Computing the Similarity Between Moving Curves

TL;DR

This paper investigates the computational complexity of measuring similarity between moving curves, such as coastlines or glaciers, using variants of the Fréchet distance, and introduces a novel max-flow min-cut approach for polynomial-time solutions.

Contribution

It characterizes the complexity of surface similarity measures for moving curves and proposes a new max-flow min-cut based method for efficient computation.

Findings

Certain variants are polynomial-time solvable.

Other variants are NP-complete.

A novel max-flow min-cut approach is introduced.

Abstract

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.