Cuttings

Authored by: Bernard Chazelle

Handbook of Data Structures and Applications

Print publication date:  March  2018
Online publication date:  February  2018

Print ISBN: 9781498701853
eBook ISBN: 9781315119335
Adobe ISBN:

10.1201/9781315119335-26

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Abstract

For divide-and-conquer purposes, it is often desirable to organize a set S of n numbers into a sorted list, or perhaps to partition it into two equal-sized groups with no element in one group exceeding any element in the other one. More generally, we might wish to break up S into k groups of size roughly n/k, with again a total ordering among the distinct groups. In the first case we sort; in the second one we compute the median; in the third one we compute quantiles. This is all well known and classical. Is it possible to generalize these ideas to higher dimension? Surprisingly the answer is yes. A geometric construction, known as an ɛ-cutting, provides a space partitioning technique that extends the classical notion of selection to any finite dimension. It is a powerful, versatile data structure with countless applications in computational geometry.

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