Optimal Planar Orthogonal Skyline Counting Queries

Brodal, Gerth Stølting; Larsen, Kasper Green

doi:10.1007/978-3-319-08404-6_10

Cited by 6 publications

(2 citation statements)

References 17 publications

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“…This problem, while rather abstract, has been shown to capture the hardness of a wealth of static data structure problems such as 2D Range Counting, 2D Rectangle Stabbing, 2D Skyline Counting and Range Mode Queries, see e.g. [26,7,16] as well as for dynamic data structure problems, including Range Selection and Median [22] and recently also all dynamic problems that the Marked Ancestor Problem reduces to [21,3], which includes 2d Range Emptiness, Partial Sums and Worst-Case Union-Find. Our work adds 3SUM-Indexing and all problems it reduces to, to the list.…”

Section: Our Contributionsmentioning

confidence: 99%

Stronger 3SUM-Indexing Lower Bounds

Chung¹,

Larsen²

2022

Preprint

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The 3SUM-Indexing problem was introduced as a data structure version of the 3SUM problem, with the goal of proving strong conditional lower bounds for static data structures via reductions. Ideally, the conjectured hardness of 3SUM-Indexing should be replaced by an unconditional lower bound. Unfortunately, we are far from proving this, with the strongest current lower bound being a logarithmic query time lower bound by Golovnev et al. from STOC'20. Moreover, their lower bound holds only for non-adaptive data structures and they explicitly asked for a lower bound for adaptive data structures. Our main contribution is precisely such a lower bound against adaptive data structures. As a secondary result, we also strengthen the non-adaptive lower bound of Golovnev et al. and prove strong lower bounds for 2-bit-probe non-adaptive 3SUM-Indexing data structures via a completely new approach that we find interesting in its own right.

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Section: Our Contributionsmentioning

confidence: 99%

Stronger 3SUM-Indexing Lower Bounds

Chung¹,

Larsen²

2022

Preprint

Self Cite

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“…We remark that the ball-inheritance problem also has been used to improve the upper bounds for various other problems with a range reporting flavour to them, see e.g. [6,4]. Thus in light of the lack of progress in proving tight lower bounds for range reporting, it seems like a natural goal to understand the complexity of the ball-inheritance problem.…”

Section: Range Reporting In the Word-rammentioning

confidence: 99%

Towards Tight Lower Bounds for Range Reporting on the RAM

Grønlund,

Larsen

2014

Preprint

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In the orthogonal range reporting problem, we are to preprocess a set of n points with integer coordinates on a U × U grid. The goal is to support reporting all k points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exists:

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An Efficient Algorithm to Count the Relations in a Range of Binary Relations Represented by a k²-Tree

2021

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Two sets A and B, whose elements fulfill a total order on operator ≤, can have a binary relation R ⊆ A × B represented by the k 2 -tree compact data structure, which greatly improves storage space. Currently, Count query is managed by either using Range query or to modify the structure to have aggregate information, implying additional time or space in order to perform the query. This article presents Compact Count, which exploits the k 2 -tree properties to reduce the paths to be scanned to count the numbers in a range r, thus ensuring an expected runtime of O(log k r log k n) and storage of O(log k r) with the k 2 -tree parameters n and k. Our algorithm was compared through a series of experiments that consider both synthetic data with different distributions and real data, with a solution based on the Range algorithm. Experimental results show that Compact Count is 250 to 1,000 times faster than Range on synthetic and real data, respectively, with a small additional storage cost, as expected by the theoretical analysis.

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Optimal Planar Orthogonal Skyline Counting Queries

Cited by 6 publications

References 17 publications

Stronger 3SUM-Indexing Lower Bounds

Stronger 3SUM-Indexing Lower Bounds

Towards Tight Lower Bounds for Range Reporting on the RAM

An Efficient Algorithm to Count the Relations in a Range of Binary Relations Represented by a k²-Tree

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