Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1409.2456

Intermittent Map Matching with the Discrete Fréchet Distance

Tim Wylie·Binhai Zhu
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 Fréchet distance matching the edges of the path as well. Here, we formally define the discrete map matching problem based on the discrete Fréchet 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.

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tim Wylie, Binhai Zhu. 2016-05-17. Intermittent Map Matching with the Discrete Fréchet Distance. https://arxiv.org/abs/1409.2456

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimal Trajectories in Discrete Space with Acceleration Constraints

In a recreational column of the Scientific American, Martin Gardner presented in 1973 a game called \textsc{Racetrack}, consisting of computing an optimal trajectory for a vehicle on a race circuit, subject to acceleration constraints in discrete space~$\mathbb{Z}^2$. In this model, each step consists of changing the position of the vehicle by a vector in $\mathbb{Z}^2$, with the constraints that two consecutive vectors differ by at most one unit in each dimension. We investigate two problems related to this model in arbitrary dimension in open space (no obstacles), where a \emph{configuration} of the vehicle consists of its current position and the last-used vector (concretely, a value in $\mathbb{Z}^d \times \mathbb{Z}^d$). The two problems are the following. In BRANCHING COST, two configurations are given and the goal is to compute the minimum number of moves (length of a trajectory) between the two configurations. BRANCHING TRAJECTORY has the same input and asks for a description of the trajectory. We obtain various results. First, we revisit known approaches solving BRANCHING COST in 2D, clarifying the analysis and showing that this problem can be solved in constant time in any fixed number of dimensions $d$ (more generally, in $O(d \log d)$ time). We also show that BRANCHING TRAJECTORY can also be solved in constant time for any fixed $d$, despite the fact that the length of the trajectory is not constant. The main ingredient is to show that there always exists \emph{at least one} optimal trajectory that can be compactly represented using only $O(1)$ intermediate configurations, with monotonic evolution between them. Among other uses, the latter implies that computing an optimal trajectory that visits a sequence of $n$ points at prescribed velocities in 2D or 3D can be done in linear time in the number of points.

cs.CG↗
arXiv2026

Prismatoid Band-Unfolding Revisited

It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.

cs.CG↗
arXiv2026

On (Directed) Width-Parameters of Geometric Spanners

To speed up algorithms on geometric graphs, it is common to approximate the complete Euclidean graph while maintaining certain geometric properties. A (directed) $t$-spanner $G$ for a point set $P$ in the Euclidean space is a (directed) graph such that for every pair of points, the shortest path in $G$ is at most a factor $t$ longer than the Euclidean distance between those points. In this paper, we investigate $t$-spanners that are bounded by certain graph parameters. Let $κ$ be a graph parameter. We show that for path-width, branch-width and cut-width there is an $\mathcal{O}(n/k^{d/(d-1)})$-spanner $G$ on $P$ with $κ(G)=k$ and that this is asymptotically worst-case optimal. In $\mathbb{R}^2$ we show the same bounds for planar graphs of clique-width or rank-width $k$. In contrast, for tree-depth, we show that there are sets of points for which the dilation cannot be bounded. Therefore, we investigate computing a spanner with tree-depth $k$ and minimum dilation. We show that already for tree-depth $3$ this problem is NP-hard to approximate within any factor strictly less than $\sqrt{2}$, and present an XP-algorithm to compute for a given tree-depth $k$ a graph with dilation at most $2t^*$, where $t^*$ is the minimum dilation. We further extend these results to obtain directed $\mathcal{O}(n/k^{d/(d-1)})$-spanners $G$ with $κ(G)=k$ for $κ$ being directed tree-width, directed path-width or DAG-width and show that also in the directed case, this is asymptotically worst-case optimal.

cs.CG↗