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• W76887364 abstract "This thesis consists of two parts dealing with combinatorial and computational problems in geometry, respectively. In the first part three independent problems are considered: (1) We determine an upper bound $LF 11n/6 RF +1$ for the number of extreme triples of $n$ points in the plane, almost matching a known lower bound $LF 11n/6 RF$; (2) we determine some bounds for the smallest dimension $d=Delta(j,k)$ such that for any $j$ mass distributions in $R d$, there are $k$ hyperplanes so that each orthant contains a fraction $1/2^k$ of each of the masses; it is easily shown that $j(2^k-1)/kleqDelta(j,k)leq j2^{k-1}$; we believe the lower bound is tight, but can only prove it in a few cases (as a tool we prove a Borsuk-Ulam theorem on a product of balls, which is of independent interest); (3) for a collection $B$ of pseudo-disks in the plane, we show the existence of a two-dimensional abstract simplicial complex, $XX subseteq 2^B$, which has some nice topological properties, such that the inclusion-exclusion relation $measure{union{B}} = sum_{simplex in 2^B-{emptyset}} (-1)^{card{simplex}-1} measure{bigcap simplex}$ holds when $XX$ is substituted for $2^B$. In the second part, using geometric sampling techniques, we give algorithms for three similar problems: (4) Computing the intersection of halfspaces in $R 3$; (5) computing the intersection of balls of equal radius in $R 3$; and (6) computing the Voronoi diagram of line segments in $R 2$; in each case we obtain a deterministic parallel algorithm for the EREW PRAM model that runs in time $O(log^2 n)$ and uses work $O(nlog n)$ for a problem of size $n$ (for ball intersection this is also the first optimal deterministic and sequential algorithm, using the Dobkin-Kirkpatrick decomposition, we can only achieve time $O(nlog^2 n)$). Using the parallel algorithm for ball intersection, one obtains (7) a sequential deterministic algorithm for computing the diameter of a point set in $R 3$ that runs in time $O(nlog^3 n)$. Using also geometric sampling techniques, (8) we describe an algorithm for computing the arrangement of $n$ segments in the plane in time $O(log^2 n)$ and using work $O(nlog n +k)$ where $k$ is the number of pairwise intersections, also in the EREW PRAM model (sequentially this results in an algorithm that outputs all the intersections in optimal time using $O(n)$ space); and (9) assuming that certain sampling result can be derandomized in polynomial time, we describe a sequential algorithm for computing one face in an arrangement of segments that runs in time $O(nalpha^2(n)log n)$ where $alpha(n)$ is a very slowly growing function." @default.
• W76887364 created "2016-06-24" @default.
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• W76887364 date "1995-10-01" @default.
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• W76887364 title "Topics in combinatorial and computational geometry" @default.
• W76887364 hasPublicationYear "1995" @default.
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