55 research outputs found
Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment
Let be a planar polygonal environment
(i.e., a polygon potentially with holes) with a total of vertices, and let
be two robots, each modeled as an axis-aligned unit square, that can
translate inside . Given source and target placements
of and , respectively, the goal is to
compute a \emph{collision-free motion plan} , i.e., a motion
plan that continuously moves from to and from to
so that and remain inside and do not collide with
each other during the motion. Furthermore, if such a plan exists, then we wish
to return a plan that minimizes the sum of the lengths of the paths traversed
by the robots, . Given and a parameter , we present an
-time -approximation algorithm
for this problem. We are not aware of any polynomial time algorithm for this
problem, nor do we know whether the problem is NP-Hard. Our result is the first
polynomial-time -approximation algorithm for an optimal motion
planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n â„ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Straight Line Movement in Morphing and Pursuit Evasion
Piece-wise linear structures are widely used to define problems and to represent simplified
solutions in computational geometry. A piece-wise linear structure consists of straight-line
or linear pieces connected together in a continuous geometric environment like 2D or 3D
Euclidean spaces. In this thesis two different problems both with the approach of finding
piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion
and straight-line morphing.
Straight-line pursuit evasion is a geometric version of the famous cops and robbers game
that is defined in this thesis for the first time. The game is played in a simply connected
region in 2D. It is a full information game where the players take turns. The copâs goal
is to catch the robber. In a turn, each player may move any distance along a straight
line as long as the line segment connecting their current location to the new location is
not blocked by the regionâs boundary. We first prove that the cop can always win the
game when the players move on the visibility graph of a simple polygon. We prove this by
showing that the visibility graph of a simple polygon is âdismantlableâ (the known class of
cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other
settings of the game are also studied in this thesis: when the players are free to move on
the infinitely many points inside a simple polygon, and inside a splinegon. In both cases
we show that the cop can always win the game. For the case of polygons, the proposed cop
strategy gives an asymptotically tight linear bound on the number of steps the cop needs
to catch the robber. For the case of splinegons, the cop may need a quadratic number of
steps with the proposed strategy, while our best lower bound is linear.
Straight-line morphing is a type of morphing first defined in this thesis that provides a
nice and smooth transformation between straight-line graph drawings in 2D. In straight-
line morphing, each vertex of the graph moves forward along the line segment connecting
its initial position to its final position. The vertex trajectories in straight-line morphing
are very simple, but because the speed of each vertex may vary, straight-line morphs are
more general than the commonly used âlinear morphsâ where each vertex moves at uniform
speed. We explore the problem of whether an initial planar straight-line drawing of a graph
can be morphed to a final straight-line drawing of the graph using a straight-line morph
that preserves planarity at all times. We prove that this problem is NP-hard even for
the special case where the graph drawing consists of disjoint segments. We then look at
some restricted versions of the straight-line morphing: when only one vertex moves at a
time, when the vertices move one by one to their final positions uninterruptedly, and when
the edges morph one by one to their final configurations in the case of disjoint segments.
Some of the variations are shown to be still NP-complete while some others are solvable
in polynomial time. We conjecture that the class of planar straight-line morphs is as
powerful as the class of planar piece-wise linear straight-line morphs. We also explore
a simpler problem where for each edge the quadrilateral formed by its initial and final
positions together with the trajectories of its two vertices is convex. There is a necessary
condition for this case that we conjecture is also sufficient for paths and cycles
Facets of Planar Graph Drawing
This thesis makes a contribution to the field of Graph Drawing, with a focus on the planarity drawing convention. The following three problems are considered.
(1) Ordered Level Planarity:
We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. With reductions from Ordered Level Planarity, we show NP-hardness of multiple problems whose complexity was previously open, and strengthen several previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two nontrivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli [2015]. We settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Stefankovic [2013]. We also present a reduction to Manhattan Geodesic Planarity, showing that a previously [2009] claimed polynomial time algorithm is incorrect unless P=NP.
(2) Two-page book embeddings of triconnected planar graphs:
We show that every triconnected planar graph of maximum degree five is a subgraph of a Hamiltonian planar graph or, equivalently, it admits a two-page book embedding. In fact, our result is more general: we only require vertices of separating 3-cycles to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: we describe a family of triconnected planar graphs that cannot be realized on two pages and where every vertex of a separating 3-cycle has degree at most six. Our results strengthen earlier work by Heath [1995] and by Bauernöppel [1987] and, independently, Bekos, Gronemann, and Raftopoulou [2016], who showed that planar graphs of maximum degree three and four, respectively, can always be realized on two pages. The proof is constructive and yields a quadratic time algorithm to realize the given graph on two pages.
(3) Convexity-increasing morphs:
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.Diese Arbeit behandelt drei unterschiedliche Problemstellungen aus der Disziplin des Graphenzeichnens (Graph Drawing). Bei jedem der behandelten Probleme ist die gesuchte Darstellung planar.
(1) Ordered Level Planarity:
Wir fĂŒhren das Problem Ordered Level Planarity ein, bei dem es darum geht, einen Graph so zu zeichnen, dass jeder Knoten an einer vorgegebenen Position der Ebene platziert wird und die Kanten als y-monotone Kurven dargestellt werden. Dies kann als eine Variante von Level Planarity interpretiert werden, bei der die Knoten jedes Levels in einer vorgeschriebenen Reihenfolge platziert werden mĂŒssen. Wir klassifizieren die Eingaben bezĂŒglich ihrer KomplexitĂ€t in AbhĂ€ngigkeit von sowohl dem Maximalgrad, als auch der maximalen Anzahl von Knoten, die demselben Level zugeordnet sind. Wir motivieren die Ergebnisse, indem wir Verbindungen zu einigen anderen Graph Drawing Problemen herleiten: Mittels Reduktionen von Ordered Level Planarity zeigen wir die NP-Schwere einiger Probleme, deren KomplexitĂ€t bislang offen war. Insbesondere wird gezeigt, dass Clustered Level Planarity bereits fĂŒr Instanzen mit zwei nichttrivialen Clustern NP-schwer ist, was eine Frage von Angelini, Da Lozzo, Di Battista, Frati und Roselli [2015] beantwortet. Wir zeigen die NP-Schwere des Bi-Monotonicity Problems und beantworten damit eine Frage von Fulek, Pelsmajer, Schaefer und Stefankovic [2013]. AuĂerdem wird eine Reduktion zu Manhattan Geodesic Planarity angegeben. Dies zeigt, dass ein bestehender [2009] Polynomialzeitalgorithmus fĂŒr dieses Problem inkorrekt ist, es sei denn, dass P=NP ist.
(2) Bucheinbettungen von dreifach zusammenhÀngenden planaren Graphen mit zwei Seiten:
Wir zeigen, dass jeder dreifach zusammenhĂ€ngende planare Graph mit Maximalgrad 5 Teilgraph eines Hamiltonischen planaren Graphen ist. Dies ist Ă€quivalent dazu, dass ein solcher Graph eine Bucheinbettung auf zwei Seiten hat. Der Beweis ist konstruktiv und zeigt in der Tat sogar, dass es fĂŒr die Realisierbarkeit nur notwendig ist, den Grad von Knoten separierender 3-Kreise zu beschrĂ€nken - die ĂŒbrigen Knoten können beliebig hohe Grade aufweisen. Dieses Ergebnis ist bestmöglich: Wenn die Gradschranke auf 6 abgeschwĂ€cht wird, gibt es Gegenbeispiele. Diese Ergebnisse verbessern Resultate von Heath [1995] und von Bauernöppel [1987] und, unabhĂ€ngig davon, Bekos, Gronemann und Raftopoulou [2016], die gezeigt haben, dass planare Graphen mit Maximalgrad 3 beziehungsweise 4 auf zwei Seiten realisiert werden können.
(3) KonvexitÀtssteigernde Deformationen:
Wir zeigen, dass jede planare geradlinige Zeichnung eines intern dreifach zusammenhÀngenden planaren Graphen stetig zu einer solchen deformiert werden kann, in der jede FlÀche ein konvexes Polygon ist. Dabei erhÀlt die Deformation die PlanaritÀt und ist konvexitÀtssteigernd - sobald ein Winkel konvex ist, bleibt er konvex. Wir geben einen effizienten Algorithmus an, der eine solche Deformation berechnet, die aus einer asymptotisch optimalen Anzahl von Schritten besteht. In jedem Schritt bewegen sich entweder alle Knoten entlang horizontaler oder entlang vertikaler Geraden
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
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