I/O-Efficient Data Structures for Colored Range and Prefix Reporting

Larsen, Kasper Green; Pagh, Rasmus

doi:10.1137/1.9781611973099.49

Cited by 12 publications

(13 citation statements)

References 29 publications

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“…When the twodimensional points are on the [n]×[n] grid, Larsen et. al [17] achieve improved query bound of O(1 + k/B) I/Os.…”

Section: Three-sided Orthogonal Range Reportingmentioning

confidence: 99%

“…The problem has been extensively studied in computational geometry and database communities [14,12,6,20,16,23,17,18].…”

Section: Categorical Range Reporting Without Duplicatesmentioning

confidence: 99%

“…Then we apply rank space reduction to these two-dimensional points and maintain a three-sided range reporting structure T Sj by Larsen et al [17] on this collection. All those type-j interval pairs within i ∈ [a, b] and A[i] ≥ τ for any given a, b, and τ can be answered in optimal I/Os using T Sj.…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…All those type-j interval pairs within i ∈ [a, b] and A[i] ≥ τ for any given a, b, and τ can be answered in optimal I/Os using T Sj. Further, we associate each two-dimensional point with its corresponding interval-pair, so that the interval-pairs corresponding to the points reported by structure from [17] can be obtained without spending any additional I/Os. Moreover, to be able to query data structure in [17] we need to map the boundary points (a and b) and the threshold τ to rank space.…”

Section: Proof Of Lemmamentioning

confidence: 99%

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Categorical range maxima queries

Patil

Thankachan

Shah

et al. 2014

Proceedings of the 33rd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems

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Given an arrayA[1...n] of n distinct elements from the set {1, 2, ..., n} a range maximum query RMQ(a, b) returns the highest element in A[a...b] along with its position. In this paper, we study a generalization of this classical problem called Categorical Range Maxima Query (CRMQ) problem, in which each element A[i] in the array has an associated category (color) given by C[i] ∈ [σ]. A query then asks to report each distinct color c appearing in C[a...b] along with the highest element (and its position) in A[a...b] with color c. Let pc denote the position of the highest element in A[a...b] with color c. We investigate two variants of this problem: a threshold version and a top-k version. In threshold version, we only need to output the colors with A[pc] more than the input threshold τ , whereas top-k variant asks for k colors with the highest A[pc] values.In the word RAM model, we achieve linear space structure along with O(k) query time, that can report colors in sorted order of A [·]. In external memory, we present a data structure that answers queries in optimal O(1+ k B ) I/O's using almost-linear O(n log * n) space, as well as a linear space data structure with O(log * n + k B ) query I/Os. Here k represents the output size, log * n is the iterated logarithm of n and B is the block size. CRMQ has applications to document retrieval and categorical range reporting -giving a one-shot framework to obtain improved results in both these problems. Our results for CRMQ not only improve the existing best known results for three-sided categorical range reporting but also overcome the hurdle of maintaining color uniqueness in the output set.

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“…When the twodimensional points are on the [n]×[n] grid, Larsen et. al [17] achieve improved query bound of O(1 + k/B) I/Os.…”

Section: Three-sided Orthogonal Range Reportingmentioning

confidence: 99%

“…The problem has been extensively studied in computational geometry and database communities [14,12,6,20,16,23,17,18].…”

Section: Categorical Range Reporting Without Duplicatesmentioning

confidence: 99%

Section: Proof Of Lemmamentioning

confidence: 99%

Section: Proof Of Lemmamentioning

confidence: 99%

See 2 more Smart Citations

Categorical range maxima queries

Patil

Thankachan

Shah

et al. 2014

Proceedings of the 33rd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems

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show abstract

“…Very recently, several results were presented for categorical range reporting in the I/O-model [19,21]. In [21], Nekrich presents a number of tradeoffs for threesided queries: His first data structure uses linear space and answers queries in O((N/B) ε + K/B) time, where ε > 0 is an arbitrarily small constant.…”

Section: Previousmentioning

confidence: 99%

Near-Optimal Range Reporting Structures for Categorical Data

Larsen¹,

Walderveen²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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Range reporting on categorical (or colored) data is a well-studied generalization of the classical range reporting problem in which each of the N input points has an associated color (category). A query then asks to report the set of colors of the points in a given rectangular query range, which may be far smaller than the set of all points in the query range.We study two-dimensional categorical range reporting in both the word-RAM and I/O-model. For the I/O-model, we present two alternative data structures for three-sided queries. The first answers queries in optimal O(lg B N + K/B) I/Os using O(N lg * N ) space, where K is the number of distinct colors in the output, B is the disk block size, and lg * N is the iterated logarithm of N . Our second data structure uses linear space and answers queries in O(lg B N + lg (h) N + K/B) I/Os for any constant integer h ≥ 1. Here lg (1) N = lg N and lg (h) N = lg(lg (h−1) N ) when h > 1. Both solutions use only comparisons on the coordinates. We also show that the lg B N terms in the query costs can be reduced to optimal lg lg B U when the input points lie on a U × U grid and we allow word-level manipulations of the coordinates. We further reduce the query time to just O(1) if the points are given on an N × N grid. Both solutions also lead to improved data structures for four-sided queries. For the word-RAM, we obtain optimal data structures for three-sided range reporting, as well as improved upper bounds for four-sided range reporting.Finally, we show a tight lower bound on onedimensional categorical range counting using an elegant reduction from (standard) two-dimensional range counting.

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Partial Sums on the Ultra-Wide Word RAM

Bille

Gørtz

Skjoldjensen

2020

Lecture Notes in Computer Science

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We consider the dynamic range minimum problem on the ultra-wide word RAM model of computation. This model extends the classic w-bit word RAM model with special ultrawords of length w 2 bits that support standard arithmetic and boolean operation and scattered memory access operations that can access w (non-contiguous) locations in memory. The ultra-wide word RAM model captures (and idealizes) modern vector processor architectures.Our main result is a linear space data structure that supports range minimum queries and updates in O(log log log n) time. This exponentially improves the time of existing techniques. Our result is based on a simple reduction to prefix minimum computations on sequences O(log n) words combined with a new parallel, recursive implementation of these.

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I/O-Efficient Data Structures for Colored Range and Prefix Reporting

Cited by 12 publications

References 29 publications

Categorical range maxima queries

Categorical range maxima queries

Near-Optimal Range Reporting Structures for Categorical Data

Partial Sums on the Ultra-Wide Word RAM

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