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Using Sparsification for Parametric Minimum Spanning Tree Problems
 Nordic J. Computing
, 1996
"... Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning t ..."
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Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an nnode, medge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...
Halving Point Sets
, 1998
"... Given n points in R d , a hyperplane is called halving if it has at most bn=2c points on either side. How many partitions of a point set (into the points on one side, on the hyperplane, and on the other side) by halving hyperplanes can be realized by an npoint set in R d? ..."
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Given n points in R d , a hyperplane is called halving if it has at most bn=2c points on either side. How many partitions of a point set (into the points on one side, on the hyperplane, and on the other side) by halving hyperplanes can be realized by an npoint set in R d?
Geometry © 1998 SpringerVerlag New York Inc. How to Cut Pseudoparabolas into Segments
"... Abstract. LetƔ be a collection of unbounded xmonotone Jordan arcs intersecting at most twice each other, which we call pseudoparabolas, since two axis parallel parabolas intersect at most twice. We investigate how to cut pseudoparabolas into the minimum number of curve segments so that each pair of ..."
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Abstract. LetƔ be a collection of unbounded xmonotone Jordan arcs intersecting at most twice each other, which we call pseudoparabolas, since two axis parallel parabolas intersect at most twice. We investigate how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once. We give an �(n 4/3) lower bound and O(n 5/3) upper bound on the number of cuts. We give the same bounds for an arrangement of circles. Applying the upper bound, we give an O(n 23/12) bound on the complexity of a level in an arrangement of pseudoparabolas, and an O(n 11/6) bound on the complexity of a combinatorially concave chain of pseudoparabolas. We also give some upper bounds on the number of transitions of the minimum weight matroid base when the weight of each element changes as a quadratic function of a single parameter. 1.