Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.
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Fast Fréchet Distance for Curves with Long Edges
Computing Fréchet distance between two curves takes roughly quadratic time. The only strongly subquadratic time algorithm has been proposed in [7] for c-packed curves. In this paper, we show that for curves with long edges the Fréchet distance computations become easier. Let P and Q be two polygonal curves in Rd with n and m vertices, respectively. We prove three main results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, and (3) a linear-time [EQUATION]-approximation algorithm for approximating the Fréchet distance between two curves.
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- Award ID(s):
- 1637576
- NSF-PAR ID:
- 10072476
- Date Published:
- Journal Name:
- 3rd International Workshop on Interactive and Spatial Computing
- Page Range / eLocation ID:
- 52-58
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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