## New applications of random sampling in computational geometry.(English)Zbl 0615.68037

For a set $$X$$ and integer $$i$$, let $$X^i$$ be the set of $$i$$-tuples of $$X$$. For a positive integer $$n$$, let $$\mathfrak n$$ denote the set of integers $$\{1,2,\ldots,n\}$$. Let $$b(j;t,\alpha)$$ be the probability of $$j$$ successes out of $$t$$ Bernoulli trials with probability of success $$\alpha$$. For region $$A$$ and for $$B$$ a set of regions (subsets) of $$E^d$$, let $$\#(A,B)$$ be the number of elements of $$B$$ that have nonempty intersection with $$A$$. The following theorems are proved:
Theorem 1: Let $$S$$ and $$\mathfrak F$$ be sets of regions of $$E^d$$ with $$|S| = s$$. For fixed positive integers $$i$$ and $$n$$, let $$\nu_k$$, $$k\in\mathfrak n$$, be a collection of mappings from $$S^i$$ to $$\mathfrak F$$. Let $$R$$ be a random sample of $$S$$, of size $$r$$, and let $$\mathfrak F_R$$ be $$\{\nu_k(R^*)$$; $$k\in\mathfrak n$$, $$R^*\in R^i\}$$ the union of the images of $$R^i$$ under the $$\nu_k$$’s. Then for integer $$m$$ and $$\alpha\in [0,1]$$, with $$m\le (r-i)\alpha$$,
$\operatorname{Prob}\{\exists A\in\mathfrak F_ R \text{ with } \#(A,R)\le m\text{ and }\#(A,S)>\alpha s\}\le O(r^i)\sum_{j\le m}b(j;r-i,\alpha)$
as $$r\to \infty$$. Similarly, for integer $$m$$ and $$\alpha\in [0,1]$$ with $$m\ge (r-i)\alpha$$,
$\operatorname{Prob}\{\exists A\in\mathfrak F_ R \text{ with } \#(A,R)\ge m\text{ and } \#(A,S)<\alpha s\}\le O(r^i)\sum_{j\ge m}b(j;r-i,\alpha)$
as $$r\to \infty$$. When $$m$$ is much larger than the mean $$\alpha r$$, with high probability, every $$\#(A,R)$$ is a good estimate of $$\#(A,S)$$.
A $$k$$-set of a set of sites (points) $$S$$ in $$E^d$$ is a subset of $$S$$ of size $$k$$ that is all on one side of some hyperplane, while the other sites are all on the other side of the hyperplane.
Theorem 2: Let $$g_{k,3}(s)$$ be the maximum total number of $$(\le k)$$-sets of any set of $$s$$ sites in $$E^3$$. Then
$g_{k,3}(s) = O(sk^2 \log^8s/(\log \log s)^6)$
$\text{(conjecture}\quad g_{k,d}(s) = O(s^{[d/2]}k^{[d/2]})).$
Theorem 3: One new algorithm is given for searching an arrangement of $$s$$ hyperplanes in $$E^d$$ in $$O(s^{d+\varepsilon})$$ expected time and $$O(s^{d+\varepsilon})$$ worst-case space, so that queries may be answered in $$O(\log s)$$ time, as $$s\to \infty$$, for fixed $$d$$ and for any fixed $$\varepsilon >0$$.
Theorem 4: The separation of two polytopes $$A,B\subset E^d$$ (minimum distance from a point of one to a point of the other) may be computed in expected time $$O((\vert A)^{[d/2]}+(\vert B)^{[d/2]})$$ where $$\vert =$$ number of vertices and the expectation is with respect to the random behavior of the algorithm. Several lemmas preceding the theorems are interesting by themselves.

### MSC:

 68Q25 Analysis of algorithms and problem complexity 68P10 Searching and sorting 52Bxx Polytopes and polyhedra 52A22 Random convex sets and integral geometry (aspects of convex geometry)

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### References:

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