منابع مشابه
Approximating Majority Depth
We consider the problem of approximating the majority depth (Liu and Singh, 1993) of a point q with respect to an n-point set, S, by random sampling. At the heart of this problem is a data structures question: How can we preprocess a set of n lines so that we can quickly test whether a randomly selected vertex in the arrangement of these lines is above or below the median level. We describe a M...
متن کاملBounds on the Size of Small Depth Circuits for Approximating Majority
In this paper, we show that for every constant 0 < ǫ < 1/2 and for every constant d ≥ 2, the minimum size of a depth d Boolean circuit that ǫ-approximates Majority function on n variables is exp(Θ(n)). The lower bound for every d ≥ 2 and the upper bound for d = 2 have been previously shown by O’Donnell and Wimmer [ICALP’07], and the contribution of this paper is to give a matching upper bound f...
متن کاملApproximating the Simplicial Depth
Let P be a set of n points in d-dimensions. The simplicial depth, σP (q) of a point q is the number of d-simplices with vertices in P that contain q in their convex hulls. The simplicial depth is a notion of data depth with many applications in robust statistics and computational geometry. Computing the simplicial depth of a point is known to be a challenging problem. The trivial solution requi...
متن کاملAlgorithms for Bivariate Majority Depth
The majority depth of a point with respect to a point set is the number of major sides it is in. An algorithm for majority depth in R is given in this paper, and it is the first algorithm to compute the majority depth. This algorithm runs in O((n+m) log n) time with Brodal and Jacob’s data structure, and in O ( (n + m) log n log log n ) time in the word RAM model.
متن کاملSeparating Ac from Depth-2 Majority Circuits∗
We construct a function in AC that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n1/5)). This solves an open problem due to Krause and Pudlák (1997) and matches Allender’s classic result (1989) that AC can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This...
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ژورنال
عنوان ژورنال: Computational Geometry
سال: 2013
ISSN: 0925-7721
DOI: 10.1016/j.comgeo.2013.06.005