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RayShooting Depth: Computing Statistical Data Depth of Point Sets in the Plane
"... Abstract. Over the past several decades, many combinatorial measures have been devised for capturing the statistical data depth of a set of n points in R 2. These include Tukey depth [15], Oja depth [12], Simplicial depth [10] and several others. Recently Fox et al. [7] have defined the RayShooting ..."
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Abstract. Over the past several decades, many combinatorial measures have been devised for capturing the statistical data depth of a set of n points in R 2. These include Tukey depth [15], Oja depth [12], Simplicial depth [10] and several others. Recently Fox et al. [7] have defined the RayShooting depth of a point set, and given a topological proof for the existence of points with high RayShooting depth in R 2. In this paper, we present an O(n 2 log 2 n)time algorithm for computing a point of high RayShooting depth. We also present a linear time approximation algorithm. 1
Hitting and piercing rectangles induced by a point set
"... Abstract. We consider various hitting and piercing problems for the family of axisparallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon ..."
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Abstract. We consider various hitting and piercing problems for the family of axisparallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number of induced objects that are hit/pierced by a single point. First, we prove an exact result on the strong and the weak variant of the First Selection Lemma for rectangles. We also show bounds for the Second Selection Lemma which improve upon previous bounds when there are nearquadratic number of induced rectangles. Next, we consider the hitting set problem for induced rectangles. This is a special case of the geometric hitting set problem which has been extensively studied. We give efficient algorithms and show exact combinatorial bounds on the hitting set problem for two special classes of induced axisparallel rectangles. Finally, we show that the minimum hitting set problem for all induced lines is NPComplete.
A Proof of the OjaDepth Conjecture in the Plane
"... Given a set P of n points in the plane, the Oja depth of a point x ∈ R 2 is defined to be the sum of the areas of all triangles defined by x and two points from P, normalized by the area of convexhull of P. The Ojadepth of P is the minimum Ojadepth of any point in R 2. The Ojadepth conjecture st ..."
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Given a set P of n points in the plane, the Oja depth of a point x ∈ R 2 is defined to be the sum of the areas of all triangles defined by x and two points from P, normalized by the area of convexhull of P. The Ojadepth of P is the minimum Ojadepth of any point in R 2. The Ojadepth conjecture states that any set P of n points in the plane has Ojadepth at most n 2 /9 (this would be optimal as there are examples where it is not possible to do better). We present a proof of this conjecture. We also improve the previously best bounds for all R d, d ≥ 3, via a different, more combinatorial technique. 1
Selection Lemmas for various geometric objects
, 2014
"... Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many object ..."
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Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set P. This question has been widely explored for simplices in Rd, with tight bounds in R2. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from P. We prove an exact result on the strong and the weak variant of the first selection lemma for axisparallel rectangles, special subclasses of axisparallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show nontrivial bounds on the first selection lemma for axisparallel boxes and hyperspheres in Rd. In the second selection lemma, we consider an arbitrary m sized subset of the set of all objects induced by P. We study this problem for axisparallel rectangles and show that there exists an point in the plane that is contained in m 3 24n4 rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir [20] when m is almost quadratic. 1