Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements

TL;DR

This paper extends fast convolution algorithms to arbitrary convex polygons and introduces efficient methods for counting specific 3-element sub-cadences, broadening the applicability of geometric convolution techniques.

Contribution

It generalizes the convolution approach from rectangles to arbitrary convex polygons and develops fast algorithms for counting 3-element sub-cadences.

Findings

Extended convolution algorithm to convex polygons with k vertices.

Achieved O(k + p(log p)^2 log k) time complexity.

Developed fast algorithms for counting 3-element sub-cadences.

Abstract

The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in [0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle P with perimeter p in O(p log p) time. This paper extends this geometric interpretation in order to allow arbitrary convex polygons P with k vertices and perimeter p. Also, this extended algorithm only needs O(k + p(log p)^2 log k) time. Additionally, this paper presents fast algorithms for counting sub-cadences and cadences with 3 elements using this extended method.