## Partitioning Point Sets in Arbitrary Dimension (Classic Reprint)

### Richard Cole

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Excerpt from Partitioning Point Sets in Arbitrary Dimension

1. Introduction

The half-space retrieval problem is the following. Given a set of n points in d dimensional Euclidean space, preprocess them so as to be able to quickly answer the query; how many points lie in the query half space H. (A variant of the problem, the listing problem, is to ask for a list of the points in the half space.) It is assumed that many such queries will be made. Thus it is reasonable to preprocess the set of points and to amortize the cost of this preprocessing over the (many) queries. Our first concern here is with the query time and the space used by the data structure for holding the preprocessed information. We note that the naive searching algorithm takes linear time. Thus, we aim for a sublinear search time and a data structure using linear space. A second concern is to find relatively efficient algorithms for building the data structures.

Recently, an elegant approach to this problem was discovered by Willard [ᵂ]. There are two basic lemmas underlying his construction. First, given a set of n points, the plane can be partitioned by two straight lines into 4 quadrants, so that each (open) quadrant holds at most n/4 points. This leads to a recursive storage of the points in a 4-way balanced tree (ignoring, for now, points lying on the partitioning lines). Second, any line intersects only 3 of the 4 quadrants defined by the two partitioning lines. This implies that in carrying out a half-space query at most three of the four subtrees of the root need be explored further. Recursive application of this observation yields a sublinear search time.

Using the same approach Yao [Y1,Y2] obtained a similar result in 3 dimensions. She proved that, given any set of n points, the space can be divided by 3 planes into 8 regions, so that each open region contains at most one eighth of the points. This yielded an algorithm for the half-space retrieval problem with a sublinear query time.

Such a partition would yield an algorithm for half space retrieval, with sublinear query time, similar to Willard's. It is natural to try to achieve this partition with d
planes.

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## 1.Partitioning Point Sets in Arbitrary Dimension (Classic Reprint) (Paperback)

ISBN 10: 1332175929 ISBN 13: 9781332175925
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Book Description Forgotten Books, United States, 2015. Paperback. Book Condition: New. Language: English . Brand New Book ***** Print on Demand *****. Excerpt from Partitioning Point Sets in Arbitrary Dimension 1. Introduction The half-space retrieval problem is the following. Given a set of n points in d dimensional Euclidean space, preprocess them so as to be able to quickly answer the query; how many points lie in the query half space H. (A variant of the problem, the listing problem, is to ask for a list of the points in the half space.) It is assumed that many such queries will be made. Thus it is reasonable to preprocess the set of points and to amortize the cost of this preprocessing over the (many) queries. Our first concern here is with the query time and the space used by the data structure for holding the preprocessed information. We note that the naive searching algorithm takes linear time. Thus, we aim for a sublinear search time and a data structure using linear space. A second concern is to find relatively efficient algorithms for building the data structures. Recently, an elegant approach to this problem was discovered by Willard [ ]. There are two basic lemmas underlying his construction. First, given a set of n points, the plane can be partitioned by two straight lines into 4 quadrants, so that each (open) quadrant holds at most n/4 points. This leads to a recursive storage of the points in a 4-way balanced tree (ignoring, for now, points lying on the partitioning lines). Second, any line intersects only 3 of the 4 quadrants defined by the two partitioning lines. This implies that in carrying out a half-space query at most three of the four subtrees of the root need be explored further. Recursive application of this observation yields a sublinear search time. Using the same approach Yao [Y1, Y2] obtained a similar result in 3 dimensions. She proved that, given any set of n points, the space can be divided by 3 planes into 8 regions, so that each open region contains at most one eighth of the points. This yielded an algorithm for the half-space retrieval problem with a sublinear query time. Such a partition would yield an algorithm for half space retrieval, with sublinear query time, similar to Willard s. It is natural to try to achieve this partition with d planes. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. Bookseller Inventory # APC9781332175925

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## 2.Partitioning Point Sets in Arbitrary Dimension (Classic Reprint) (Paperback)

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Book Description Forgotten Books, United States, 2015. Paperback. Book Condition: New. Language: English . Brand New Book ***** Print on Demand *****.Excerpt from Partitioning Point Sets in Arbitrary Dimension 1. Introduction The half-space retrieval problem is the following. Given a set of n points in d dimensional Euclidean space, preprocess them so as to be able to quickly answer the query; how many points lie in the query half space H. (A variant of the problem, the listing problem, is to ask for a list of the points in the half space.) It is assumed that many such queries will be made. Thus it is reasonable to preprocess the set of points and to amortize the cost of this preprocessing over the (many) queries. Our first concern here is with the query time and the space used by the data structure for holding the preprocessed information. We note that the naive searching algorithm takes linear time. Thus, we aim for a sublinear search time and a data structure using linear space. A second concern is to find relatively efficient algorithms for building the data structures. Recently, an elegant approach to this problem was discovered by Willard [ ]. There are two basic lemmas underlying his construction. First, given a set of n points, the plane can be partitioned by two straight lines into 4 quadrants, so that each (open) quadrant holds at most n/4 points. This leads to a recursive storage of the points in a 4-way balanced tree (ignoring, for now, points lying on the partitioning lines). Second, any line intersects only 3 of the 4 quadrants defined by the two partitioning lines. This implies that in carrying out a half-space query at most three of the four subtrees of the root need be explored further. Recursive application of this observation yields a sublinear search time. Using the same approach Yao [Y1, Y2] obtained a similar result in 3 dimensions. She proved that, given any set of n points, the space can be divided by 3 planes into 8 regions, so that each open region contains at most one eighth of the points. This yielded an algorithm for the half-space retrieval problem with a sublinear query time. Such a partition would yield an algorithm for half space retrieval, with sublinear query time, similar to Willard s. It is natural to try to achieve this partition with d planes. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. Bookseller Inventory # APC9781332175925

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## 4.Partitioning Point Sets in Arbitrary Dimension (Classic Reprint) (Paperback)

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Book Description Forgotten Books, United States, 2015. Paperback. Book Condition: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. Brand New Book. Excerpt from Partitioning Point Sets in Arbitrary Dimension 1. Introduction The half-space retrieval problem is the following. Given a set of n points in d dimensional Euclidean space, preprocess them so as to be able to quickly answer the query; how many points lie in the query half space H. (A variant of the problem, the listing problem, is to ask for a list of the points in the half space.) It is assumed that many such queries will be made. Thus it is reasonable to preprocess the set of points and to amortize the cost of this preprocessing over the (many) queries. Our first concern here is with the query time and the space used by the data structure for holding the preprocessed information. We note that the naive searching algorithm takes linear time. Thus, we aim for a sublinear search time and a data structure using linear space. A second concern is to find relatively efficient algorithms for building the data structures. Recently, an elegant approach to this problem was discovered by Willard [ ]. There are two basic lemmas underlying his construction. First, given a set of n points, the plane can be partitioned by two straight lines into 4 quadrants, so that each (open) quadrant holds at most n/4 points. This leads to a recursive storage of the points in a 4-way balanced tree (ignoring, for now, points lying on the partitioning lines). Second, any line intersects only 3 of the 4 quadrants defined by the two partitioning lines. This implies that in carrying out a half-space query at most three of the four subtrees of the root need be explored further. Recursive application of this observation yields a sublinear search time. Using the same approach Yao [Y1, Y2] obtained a similar result in 3 dimensions. She proved that, given any set of n points, the space can be divided by 3 planes into 8 regions, so that each open region contains at most one eighth of the points. This yielded an algorithm for the half-space retrieval problem with a sublinear query time. Such a partition would yield an algorithm for half space retrieval, with sublinear query time, similar to Willard s. It is natural to try to achieve this partition with d planes. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. Bookseller Inventory # LIE9781332175925