# External Memory Planar Point Location with Fast Updates

## TL;DR

This paper introduces a new external memory data structure for dynamic planar point location that significantly improves update times while maintaining efficient query performance, especially for large datasets.

## Contribution

It presents a novel data structure achieving faster amortized update times in the external memory model for dynamic planar point location with constant face size.

## Key findings

- Update time reduced by a factor of B^{1-ε}
- Query time remains polylogarithmic in N
- Supports efficient vertical ray-shooting queries

## Abstract

We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with $O( \log_B^2 N)$ query time and $O(\frac{1}{ B^{1-\epsilon}} \log_B N)$ amortized update time, where $N$ is the number of segments, $B$ the block size and $\epsilon$ is a small positive constant, under the assumption that all faces have constant size. This is a $B^{1-\epsilon}$ factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of $N$ and $B$. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size.