TL;DR
This paper presents a Monte-Carlo data structure for efficiently approximating the majority depth of a point relative to a set of points, improving query times with probabilistic guarantees.
Contribution
It introduces a novel Monte-Carlo data structure that preprocesses lines in O(n log n) time for fast approximate majority depth queries.
Findings
Construction time is O(n log n)
Query time is O((log n)^{4/3}) expected
Answers are correct with high probability
Abstract
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 Monte-Carlo data structure for this problem that can be constructed in O(nlog n) time, can answer queries O((log n)^{4/3}) expected time, and answers correctly with high probability.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Algorithms and Data Compression · Machine Learning and Algorithms