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Approximating the Fréchet distance when only one curve is $c$-packed

Joachim Gudmundsson, Tiancheng Mai, Sampson Wong

Abstract

One approach to studying the Fréchet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is $c$-packedness. Existing algorithms for computing the Fréchet distance between $c$-packed curves require both curves to be $c$-packed. In this paper, we only require one of the two curves to be $c$-packed. Our result is a nearly-linear time algorithm that $(1+\varepsilon)$-approximates the Fréchet distance between a $c$-packed curve and a general curve in $\mathbb R^d$, for constant values of $\varepsilon$, $d$ and $c$.

Approximating the Fréchet distance when only one curve is $c$-packed

Abstract

One approach to studying the Fréchet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is -packedness. Existing algorithms for computing the Fréchet distance between -packed curves require both curves to be -packed. In this paper, we only require one of the two curves to be -packed. Our result is a nearly-linear time algorithm that -approximates the Fréchet distance between a -packed curve and a general curve in , for constant values of , and .
Paper Structure (14 sections, 6 theorems, 2 equations, 3 figures)

This paper contains 14 sections, 6 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Decision algorithm
  5. Simplification
  6. Fuzzy decider
  7. Constructing the vertices
  8. Constructing the edges
  9. Returning either (i) or (ii)
  10. Complete approximate decider
  11. Optimisation algorithm
  12. Approximating the optimal simplification
  13. Approximating the Fréchet distance
  14. Conclusion

Key Result

Lemma 5

Let $\delta > 0$ be fixed. Let $P$ be a $c$-packed curve with~$n$ vertices in $\mathbb R^d$. Let $K = \mathop{\mathrm{simpl}}\nolimits(P,\delta r)$. We can preprocess $K$ into a data structure of $O(n \log^d n + c \delta^{-1} n)$ size, so that given a query point $q \in \mathbb R^d$, the data struct

Figures (3)

  • Figure 1: A polygonal trajectory $P$ (blue) and its $\mu$-simplification (red dashed). The vertices marked with blue squares are on $P$ but not included in the simplification.
  • Figure 2: The general curve $Q$ (black), the $(\delta r)$-simplification $K$ (blue) and two candidate sets $W_i$ and $W_{i+1}$ (red dots). The coloured arrows indicate the order of vertices on the curve. A candidate set $W_i$ (red dots) contains evenly spaced points on $K$ chords that are at most a distance $2r$ away from $q_i$, i.e., in the violet shading. The point $w_{i,j}$ is on the edge $a_{i,j}b_{i,j}$ and the point $w_{i+1,k}$ is on the edge $a_{i+1,k}b_{i+1,k}$.
  • Figure 3: The Fréchet distance (purple), between a segment $(q_i, q_{i+1})$ (black) and subcurve $K \langle w_{i, j}, w_{i+1, k} \rangle$ (blue). A candidate set $W_i$ (red dots) contains evenly spaced points on $K$ chords that are at most a distance $2r$ away from $q_i$, i.e., in the green shading.

Theorems & Definitions (12)

  • proof
  • Definition 3
  • Lemma 5
  • proof
  • Theorem 7: Fuzzy decider
  • proof
  • Theorem 8: Complete approximate decider
  • proof
  • Corollary 10
  • Theorem 11
  • ...and 2 more