polynomial-time
approximation scheme.
Concretely, our results resolve several open problems. We prove that computing the
minimum-link diffuse reflection path in polygons with holes [28] is NP-hard, and we prove
∗
An abridged version of this paper appeared in the proceedings of the 32nd International Symposium on
Computational Geometry in 2016.
†
Université libre de Bruxelles, irina.kostitsyna@ulb.ac.be
‡
Utrecht University, m.loffler@uu.nl
§
Linköping University, valentin.polishchuk@liu.se
¶
Aarhus University, f.staals@cs.au.dk
JoCG 8(2), 80–108, 2017 81
Journal of Computational Geometry jocg.org
t
s
s
t
s
t
Figure 1: Left: MinLinkPath
2,2
in a polygon with holes. Middle: MinLinkPath
1,2
on a
polyhedron. Right: MinLinkPath
0,3
on a polyhedral terrain.
that the minimum-link path problem in 3-space [ 29] (Chapter 27.5, Open problem 3) is
NP-hard (even for terrains). In both cases, there is no FPTAS unless P=NP, but there is a
PTAS.
We use terms links and bends for edges and vertices of the path, saving the terms
edges and vertices for those of the domain (also historically, minimum-link paths used to be
called minimum-bend [51–53]).
✶✳✶ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t✱ ❉♦♠❛✐♥s ❛♥❞ ❈♦♥str❛✐♥ts
Due to their diverse applications, many different variants of minimum-link paths have been
considered in the literature. These variants can be categorized by two aspects. Firstly, the
domain can take very different forms. We select several common domains, ranging from a
simple polygon in 2D to complex scenes in full 3D or even in higher dimensions. Secondly,
the links and bends of the solution paths are sometimes constrained to lie on the boundary
of the domain, or bends may be restricted to vertices or edges of the domain. We now survey
these settings in more detail.
Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t✳
Let D be a closed connected d-dimensional polyhedral domain. For
0
≤ a ≤ d
we denote by
D|
a
the
a
-skeleton of
D
, that is, its
a
-dimensional subcomplex. For
instance,
D|
d−1
is the boundary of
D
;
D|
0
is the set of vertices of
D
. Note that
D|
a
is not
necessarily connected.
Definition 1.
We define
MinLinkPath
a,b
(
D, s, t
), for 0
≤ a ≤ b ≤ d
and 1
≤ b
, to be the
problem of finding a minimum-link polygonal path in
D
between two given points
s
and
t
,
where the bends of the solution (and
s
and
t
) are restricted to lie in
D|
a
and the links of the
solution are restricted to lie in D|
b
.
Figure
1 illustrates several instances of the problem in different domains.
❉♦♠❛✐♥s✳
We recap the various settings that have been singled out for studies in compu-
tational geometry. We remark that we will not survey the rich field of path planning in
rectilinear, or more generally,
C
-oriented worlds [1]; all our paths will be assumed to be
unrestricted in terms of orientations of their links.
JoCG 8(2), 80–108, 2017 82
Journal of Computational Geometry jocg.org
One classical distinction between working setups in 2D is simple polygons vs. polygonal
domains. The former are a special case of the latter: simple polygons are domains without
holes. Many problems admit more efficient solutions in simple polygons—loosely speaking,
the golden standard is running time of
O
(
n
) for simple polygons and of
O
(
n log n
) for
polygonal domains of complexity
n
. This is the case, e.g., for the shortest path problem
[31, 32]. For minimum-link paths,
O
(
n
)-time algorithms are known for simple polygons [ 27 ,
33, 49], but for polygonal domains with holes the fastest known algorithm runs in nearly
quadratic time [41], which may be close to optimal due to 3SUM-hardness of the problem [38].
Even more striking is the difference in the watchman route problem (find a shortest path
to see all of the domain), which combines path planning with visibility: in simple polygons
the optimal route can be found in polynomial time [15, 19] while for domains with holes the
problem cannot be approximated to within a logarithmic factor unless P=NP [40]. Finding
minimum-link watchman route is NP-hard even for simple polygons [ 4].
In 3D, a terrain is a polyhedral surface (often restricted to a bounded region in the
xy
-projection) that is intersected only once by any vertical line. Terrains are traditionally
studied in GIS applications and are ubiquitous in computational geometry [11, 39]. Minimum-
link paths are closely related to visibility problems, which have been studied extensively on
terrains [8, 9, 22, 34, 36, 48 ]. One step up from terrains, we may consider simple polyhedra
(surfaces of genus 0), or full 3D scenes. Visibility has been studied in full 3D as well [20,
42, 50]. To our knowledge, minimum-link paths in higher dimensions have not been studied
before (with the exception of [10] that considered rectilinear paths).
❈♦♥str❛✐♥ts✳
In path planning on polyhedral surfaces or terrains, it is standard to restrict
paths to the (terrain) surface. Minimum-link paths, on the other hand, have various
geographic applications, ranging from feature simplification [30] to visibility in terrains [22].
In some of these applications, paths are allowed to live in free space, while bends are still
restricted to the terrain. In the GIS literature, out of simplicity and/or efficiency concerns, it
is common to constrain bends even further to vertices of the domain (or, even more severely,
the terrain itself may restrict vertices to grid points, as in the popular digital elevation map
(DEM) model; this, however, may lead to an arbitrarily high increase in the link distance).
In a vanilla minimum-link path problem the location of vertices (bends) of the path are
unconstrained, i.e., they can occur anywhere in the free space. In the diffuse reflection model
[5–7, 12, 28, 45] the bends are restricted to occur on the boundary of the domain. Studying
this kind of paths is motivated by ray tracing in realistic rendering of 3D scenes in graphics,
as light sources that can reach a pixel with fewer reflections make higher contributions
to intensity of the pixel [
11, 23]. Despite the 3D graphics motivation, all work on diffuse
reflection has b een confined to 2D polygonal domains, where the path bends are restricted
to edges of the domain.
✶✳✷ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥
In computational geometry, the standard model of computation is the real RAM, which
represents data as an infinite sequence of storage cells which can store any real number or
JoCG 8(2), 80–108, 2017 83
Journal of Computational Geometry jocg.org
integer. The model supports standard operations (such as addition, multiplication, or taking
square-roots) in constant time. The real RAM is preferred for its elegance, but may not
always be the best representation of physical computers. For example, the floor function
is often allowed, which can be used to truncate a real number to the nearest integer, but
points at a flaw in the model: if we were allowed to use it arbitrarily, the real RAM could
solve PSPACE-complete problems in polynomial time [47]. In contrast, the word RAM
stores a sequence of
w
-bit words, where
w ≥ log n
(and
n
is the problem size). Data can
be accessed arbitrarily, and standard operations, such as Boolean operations (
and
,
xor
,
shl
,
. . .
), addition, or multiplication take constant time. There are many variants of the
word RAM, depending on precisely which instructions are supported in constant time. The
general consensus seems to be that any function in
AC
0
is acceptable.
1
However, it is always
preferable to rely on a set of operations as small, and as non-exotic, as possible. Note that
multiplication is not in
AC
0
[25]. Nevertheless, it is usually included in the word RAM
instruction set [24]. The word RAM is much closer to reality, but complicates the analysis of
geometric problems.
In many cases, the difference between the models is unimportant, as the real numbers
involved in solving geometric problems are in fact algebraic numbers of low degree in a
bounded domain, which can be described exactly with constantly many words. Path planning
is notoriously different in this respect. Indeed, in the real RAM both the Euclidean shortest
paths and the minimum-link paths in 2D can b e found in optimal times. On the contrary,
much less is known about the complexity of the problems in other models. For
L
2
-shortest
paths the issue is that their length is represented by the sum of square roots and it is not
known whether comparing the sum to a number can be done efficiently (if yes, one may hope
that the difference between the models vanishes). Slightly more is known about minimum-link
paths, for which the models are provably different: Kahan and Snoeyink [35] observed that
the region of points reachable by
k
-link paths may have vertices needing Ω(
k log n
) bits to
describe. One of the results in this paper is the matching upper bound on the bit complexity
of minimum-link paths.
Relatedly, when studying the computational complexity of geometric problems, it is
often not trivial to show a problem is in NP. Even if a potential solution can be verified in
polynomial time, if such a solution requires real numbers that cannot be described succinctly,
the set of solutions to try may be too large. Recently, there has been some interest in
computational geometry in showing problems are in NP [21] (see also [46]).
A common practical approach to avoiding bit complexity issues is to approximate the
problem by restricting solutions to use only vertices of the input. In minimum-link paths,
this corresponds to MinLinkPath
0,b
. In this case, one can easily compute a minimum-link
path by a breadth-first search in the visibility graph of the vertices. This results in an
O
(
n
2
)
time algorithm in 2D (using [43]), and an
O
(
n
7/3
polylog n
) time algorithm in 3D (using [2];
for terrains this can be improved slightly [16]). In both cases the running time is dominated
1
AC
0
is the class of all functions
f
:
{
0
,
1
}
∗
→ {
0
,
1
}
∗
that can be computed by a family of circuits
(
C
n
)
n∈N
with the following properties: (i) each
C
n
has
n
inputs; (ii) there exist constants
a, b
, such that
C
n
has at most
an
b
gates, for
n ∈ N
; (iii) there is a constant
d
such that for all
n
the length of the longest path
from an input to an output in
C
n
is at most
d
(i.e., the circuit family has bounded depth); (iv) each gate has
an arbitrary number of incoming edges (i.e., the fan-in is unbounded).