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Constant Approximation of Fréchet Distance in Strongly Subquadratic Time

Siu-Wing Cheng, Haoqiang Huang, Shuo Zhang

TL;DR

This work addresses the problem of approximating the Fréchet and discrete Fréchet distances between polygonal curves in fixed dimension in strongly subquadratic time. It introduces a randomized framework that partitions input curves into short subcurves, classifies reachability targets into four types, and propagates approximate reachability via curve simplification and WaveFront-based procedures to obtain a constant-factor $(7+\varepsilon)$-approximation. For both the continuous and discrete variants, the authors achieve subquadratic running time with high probability, representing the first such results for constant-factor approximation. The techniques rely on careful handling of Type-4 targets and leverage surrogate curves and data-structure-assisted coverage; these methods yield practical subquadratic algorithms with provable guarantees, and they open avenues for further deterministic improvements and tighter approximation factors.

Abstract

Let $τ$ and $σ$ be two polygonal curves in $\mathbb{R}^d$ for any fixed $d$. Suppose that $τ$ and $σ$ have $n$ and $m$ vertices, respectively, and $m\le n$. While conditional lower bounds prevent approximating the Fréchet distance between $τ$ and $σ$ within a factor of 3 in strongly subquadratic time, the current best approximation algorithm attains a ratio of $n^c$ in strongly subquadratic time, for some constant $c\in(0,1)$. We present a randomized algorithm with running time $O(nm^{0.99}\log(n/\varepsilon))$ that approximates the Fréchet distance within a factor of $7+\varepsilon$, with a success probability at least $1-1/n^6$. We also adapt our techniques to develop a randomized algorithm that approximates the \emph{discrete} Fréchet distance within a factor of $7+\varepsilon$ in strongly subquadratic time. They are the first algorithms to approximate the Fréchet distance and the discrete Fréchet distance within constant factors in strongly subquadratic time.

Constant Approximation of Fréchet Distance in Strongly Subquadratic Time

TL;DR

This work addresses the problem of approximating the Fréchet and discrete Fréchet distances between polygonal curves in fixed dimension in strongly subquadratic time. It introduces a randomized framework that partitions input curves into short subcurves, classifies reachability targets into four types, and propagates approximate reachability via curve simplification and WaveFront-based procedures to obtain a constant-factor -approximation. For both the continuous and discrete variants, the authors achieve subquadratic running time with high probability, representing the first such results for constant-factor approximation. The techniques rely on careful handling of Type-4 targets and leverage surrogate curves and data-structure-assisted coverage; these methods yield practical subquadratic algorithms with provable guarantees, and they open avenues for further deterministic improvements and tighter approximation factors.

Abstract

Let and be two polygonal curves in for any fixed . Suppose that and have and vertices, respectively, and . While conditional lower bounds prevent approximating the Fréchet distance between and within a factor of 3 in strongly subquadratic time, the current best approximation algorithm attains a ratio of in strongly subquadratic time, for some constant . We present a randomized algorithm with running time that approximates the Fréchet distance within a factor of , with a success probability at least . We also adapt our techniques to develop a randomized algorithm that approximates the \emph{discrete} Fréchet distance within a factor of in strongly subquadratic time. They are the first algorithms to approximate the Fréchet distance and the discrete Fréchet distance within constant factors in strongly subquadratic time.

Paper Structure

This paper contains 12 sections, 25 theorems, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of the decision procedure
  4. Technical overview of Reach
  5. Classification of target points
  6. Covering points of each type
  7. Subquadratic decision algorithm for Fréchet distance
  8. Preprocessing
  9. Implementation of Reach
  10. Subquadratic decision algorithm for discrete Fréchet distance
  11. Discussion and future works
  12. Data structure for answering the query Cover

Key Result

Lemma 1

Take $x_1, x_2, y_1, y_2\in \tau$ and $p, q\in \sigma$. Suppose that $x_1\le_\tau x_2\le_\tau y_2\le_\tau y_1$, and $p\le_\sigma q$. If $(y_1, q)$ is $r$-reachable from $(x_1, p)$, and $(y_2, q)$ is $r$-reachable from $(x_2, p)$, then $(y_2,q)$ is $r$-reachable from $(x_1,p)$ as well.

Figures (3)

  • Figure 1: The free space diagram with respect to $r$. The free space is the region in white. The pair $(y,q)$ is $r$-reachable from $(x,p)$, which implies $d_F(\tau[x,y], \sigma[p,q])\le r$.
  • Figure 2: Illustration of proof of Lemma \ref{['lem:planarity']}
  • Figure 3: Free space diagram induced by $\tau_k$ and $\sigma_l$ with respect to $\delta$. The input arrays $\mathcal{A}^{a_k}_{l}$ and $\mathcal{A}^{b_l}_{k}$ correspond to intervals on the left and bottom boundaries. The targeted output arrays correspond to intervals on the right and upper boundaries where we can reach from $\mathcal{A}^{a_k}_{l}$ and $\mathcal{A}^{b_l}_{k}$ via bi-monotone paths in the free space. These paths can be classified into four types depending on where they start and end.

Theorems & Definitions (39)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: SHJ
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • ...and 29 more