Colored Range Searching in Linear Space

Grossi, Roberto; Vind, Søren

doi:10.1007/978-3-319-08404-6_20

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…In computational geometry, there have been extensive studies on problems over points associated with information represented as colors [13,14,20,19,21,11,23,10,12,5,3,16,24]. Among them, the 2D orthogonal range counting query problem is one of the most fundamental.…”

Section: Introductionmentioning

confidence: 99%

“…Setting X = √ n lg n minimizes space usage, achieving an O(n lg 4 n)-word solution with O( √ n lg 8 n) query time. When only linear space is allowed, Grossi and Vind [11] showed how to answer a query in O(n/ polylog(n)) time. Though not explicitly stated anywhere, by combining an approach that Kaplan et al [19] presented for dimensions of 3 or higher (which also works for 2D) and a linear space solution to 2D orthogonal range emptiness [4], the query time can be improved to O(n 3/4 lg ϵ n) for any constant ϵ using a linear space structure.…”

Section: Introductionmentioning

confidence: 99%

“…One of their solutions to a path query problem generalized from 2D orthogonal colored range counting also uses linear space and provides O(n 3/4 lg ϵ n) query time. We end this brief survey by commenting that, after a long serious of work [18,13,19,23,11,12], Chan and Nekrich [5] solved the related 2D orthogonal range reporting problem in O(n lg 3/4+ϵ n) words of space and O(lg lg n + k) query time for points in rank space, where k is output size. This almost matches the bounds of the optimal solution to (uncolored) 2D orthogonal range reporting over points in rank space, which uses O(n lg ϵ n) words to answer queries in O(lg lg n + k) time.…”

Section: Introductionmentioning

confidence: 99%

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Space Efficient Two-Dimensional Orthogonal Colored Range Counting

Gao,

2021

Preprint

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In the two-dimensional orthogonal colored range counting problem, we preprocess a set, P , of n colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space are in words and ϵ is an arbitrary constant in (0, 1)):and O( √ n lg 5/2 n lg lg n) query time; O(n lg 2 n) space and O( √ n lg 4+ϵ n) query time; O(n lg 2 n lg lg n ) space and O( √ n lg 5+ϵ n) query time;O(n lg n) space and O(n 1/2+ϵ ) query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than √ n by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses O(n lg 4 n) space to answer queries in O( √ n lg 8 n) time (SIAM. J. Comp. 2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Space Efficient Two-Dimensional Orthogonal Colored Range Counting

Gao,

2021

Preprint

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“…Colored range searching (also known as "categorical range searching", or "generalized range searching") have been extensively studied in computational geometry since the 1990s. For example, see the papers [6,12,19,20,21,22,24,25,26,27,29,30,31,32,33,35,37,38,39,41,47] and the survey by Gupta et al [23]. Given a set of n colored data points (where the color of a point represents its "category"), the objective is to build data structures that can provide statistics or some kind of summary about the colors of the points inside a query range.…”

Section: Introductionmentioning

confidence: 99%

Further Results on Colored Range Searching

Chan¹,

He²,

Nekrich³

2020

Preprint

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We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1, . . . , n}. Previous data structures require O( log n log log n + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log 2 n) query time. 3. For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n)space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time iswhere k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k log n log log n ) time.Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

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Adaptive Data Structures for 2D Dominance Colored Range Counting

Gao¹

2023

Lecture Notes in Computer Science

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Colored Range Searching in Linear Space

Cited by 5 publications

References 19 publications

Space Efficient Two-Dimensional Orthogonal Colored Range Counting

Space Efficient Two-Dimensional Orthogonal Colored Range Counting

Further Results on Colored Range Searching

Adaptive Data Structures for 2D Dominance Colored Range Counting

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