Computing The Packedness of Curves

TL;DR

This paper introduces new algorithms and data structures for approximating and exactly computing the packedness of polygonal curves, improving efficiency and accuracy over previous methods, with applications to disks and other shapes.

Contribution

It presents the first exact algorithm for computing curve packedness with disks, improves approximation algorithms, and develops a data structure for efficient curve-length queries inside disks.

Findings

Approximation factor improved to 4+ε with faster algorithms.

Exact algorithm for c-packedness of disks in O(n^5) time.

Data structure enables efficient curve-length queries inside disks.

Abstract

A polygonal curve $P$ with $n$ vertices is $c$-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius $r$ is at most $cr$, for any $r>0$. Similarly, the concept of $c$-packedness can be defined for any scaling of a given shape. Assuming $L$ is the diameter of $P$ and $\delta$ is the minimum distance between points on disjoint edges of $P$, we show the approximation factor of the existing $O(\frac{\log (L/\delta)}{\epsilon}n^3)$ time algorithm is $1+\epsilon$-approximation algorithm. The massively parallel versions of these algorithms run in $O(\log (L/\delta))$ rounds. We improve the existing $O((\frac{n}{\epsilon^3})^{\frac 4 3}\polylog \frac n \epsilon)$ time $(6+\epsilon)$-approximation algorithm by providing a $(4+\epsilon)$-approximation $O(n(\log^2 n)(\log^2 \frac{1}{\epsilon})+\frac{n}{\epsilon})$ time algorithm, and the existing $O(n^2)$ time $2$-approximation algorithm improving the existing $O(n^2\log n)$ time $2$-approximation algorithm. Our exact $c$-packedness algorithm takes $O(n^5)$ time, which is the first exact algorithm for disks. We show using $\alpha$-fat shapes instead of disks adds a factor $\alpha^2$ to the approximation. We also give a data-structure for computing the curve-length inside query disks. It has $O(n^6\log n)$ construction time, uses $O(n^6)$ space, and has query time $O(\log n+k)$, where $k$ is the number of intersected segments with the query shape. We also give a massively parallel algorithm for relative $c$-packedness with $O(1)$ rounds.