On the complexity of range searching among curves

TL;DR

This paper analyzes the computational complexity of range searching among polygonal curves using the Fréchet distance, establishing lower bounds and demonstrating the impact of input and query complexity on data structure efficiency.

Contribution

It provides the first comprehensive lower bounds for space and query time trade-offs in curve range searching, and introduces data structures based on semialgebraic range searching.

Findings

Lower bounds show exponential worsening with input/query complexity.

More complex queries increase logarithmic factors in bounds.

Data structures for Fréchet distance align with theoretical lower bounds.

Abstract

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given $n$ polygonal curves $S$ in $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $\rho$ for the curves of $S$ that have \Frechet distance at most $\rho$ to $q$. We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves $Q(n) + O(k)$ query time, where $k$ is the output size, has to use roughly $\Omega\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question in the range searching literature: whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively. On the positive side, we show we can build data structures for the \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet distance is in line with the lower bound, as the number of levels in the data structure is $O(t)$, where $t$ denotes the maximal number of vertices of a curve. For the continuous \Frechet distance, the number of levels increases to $O(t^2)$.