Necklaces, Convolutions, and X+Y

TL;DR

This paper presents subquadratic algorithms for aligning necklaces based on different p norms, reducing the problem to various convolutions, and solving these convolutions efficiently, advancing understanding of related sorting and order statistic problems.

Contribution

The paper introduces novel subquadratic algorithms for necklace alignment problems by reducing them to convolution problems and solving these convolutions efficiently.

Findings

Algorithms run in o(n^2) time, faster than naive methods.

Reductions to standard, (min,+), and median,+ convolutions.

Insights into the sorting X+Y problem and order statistics.

Abstract

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.