69,419 research outputs found

    Triangle areas in line arrangements

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    A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set PP of cardinality nn in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of nn lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these nn lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2)\Omega(n^2) and O(n9/4)O(n^{9/4}). This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.Comment: Title is shortened. Some typos and small errors were correcte

    On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

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    We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by nn points in 3-space, and in general in dd dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by nn points in \RR^3 is at most 2/3n3O(n2){2/3}n^3-O(n^2), and there are point sets for which this number is 3/16n3O(n2){3/16}n^3-O(n^2). We also present an O(n3)O(n^3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k,d\in \NN, 1kd1\leq k \leq d, the maximum number of kk-dimensional simplices of minimum (nonzero) volume spanned by nn points in \RR^d is Θ(nk)\Theta(n^k). (ii) The number of unit-volume tetrahedra determined by nn points in \RR^3 is O(n7/2)O(n^{7/2}), and there are point sets for which this number is Ω(n3loglogn)\Omega(n^3 \log \log{n}). (iii) For every d\in \NN, the minimum number of distinct volumes of all full-dimensional simplices determined by nn points in \RR^d, not all on a hyperplane, is Θ(n)\Theta(n).Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 200

    Geometric versions of the 3-dimensional assignment problem under general norms

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    We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles measured according to a certain norm. (All our results also carry over to the corresponding special cases of the 3-dimensional matching problem.) The minimization version is NP-hard for every norm, even if the underlying Cartesian space is 2-dimensional. The maximization version is polynomially solvable, if the dimension of the Cartesian space is fixed and if the considered norm has a polyhedral unit ball. If the dimension of the Cartesian space is part of the input, the maximization version is NP-hard for every LpL_p norm; in particular the problem is NP-hard for the Manhattan norm L1L_1 and the Maximum norm LL_{\infty} which both have polyhedral unit balls.Comment: 21 pages, 9 figure
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