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A new lower bound for semigroup orthogonal range searching

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We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [14]: using m units of space, the query time Q(n) must be Ω(α(m, n) + (Formula presented.) where α(·, ·) is the inverse Ackermann’s function, a very slowly growing function. In d dimensions, Bernard Chazelle [9] proved that the query time must be Q(n) = Ω((logβ n)d−1) where β = 2m/n. Chazelle’s lower bound is known to be tight for when space consumption is “high” i.e., m = Ω(n logd+ε n). We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for “low space”: we prove that we must have mQ(n) = Ω(n(log n log log n)d−1). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.

Original languageEnglish
Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
EditorsGill Barequet, Yusu Wang
Number of pages14
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Publication yearJun 2019
Article number3
ISBN (Electronic)9783959771047
Publication statusPublished - Jun 2019
Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
Duration: 18 Jun 201921 Jun 2019


Conference35th International Symposium on Computational Geometry, SoCG 2019
LandUnited States
SeriesLeibniz International Proceedings in Informatics, LIPIcs

    Research areas

  • Data Structures, Lower bounds, Range Searching

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