Near-optimal randomized algorithms for selection in totally monotone matrices

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We revisit classical problems about searching in totally monotone matrices, which have many applications in computational geometry and other areas. In a companion paper, we gave new (near-)linear-time algorithms for a number of such problems. In the present paper, we describe new subquadratic results for more basic problems, including the following: • A randomized algorithm to select the K-th smallest element in an n × n totally monotone matrix in O(n4/3 polylog n) expected time; this improves previous O(n3/2 polylog n) algorithms by Alon and Azar [SODA'92], Mansour et al. (1993), and Agarwal and Sen (1996). • A near-matching lower bound of Ω(n4/3) for the problem (which holds even for Monge matrices). • A similar result for selecting the ki-th smallest in the i-th row for all i. • In the case when all ki's are the same, an improvement of the running time to O(n6/5 polylog n). • Variants of all these bounds that are sensitive to K (or Pi ki). These matrix searching problems are intimately related to problems about arrangements of pseudo-lines. In particular, our selection algorithm implies an O(n4/3 polylog n) algorithm for computing incidences between n points and n pseudo-lines in the plane. This improves, extends, and simplifies a previous method by Agarwal and Sharir [SODA'02].

Original languageEnglish (US)
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Number of pages13
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: Jan 10 2021Jan 13 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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