Intermittent Map Matching with the Discrete Fréchet Distance

In this paper we focus on the map matching problem where the goal is to find a path through a planar graph such that the path through the vertices closely matches a given polygonal curve. The map matching problem is usually approached with the Fr\'echet distance matching the edges of the path as well. Here, we formally define the discrete map matching problem based on the discrete Fr\'echet distance. We then look at the complexities of some variations of the problem which allow for vertices in the graph to be unique or reused, and whether there is a bound on the length of the path or the number of vertices from the graph used in the path. We prove several of these problems to be NP-complete, and then conclude the paper with some open questions.

[1]  Wouter Meulemans.  Map Matching with Simplicity Constraints , 2013, ArXiv.

[2]  Leonidas J. Guibas,et al.  A faster algorithm for matching planar maps under the weak Fréchet distance , 2008 .

[3]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[4]  Tim Wylie,et al.  The discrete Frechet distance with applications , 2013 .

[5]  Haim Kaplan,et al.  Computing the Discrete Fréchet Distance in Subquadratic Time , 2012, SIAM J. Comput..

[6]  Michael Clausen,et al.  Approximately matching polygonal curves with respect to the Fre'chet distance , 2005, Comput. Geom..

[7]  Hamid Zarrabi-Zadeh,et al.  Staying Close to a Curve , 2011, CCCG.

[8]  Jun Luo,et al.  Fréchet-Distance on Road Networks , 2010, CGGA.

[9]  Helmut Alt,et al.  Computing the Fréchet distance between two polygonal curves , 1995, Int. J. Comput. Geom. Appl..

[10]  Chengyang Zhang,et al.  Map-matching for low-sampling-rate GPS trajectories , 2009, GIS.

[11]  Günter Rote,et al.  Matching planar maps , 2003, SODA '03.

[12]  Dieter Pfoser,et al.  Addressing the Need for Map-Matching Speed: Localizing Global Curve-Matching Algorithms , 2006, 18th International Conference on Scientific and Statistical Database Management (SSDBM'06).

[13]  H. Mannila,et al.  Computing Discrete Fréchet Distance ∗ , 1994 .

[14]  Dieter Pfoser,et al.  On Map-Matching Vehicle Tracking Data , 2005, VLDB.

[15]  Sariel Har-Peled,et al.  Approximating the Fréchet Distance for Realistic Curves in Near Linear Time , 2010, Discrete & Computational Geometry.

[16]  Binhai Zhu,et al.  Following a curve with the discrete Fréchet distance , 2014, Theor. Comput. Sci..

[17]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..

[18]  Kevin Buchin,et al.  Exact algorithms for partial curve matching via the Fréchet distance , 2009, SODA.

[19]  Leonidas J. Guibas,et al.  Approximate Map Matching with respect to the Fréchet Distance , 2011, ALENEX.