820 research outputs found
Covering orthogonal polygons with star polygons: The perfect graph approach
AbstractThis paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n3) algorithm
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
Perfect domination in regular grid graphs
We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph of . In contrast, there is just one
1-perfect code in and one total perfect code in
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
with parallel total perfect codes, and the -perfect and
total perfect code partitions of and , the former
having as quotient graph the undirected Cayley graphs of with
generator set . For , generalization for 1-perfect codes is
provided in the integer lattice of and in the products of cycles,
with partition quotient graph taken as the undirected Cayley graph
of with generator set .Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi
Decomposing and packing polygons / Dania el-Khechen.
In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions
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