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Minimum Convex Partition of a Constrained Point Set
 Discrete Applied Mathematics
, 1998
"... : A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is ..."
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: A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is a convex partition of S such that the number of convex polygons is minimised. In this paper, we will present a polynomial time algorithm to find a minimum convex partition with respect to a point set S where S is constrained to lie on the boundaries of a fixed number of nested convex hulls. 1 Introduction A convex partition (convex decomposition) with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition (MCP) with respect to S is a convex partition of S such that the number of convex polygons ...
Drawing outerplanar minimum weight triangulations
, 1996
"... We consider the problem of characterizing those graphs that can be drawn as minimum weight triangulations and answer the question for maximal outerplanar graphs. We provide a complete characterization of minimum weight triangulations of regular polygons by studying the combinatorial properties of th ..."
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Cited by 3 (2 self)
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We consider the problem of characterizing those graphs that can be drawn as minimum weight triangulations and answer the question for maximal outerplanar graphs. We provide a complete characterization of minimum weight triangulations of regular polygons by studying the combinatorial properties of their dual trees. We exploit this characterization to devise a linear time (real RAM) algorithm that receives as input a maximal outerplanar graph G and produces as output a straightline drawing of G that is a minimum weight triangulation of the set of points representing the vertices of G.
Drawable and Forbidden Minimum Weight Triangulations (Extended Abstract)
"... A graph is minimum weight drawable if it admits a straightline drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. In this paper we consider the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofo ..."
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Cited by 2 (0 self)
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A graph is minimum weight drawable if it admits a straightline drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. In this paper we consider the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present nontrivial classes of triangulations that are minimum weight drawable, along with corresponding linear time (real RAM) algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but none of which is Delaunay drawablethat is, drawable as a Delaunay triangulation. 1 Introduction and Overview Recently much attention has been devoted to the study of combinatorial properties of wellknown geometric structuresoften referred to a...
Minimum Weight Convex Quadrangulation of a Constrained Point Set
, 1997
"... Summary: A convex quadrangulation with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is convex and has four points from S on its boundary. A ..."
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Summary: A convex quadrangulation with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is convex and has four points from S on its boundary. A minimum weight convex quadrangulation with respect to S is a convex quadrangulation of S such that the sum of the Euclidean lengths of the edges of the subdivision is minimised. In this extended abstract, we will present a polynomial time algorithm to determine whether a set of points S admits a convex quadrangulation if S is constrained to lie on a fixed number of nested convex polygons, where the time complexity is polynomial in the cardinality of S. This algorithm can also be used to find a minimum weight convex quadrangulation of the point set. We use a similar approach to construct a convex subdivision with respect to S using three or four points from S per face, and minimising the tota...
On Stable Line Segments in Triangulations
"... this paper appeared in the Proc. of the 8th Canadian Conference in Computational Geometry, Carleton University, Ottawa, Canada, pp. 6267, August 1996. ..."
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this paper appeared in the Proc. of the 8th Canadian Conference in Computational Geometry, Carleton University, Ottawa, Canada, pp. 6267, August 1996.
Minimum Convex KPartitions of a Linearly Constrained Point Set
, 1999
"... We present an optimization algorithm to determine a partition of the convex hull of a finite set of ponts in the plane. The partition uses the points as corners of convex polygonal cells, each cell having at most K sides. We minimize the total number of cells that are obtained. The algorithm runs ..."
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We present an optimization algorithm to determine a partition of the convex hull of a finite set of ponts in the plane. The partition uses the points as corners of convex polygonal cells, each cell having at most K sides. We minimize the total number of cells that are obtained. The algorithm runs in polynomial time when the points lie on a fixed number of (almost) parallel lines. 1 Introduction Let S be a finite set of points in the plane. A convex partition with respect to S, is a cell complex with corners at points from S, that partitions the convex hull of S into convex polygonal cells so that no cell has a point from S in its interior. We use convex Kgon to denote a convex polygon of nonzero area with at most K sides. A minimum convex partition of S is a convex partition using a minimal number of cells, and a minimum convex Kpartition of S is a minimum convex partition where all cells are convex Kgons. Note that in our reckoning a polygon is convex if it is simple and has...
On Stable Line Segments in Triangulations 1
"... Let S be a set of n points in the plane and E denote the set of all the line segments with endpoints in S. A line segment pq with p; q 2 S is called a stable line segment of all triangulations of S, if no line segment in E properly intersects pq. The intersection of all possible triangulations of S ..."
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Let S be a set of n points in the plane and E denote the set of all the line segments with endpoints in S. A line segment pq with p; q 2 S is called a stable line segment of all triangulations of S, if no line segment in E properly intersects pq. The intersection of all possible triangulations of S then is the set of all stable line segments in S, denoted by SL(S). As a combinatorial problem, various properties of stable line segments of a set of planar points have been investigated in [13]. It is shown that the maximum number of stable line segments in S is 2(n 1). There is an interesting relationship between stable line segments and socalled extreme line segments EL(S) [6]. A line segment pq with p; qS is called an extreme line segment if fp; qg = E \H for some open halfplane H [6]. Then, we have that CH(S) EL(S) SL(S): A more important property is the relationship between SL(S) and socalled koptimal triangulations. Let T (S) denote a triangulation of S. T (S) is called a koptimal triangu1
Computing a Minimum Weight Triangulation of a Sparse Point Set
, 1998
"... Abstract. Investigating the minimum weight triangulation of a point set with constraint is an important approach for seeking the ultimate solution of the minimum weight triangulation problem. In this paper, we consider the minimum weight triangulation of a sparse point set, and present an O.n4/ alg ..."
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Abstract. Investigating the minimum weight triangulation of a point set with constraint is an important approach for seeking the ultimate solution of the minimum weight triangulation problem. In this paper, we consider the minimum weight triangulation of a sparse point set, and present an O.n4/ algorithm to compute a triangulation of such a set. The property of sparse point set can be converted into a new sufficient condition for finding subgraphs of the minimum weight triangulation. A special point set is exhibited to show that our new subgraph of minimum weight triangulation cannot be found by any currently known methods.
A New Subgraph of Minimum
"... Abstract. In this paper, two sufficient conditions for identifying a subgraph of minimum weight triangulation of a planar point set are presented. These conditions are based on local geometric properties of an edge to be identified. Unlike the previous known sufficient conditions for identifying sub ..."
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Abstract. In this paper, two sufficient conditions for identifying a subgraph of minimum weight triangulation of a planar point set are presented. These conditions are based on local geometric properties of an edge to be identified. Unlike the previous known sufficient conditions for identifying subgraphs, such as Keil’s flskeleton and Yang and Xu’s double circles, The local geometric requirement in our conditions is not necessary symmetric with respect to the edge to be identified. The identified subgraph is different from all the known subgraphs including the newly discovered subgraph: socalled the intersection of localoptimal triangulations by Dickerson et al. An O.n3 / time algorithm for finding this subgraph from a set of n points is presented. Keywords: 1.