An Approximation Algorithm for Dissecting a Rectangle into Rectangles with Specified Areas

Abstract

Given a rectangle R with area α and a set of n positive reals A = {a ₁, a ₂, ..., a _n } with $\Sigma_{a_i \in A}a_i = \alpha$, we consider the problem of dissecting R into n rectangles r _i with area a _i (i = 1,2,...,n) so that the set $\mathcal{R}$ of resulting rectangles minimizes an objective function such as the sum of perimeters of the rectangles in $\mathcal{R}$, the maximum perimeter of the rectangles in $\mathcal{R}$, and the maximum aspect ratio ρ(r) of the rectangles $r \in \mathcal{R}$, where we call the problems with these objective functions PERI-SUM, PERI-MAX and ASPECT-RATIO, respectively. We propose an O(n logn) time algorithm that finds a dissection $\mathcal{R}$ of R that is a 1.25-approximation solution to the PERI-SUM, a $\frac{2}{\sqrt{3}}$-approximation solution to the PERI-MAX, and has an aspect ratio at most max$\{\rho(R),3,1 + max_{i=1,...,n-1}\frac{a_i+1}{a_i}\}$, where ρ($\mathcal{R}$) denotes the aspect ratio of R.