# Packing Boundary-Anchored Rectangles and Squares

## TL;DR

This paper presents efficient algorithms for boundary-anchored packing problems, maximizing total area of interior-disjoint rectangles or squares anchored at boundary points of a square, with solutions depending on point arrangements.

## Contribution

It introduces linear-time and polynomial-time algorithms for boundary-anchored packing problems under specific point configurations.

## Key findings

- Linear-time algorithm for points sorted along the boundary.
- An $O(n^4)$-time algorithm for points on two opposite sides.
- Effective solutions for maximizing total area of anchored rectangles and squares.

## Abstract

Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each rectangle is anchored (has a corner at some point in $P$), each point in $P$ is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point $p$ in $P$ if one of its corners coincides with $p$. In this paper, we show how to solve this problem in time linear in $n$, provided that the points of $P$ are given in sorted order along the boundary of $Q$. We also consider the problem for anchoring squares and give an $O(n^4)$-time algorithm when the points in $P$ lie on two opposite sides of $Q$.