Linear-Space Data Structures for Range Minority Query in Arrays

Chan, Timothy M.; Durocher, Stéphane; Skala, Matthew; Wilkinson, Bryan T.

doi:10.1007/s00453-014-9881-9

Cited by 6 publications

(1 citation statement)

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“…For example, for Q = N 1.6 , the lower bound is near N 1.666 (in other words, we need at least N 0.066 time per query). In contrast, the previous reduction by Chan et The same results hold for the similar problem of range minority [CDSW15] (finding a least frequent element in a range).…”

Section: Batched Range Modementioning

confidence: 51%

Hardness for triangle problems under even more believable hypotheses: reductions from real APSP, real 3SUM, and OV

Chan

Williams

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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The 3SUM hypothesis, the All-Pairs Shortest Paths (APSP) hypothesis and the Strong Exponential Time Hypothesis are the three main hypotheses in the area of fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real 3SUM" hypotheses, which assert that the APSP and 3SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts.Under the very believable hypothesis that at least one of the Integer 3SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires 𝑛 3−𝑜 (1) time on an 𝑛-node graph.The main result of this paper is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real 3SUM, the Real APSP, and the Orthogonal Vector (OV) hypotheses is true. Combined with slight modifications of prior reductions from Triangle Collection, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic versions of Max Flow, Single-Source Reachability Count, and Counting Strongly Connected Components, now under the new weaker hypothesis.Our main result is built on the following two lines of reductions. In the first line of reductions, we show Real APSP and Real 3SUM hardness for the All-Edges Sparse Triangle problem. Prior reductions only worked from the integer variants of these problems. In the second line of reductions, we show Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem.

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Section: Batched Range Modementioning

confidence: 51%

Hardness for triangle problems under even more believable hypotheses: reductions from real APSP, real 3SUM, and OV

Chan

Williams

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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Compressed Dynamic Range Majority and Minority Data Structures

Gagie

Navarro

2020

Algorithmica

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In the range α-majority query problem, we are given a sequence S[1..n] and a fixed threshold α ∈ (0, 1), and are asked to preprocess S such that, given a query range [i..j], we can efficiently report the symbols that occur more than α(j − i + 1) times in S[i..j], which are called the range α-majorities. In this article we first describe a dynamic data structure that represents S in compressed space -nH k + o(n lg σ) bits for any k = o(lg σ n), where σ is the alphabet size and H k ≤ H 0 ≤ lg σ is the k-th order empirical entropy of S -and answers queries in O lg n α lg lg n time while supporting insertions and deletions in S in O lg n α amortized time. We then show how to modify our data structure to receive some β ≥ α at query time and report the range β-majorities in O lg n β lg lg n time, without increasing the asymptotic space or update-time bounds. The best previous dynamic solution has the same query and update times as ours, but it occupies O(n) words and cannot take advantage of being given a larger threshold β at query time. Not even static data structures have previously achieved compression in terms of higher-order entropy. The smallest ones take nH 0 + o(n)(H 0 + 1) bits and answer queries in O(f (n)/α) time or take (1 + ǫ)nH 0 + o(n) bits and answer queries in optimal O(1/α) time, where f (n) is any function in ω(1) and ǫ is any constant greater than 0. By giving up updates, we can improve our query time to O((1/α) lg lg w σ) on a RAM with word size w = Ω(lg n) bits, which is only slightly suboptimal, without increasing our space bound. Finally, we design the first dynamic data structure for range α-minority -i.e., find a non-α-majority that occurs in a rangeand obtain space and time bounds similar to those for α-majorities. A static version of this structure is also the first α-minority data structure achieving compression in terms of H k .

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