Necklaces, Convolutions, and X + Y

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 ℓp 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=2, and p=∞. For p=2, we reduce the problem to standard convolution, …