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Counting All Lattice Rectangles in the Square Grid in Near-Linear Time

Authors Dmitry Babichev , Sergey Babichev

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LIPIcs.MFCS.2026.26.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Lattice rectangles
  • grid enumeration
  • floor sums
  • Möbius inversion

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Abstract

We study the exact counting problem for all lattice rectangles contained in the square [0,n)×[0,n), including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction (u,v) and two side lengths, we derive a sequence of exact algorithms of complexity O(n²), O(n^{3/2} log n), O(n^{4/3} log n), and finally O(n log³n). The main idea behind the near-linear algorithm is to reduce the geometric summation to a constant-size family of weighted floor sums closed under Euclidean-style affine and reciprocal transformations, and hence evaluable in O(log n) time per query. The intermediate algorithms expose the structural reductions leading to this final kernel and provide independent cross-checks for the implementation.

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Dmitry Babichev and Sergey Babichev. Counting All Lattice Rectangles in the Square Grid in Near-Linear Time. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.26

Author Details

Dmitry Babichev
  • Paris, France
Sergey Babichev
  • Paris, France

Supplementary Materials

References

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