# On the union complexity of families of axis-parallel rectangles with a low packing number

## TL;DR

This paper establishes tight bounds on the union and level complexity of families of axis-parallel rectangles with low packing number, showing that complexity scales linearly with the number of rectangles and quadratically with the packing number.

## Contribution

It provides the first tight bounds on union and level complexity for such rectangle families with low packing number.

## Key findings

- Union complexity is at most O(n + p^2).
- Level complexity is at most O(kn + k^2 p^2).
- Both bounds are proven to be tight.

## Abstract

Let R be a family of n axis-parallel rectangles with packing number p-1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n+p^2), and that the (<=k)-level complexity of R is at most O(kn+k^2p^2). Both upper bounds are tight.