941 research outputs found
Complexity of Interlocking Polyominoes
Polyominoes are a subset of polygons which can be constructed from
integer-length squares fused at their edges. A system of polygons P is
interlocked if no subset of the polygons in P can be removed arbitrarily far
away from the rest. It is already known that polyominoes with four or fewer
squares cannot interlock. It is also known that determining the interlockedness
of polyominoes with an arbitrary number of squares is PSPACE hard. Here, we
prove that a system of polyominoes with five or fewer squares cannot interlock,
and that determining interlockedness of a system of polyominoes including
hexominoes (polyominoes with six squares) or larger polyominoes is PSPACE hard.Comment: 18 pages, 15 figure
BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning
The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples
Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of
the same plane graph. The morph preserves the convexity of the drawing at any
time instant and moves each vertex along a piecewise linear curve with linear
complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Laminations and groups of homeomorphisms of the circle
If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that
pi_1(M) acts on a circle. Here, we show that some other classes of essential
laminations also give rise to actions on circles. In particular, we show this
for tight essential laminations with solid torus guts. We also show that
pseudo-Anosov flows induce actions on circles. In all cases, these actions can
be made into faithful ones, so pi_1(M) is isomorphic to a subgroup of
Homeo(S^1). In addition, we show that the fundamental group of the Weeks
manifold has no faithful action on S^1. As a corollary, the Weeks manifold does
not admit a tight essential lamination, a pseudo-Anosov flow, or a taut
foliation. Finally, we give a proof of Thurston's universal circle theorem for
taut foliations based on a new, purely topological, proof of the Leaf Pocket
Theorem.Comment: 50 pages, 12 figures. Ver 2: minor improvement
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
Curves of Finite Total Curvature
We consider the class of curves of finite total curvature, as introduced by
Milnor. This is a natural class for variational problems and geometric knot
theory, and since it includes both smooth and polygonal curves, its study shows
us connections between discrete and differential geometry. To explore these
ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.Comment: 25 pages, 4 figures; final version, to appear in "Discrete
Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 200
An Optimal Algorithm to Compute the Inverse Beacon Attraction Region
The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move greedily towards a beacon: if unobstructed, they move along a straight line to the beacon, and otherwise they slide on the edges of the polygon. The Euclidean distance from a moving point to a beacon is monotonically decreasing. A given beacon attracts a point if the point eventually reaches the beacon.
The problem of attracting all points within a polygon with a set of beacons can be viewed as a variation of the art gallery problem. Unlike most variations, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. It is connected and can be computed in linear time for simple polygons. By contrast, it is known that the inverse attraction region of a point - the set of beacon positions that attract it - could have Omega(n) disjoint connected components.
In this paper, we prove that, in spite of this, the total complexity of the inverse attraction region of a point in a simple polygon is linear, and present a O(n log n) time algorithm to construct it. This improves upon the best previous algorithm which required O(n^3) time and O(n^2) space. Furthermore we prove a matching Omega(n log n) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone
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