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

TL;DR

This paper develops efficient algorithms for counting all lattice rectangles within a square grid, including non-axis-parallel ones, achieving near-linear time complexity for the problem.

Contribution

The authors introduce several exact algorithms with improved complexities, including a near-linear time algorithm for counting all lattice rectangles in a grid.

Findings

Multiple algorithms with complexities from O(n^2) to O(n log^2 n) for counting lattice rectangles.

An all-values algorithm computing counts for all sizes in O(N^{3/2}) operations.

A two-term asymptotic expansion for the count function validating the algorithms.

Abstract

We study the exact counting problem for all lattice rectangles contained in the square $[0,n)\times[0,n)$, including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction $(u,v)$ and two side lengths, we derive several exact algorithms: the classical $O(n^2)$ sweep, decompositions of complexity $O(n^{3/2}\log n)$ and $O(n^{4/3}\log n)$, a ten-moment weighted-floor-sum reduction of complexity $O(n\log^3 n)$, and a divisor-layer algorithm with the complexity $O(n\log^2 n)$. We also give an all-values algorithm that computes $F(1),\ldots,F(N)$ in $O(N^{3/2})$ arithmetic operations. The main idea behind the near-linear one-value algorithms is to reduce the geometric summation to constant-size families of weighted floor sums closed under Euclidean-style affine and reciprocal transformations. Besides the exact algorithmic results, we derive a two-term asymptotic expansion, $F(n)=\frac{4\log 2-1}{\pi^2}n^4\log n+B\,n^4+o(n^4)$ with the explicit formula for $B$, which provides an independent consistency check for the large-$n$ numerical data produced by the algorithms.