Distance Measures for Embedded Graphs

TL;DR

This paper introduces new geometric distance measures for embedded graphs using Fréchet and weak Fréchet distances, providing algorithms for their computation and analyzing their computational complexity.

Contribution

It defines novel graph distance measures based on continuous mappings and develops algorithms for their computation, including polynomial-time solutions for trees and planar graphs.

Findings

Deciding these distances is NP-hard for general embedded graphs.

Polynomial time algorithms exist for trees and planar embedded graphs with weak Fréchet distance.

Approximation algorithms are provided for the NP-hard cases.

Abstract

We introduce new distance measures for comparing straight-line embedded graphs based on the Fr\'echet distance and the weak Fr\'echet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fr\'echet distance if the graphs are planar embedded. Moreover, we prove that deciding the distances based on the Fr\'echet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.

Figures

Equations

$${\delta }_F(f,g)=\underset{\sigma :[0,1]\to [0,1]}{\inf }\underset{t\in [0,1]}{\max }||f(t)-g(\sigma (t))||,$$

$${\delta }_{wF}(f,g)=\underset{\alpha ,\beta :[0,1]\to [0,1]}{\inf }\underset{t\in [0,1]}{\max }||f(\alpha (t))-g(\beta (t))||,$$

$${\vec{\delta }}_G(G_1,G_2)=\underset{s:G_1\to G_2}{\inf }\underset{e\in E_1}{\max }{\delta }_F(e,s(e))$$

$${\vec{\delta }}_{wG}(G_1,G_2)=\underset{s:G_1\to G_2}{\inf }\underset{e\in E_1}{\max }{\delta }_{wF}(e,s(e)),$$

$${\delta }_G(G_1,G_2)=\max ({\vec{\delta }}_G(G_1,G_2),{\vec{\delta }}_G(G_2,G_1))\text{ and }$$

$${\delta }_{wG}(G_1,G_2)=\max ({\vec{\delta }}_{wG}(G_1,G_2),{\vec{\delta }}_{wG}(G_2,G_1)).$$

$${\delta }_T(G_1,G_2)=\underset{f,g}{\inf }\underset{t\in [0,1]}{\max }||f(t)-g(t)||,$$

$${\delta }_T(G_1,G_2)\le {\vec{\delta }}_{wG}(G_1,G_2)\le {\vec{\delta }}_G(G_1,G_2).$$

$${\delta }_{wF\ast }(P_1,P_2)\le {\delta }_{wG}(P_1,P_2)\le {\delta }_G(P_1,P_2)\le {\delta }_F(P_1,P_2),$$

$$\frac{F}{2}{m}_2^2+\frac{F}{4}{m}_2^2{m}_2^4+\cdots +m_2^2{\left(m_2^{2^{log(F)-2}}\right)}^2=F\sum _{i=1}^{\log (F)}\frac{1}{2^i}{m}_2^{2^{i+1}-2}$$

$$F\sum _{i=1}^{\log (F)}\frac{1}{2^i}{m}_2^{2^{i+1}-2}\le F\sum _{i=1}^{\log (F)}{m}_2^{2^{i+1}-2}\le F\sum _{i=1}^{2^{\log (F)+1}-2}{m}_2^i$$

$$=F\sum _{i=1}^{2F-2}{m}_2^i=F\frac{m_2^{2F-1}-m_2}{m_2-1}\in O\left(F{m}_2^{2F-1}\right),$$

$$\sum _{i=1}^n{a}^i=\frac{a(a^n-1)}{a-1},$$

Full text

1 Introduction

There are many applications that work with graphs that are embedded in Euclidean space. One task that arises in such applications is comparing two embedded graphs. For instance, the two graphs to be compared could be two different representations of a geographic network (e.g., roads or rivers). Oftentimes these networks are not isomorphic, nor is one interested in subgraph isomorphism, but one would like to have a mapping of one graph to the other, and ideally such a mapping would be continuous. For instance, this occurs when we have a ground truth of a road network and a simplification or reconstruction of the same network and we would like to measure the error of the latter. In this case, a mapping would identify the parts of the ground truth that are reconstructed/simplified and would allow to study the local error.

We present new graph distance measures that are well-suited for comparing such graphs. Our distance measures are natural generalizations of the Fréchet distance [9] to graphs and require a continuous mapping, but they don’t require graphs to be homeomorphic. One graph is mapped continuously to a portion of the other, in such a way that edges are mapped to paths in the other graph. The graph distance is then defined as the maximum of the strong (or weak) Fréchet distances between the edges and the paths they are mapped to. This results in a directed or asymmetric notion of distance, and we define the corresponding undirected distances as the maximum of both directed distances. The directed distances naturally arise when seeking to measure subgraph similarity, which requires mapping one graph to a subgraph of the other.

For comparing two not necessarily isomorphic graphs only few measures were known previously. One such measure is the traversal distance suggested by Alt et al. [8] and another is the geometric edit distance suggested by Cheong et al. [15]. The traversal distance converts graphs into curves by traversing the graphs continuously and comparing the resulting curves using the Fréchet distance. It is also a directed distance that compares the traversal of one graph with the traversal of a part of the other graph. However, an explicit mapping between the two graphs is not established, and part of the connectivity information of the graphs is lost due to the conversion to curves. The geometric edit distance minimizes the cost of edit operations from one graph to another, where cost is measured by Euclidean lengths and distances. But again, connectivity is not well maintained. Figure 1 shows some examples of graphs where our graph distances, the traversal distance, and the geometric edit distance differ. In particular, only our graph distances capture the difference in connectivity between graphs $G_{1}$ and $G_{2}$, as well as between $H_{1}$ and $H_{2}$.

Our graph distances map one graph onto a subgraph of the other and they measure the Fréchet distance between the mapped parts (see Section 2.1 for a formal definition). Hence connectivity information is preserved and an explicit mapping between the two (sub-)graphs is established. One possible application of these new graph distances is the comparison of geographic networks, for instance evaluating the quality of map reconstructions and map simplification. In Section 5, we show some experimental results on graphs of map reconstructions that illustrate that our approach considers both, geometry and connectivity.

Related work

A few approaches have been proposed in the literature for comparing geometric embedded graphs. Subgraph-isomorphism considers only the combinatorial structure of the graphs and not its geometric embedding. It is NP-hard to compute in general, although it can be computed in linear time if both graphs are planar and the pattern graph has constant size [18]. If we consider the graphs as metric spaces, the Gromov-Hausdorff distance (GH) between two graphs is the minimum Hausdorff distance between isometric embeddings of the graphs into a common metric space. While it is unknown how to compute GH for general graphs, recently Agarwal et al. [1] gave a polynomial time approximation algorithm for the GH between a pair of metric trees. We are however interested in measuring the similarity between two specific embeddings of the graphs. Armiti et al. [10] suggest a probabilistic approach for comparing graphs that are not required to be isomorphic, using spatial properties of the vertices and their neighbors. However, they require vertices to be matched to vertices, which can result in a large graph distance when an edge in one graph is subdived in the other graph. Furthermore, the spatial properties used are invariant to translation and rotation, whereas we consider a fixed embedding. Cheong et al. [15] proposed the geometric edit distance for comparing embedded graphs, however it is NP-hard to compute. Alt et al. [8] defined the traversal distance, which is most similar to our graph distance measures, but it does not preserve connectivity. See Section 2.2 for a detailed comparison with the traversal distance.

For assessing the quality of map construction algorithms, several approaches have been proposed. One approach is to compare all paths [2] or random samples of shortest paths [19]. However, these measures ignore the local structure of the graphs. In order to capture more topological information, Biagioni and Eriksson developed a sampling-based distance [11] and Ahmed et al. introduced the local persistent homology distance [3]. The latter distance measure focuses on comparing the topology and does not encode geometric distances between the graphs. The sampling-based distance is not a formally defined distance measure, and it crucially depends on parameters (in particular $matched\_distance$, to decide if points are sufficiently close to be matched); in practice it is unclear how these parameters should be chosen. However, it captures the number of matched edges, which is useful when comparing reconstructed road networks. In contrast to these measures, our graph distances capture more topology than the path-based distance [2], and capture differences in geometry better than the local persistent homology distance [3]. Also our graph distances are well-defined distance measures that do not require specific parameters to be set, unlike [11].

Contributions

We present new graph distance measures that compare graphs based on their geometric embeddings while respecting their combinatorial structure. To the best of our knowledge, our graph distances are the first to establish a continuous mapping between the embedded graphs. In Section 2 we define several variants of our graph distances (weak, strong, directed, undirected) and study their properties. In Section 3 we develop an algorithmic approach for computing the graph distances. On the one hand, we prove that for general embedded graphs, deciding these distances is NP-hard. On the other hand, we also show that our algorithmic approach gives polynomial time algorithms in several cases, e.g., when one graph is a tree. The most interesting case is when both graphs are plane. Here, we show that our algorithmic approach yields a quadratic time algorithm for the weak Fréchet distance. In Section 4 we focus on plane graphs and the strong Fréchet distance. For this case, we show that the problem is NP-hard, even though it is polynomial time solvable for the weak Fréchet distance. Furthermore, we show how to obtain an approximation, that depends on the angle between incident edges, in polynomial time and an exact result in exponential time.

2 Graph Distance Definition and Properties

Let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two undirected graphs with vertices embedded as points in $\mathbb{R}^{d}$ (typically $\mathbb{R}^{2}$) that are connected by straight-line edges. We refer to such graphs as (straight-line) embedded graphs. Generally, we do not require the graphs to be planar. We denote a crossing free embedding of a planar graph shortly as a plane graph. Note that for plane graphs $G_{1}$ and $G_{2}$, crossings between edges of $G_{1}$ and edges of $G_{2}$ are still allowed.

2.1 Strong and Weak Graph Distance

We define distance measures between embedded graphs that are based on mapping one graph to the other. We consider a particular type of graph mappings, as defined below:

Definition 1** (Graph Mapping).**

We call a mapping $s\colon G_{1}\rightarrow G_{2}$ a graph mapping if

it maps each vertex $v\in V_{1}$ to a point $s(v)$ on an edge of $G_{2}$, and 2. 2.

it maps each edge $\{u,v\}\in E_{1}$ to a simple path from $s(u)$ to $s(v)$ in the embedding of $G_{2}$.

Note that a graph mapping results in a continuous map if we consider the graphs as topological spaces. To measure similarity between edges and mapped paths, our graph distances use the Fréchet distance or the weak Fréchet distance, which are popular distance measures for curves [9]. For two curves $f,g\colon[0,1]\rightarrow\mathbb{R}^{d}$ their Fréchet distance is defined as {ceqn}

[TABLE]

where $\sigma$ ranges over orientation preserving homeomorphisms. The weak Fréchet distance is {ceqn}

[TABLE]

where $\alpha,\beta$ range over all continuous onto functions that keep the endpoints fixed.

Typically, the Fréchet distance is illustrated by a man walking his dog. Here, the Fréchet distance equals the shortest length of a leash that allows the man and the dog to walk on their curves from beginning to end. For the weak Fréchet distance man and dog may walk backwards on their curves, for the Fréchet distance they may not. The Fréchet distance and weak Fréchet distance between two polygonal curves of complexity $n$ can be computed in $O(n^{2}\log n)$ time [9]. Now, we are ready to define our graph distance measures.

Definition 2** (Graph Distances).**

We define the directed (strong) graph distance $\vec{\delta}_{G}$ as {ceqn}

[TABLE]

and the directed weak graph distance $\vec{\delta}_{wG}$ as {ceqn}

[TABLE]

where $s$ ranges over graph mappings from $G_{1}$ to $G_{2}$, and $e$ and its image $s(e)$ are interpreted as curves in the plane. The undirected graph distances are

[TABLE]

[TABLE]

According to Definition 1, a graph mapping $s$ maps each edge of $G_{1}$ to a simple path $s(e)$ in $G_{2}$. This is justified by the following observation: Mapping $e$ to a non-simple path $s^{\prime}(e)$, where $s(e)$ and $s^{\prime}(e)$ have the same endpoints and $s(e)\subset s^{\prime}(e)$, does not decrease the (weak) graph distance because $\delta_{(w)F}(e,s(e))\leq\delta_{(w)F}(e,s^{\prime}(e))$. From this observation also follows that we cannot decrease $\vec{\delta}_{G}(G_{1},G_{2})$ by adding additional vertices to subdivide an edge $e$ of $G_{1}$: While the concatenation of the resulting mapped paths in $G_{2}$ may not be simple, it can be replaced by the image of the entire edge $e$, which by the observation has to be simple.

We state a first important property of the graph distances:

Lemma 1**.**

For embedded graphs, the strong graph distances and the weak graph distances fulfill the triangle inequality. The undirected distances are pseudo-metrics. For plane graphs they are metrics.

Proof.

Symmetry follows immediately for the undirected distances. The directed distances fulfill the triangle inequality because we can concatenate two maps and use the triangle inequality of $\R^{d}$: Let $G_{1}$, $G_{2}$ and $G_{3}$ be three embedded graphs. An edge $e$ of $G_{1}$ is mapped to a simple path $p$ in $G_{2}$. The segments of $p$ are again mapped to a sequence of simple paths in $G_{3}$. Thus, when concatenating two maps, one possible mapping maps each edge $e$ of $G_{1}$ to a sequence $S$ of simple paths in $G_{3}$. Note, that $S$ need not be simple. However, in that case we can instead map $e$ to a shortest path $\hat{p}$ in $S$ from beginning to end. As $\delta_{(w)F}(e,\hat{p})\leq\delta_{(w)F}(e,S)$ for each edge of $G_{1}$, we have $\vec{\delta}_{G}(G_{1},G_{2})+\vec{\delta}_{G}(G_{2},G_{3})\geq\vec{\delta}_{G}(G_{1},G_{3})$ and $\vec{\delta}_{wG}(G_{1},G_{2})+\vec{\delta}_{wG}(G_{2},G_{3})\geq\vec{\delta}_{G}(G_{1},G_{3})$ by definition of the directed (weak) graph distance as the maximum Fréchet distance of an edge and its mapping. Analogously, the undirected distances fulfill the triangle inequality as well.

For plane graphs, their (weak) graph distance is zero iff their embeddings are the same, hence the distances are metrics. If the (weak) graph distance is zero, every edge needs to be mapped to itself, hence the embeddings are the same. If on the other hand, the embeddings are the same, a graph mapping may map every edge to itself in the embedding. Since there are no intersections or overlapping vertices, this mapping is continuous in the target graph, and the distance is zero. ∎

Note that for non-plane graphs the (weak) graph distance does not fulfill the identity of indiscernibles. For example, if $G_{1}$ consists of two crossing line segment edges, and $G_{2}$ has visually the same embedding but consists of four edges and includes the intersection point as a vertex, then both, $\vec{\delta}_{G}(G_{1},G_{2})=\vec{\delta}_{wG}(G_{1},G_{2})=0$ and $\vec{\delta}_{G}(G_{2},G_{1})=\vec{\delta}_{wG}(G_{2},G_{1})=0$, and therefore $\delta_{G}(G_{1},G_{2})=\delta_{wG}(G_{1},G_{2})=0$. Also note that we do not require graph mappings to be injective or surjective. And an optimal graph mapping from $G_{1}$ to $G_{2}$ may be very different from an optimal graph mapping from $G_{2}$ to $G_{1}$. See Figure 2 for examples of graphs and their graph distances.

In the following, we show that the traversal distance between a graph $G_{1}$ and a graph $G_{2}$ is a lower bound for $\vec{\delta}_{wG}(G_{1},G_{2})$, which follows from the observation that the traversal distance captures the combinatorial structure of the graphs to a lesser extent than our graph distances. Furthermore, we apply the graph distances to measure the similarity between two polygonal paths to examine how these new definitions are generalizations of the (weak) Fréchet distance for curves to graphs.

2.2 Relation to Traversal Distance

A related distance measure for graphs was proposed by Alt et. al. [8]. They define the traversal distance of two connected embedded graphs $G_{1},G_{2}$ as {ceqn}

[TABLE]

where $f$ ranges over all traversals of $G_{1}$ and $g$ over all partial traversals of $G_{2}$. A traversal of $G_{1}$ is a continuous, surjective map $f\colon[0,1]\rightarrow G_{1}$, and a partial traversal of $G_{2}$ is a continuous map $g\colon[0,1]\rightarrow G_{2}$.

Thus, graphs $G_{1},G_{2}$ have small traversal distance if there is a traversal of $G_{1}$ and a partial traversal of $G_{2}$ that stay close together. This could also be used for comparing a graph $G_{1}$ to a larger graph $G_{2}$. However, as we observe below, the traversals do not require to maintain the combinatorial structure of $G_{1}$ within $G_{2}$. First, we observe that our distance measures are stronger distances in the sense that {ceqn}

[TABLE]

This follows because a graph mapping that realizes $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ maps any traversal of $G_{1}$ to a partial traversal of $G_{2}$ with distance at most $\varepsilon$. For the weak graph distance, the traversal might need to be adjusted, so that it moves back along an already traversed path where the weak Fréchet matching requires it. Note that a traversal need not be injective.

However, the traversal distance captures the combinatorial structure of the graphs to a lesser extent than our measures. Figure 2 (c) shows two graphs that have large graph distance (in particular the directed distance from $G_{1}$ to $G_{2}$ is large) but small traversal distance. If indeed $G_{1}$ is a map reconstruction and $G_{2}$ the ground truth we are comparing to, then the distance from $G_{1}$ to $G_{2}$ should be large.

2.3 Graph Distance for Paths

Consider the simple case that the graphs are paths embedded as polygonal curves. In this case, the (weak) graph distance is closely related to the (weak) Fréchet distance. If the graphs are single edges embedded as polygonal curves, the graph and curve distances are in fact equal except for orientation of the curves. If the graphs are paths, such that each edge is embedded as a straight segment, the curve and graph distances are related, but not identical, as we show next.

A graph mapping between two paths maps vertices from one path to points on the other path, and it maps edges to the corresponding subpaths. In this case, we can characterize the graph distance in the free space [9], the geometric structure used for computing the Fréchet distance. Recall that for curves $f,g\colon[0,1]\rightarrow\mathbb{R}^{d}$ the free space is defined as $F_{\varepsilon}(f,g)=\{(s,t)\,|\,d(f(s),g(t))\leq\varepsilon\}$, i.e., the subset of the product of parameter spaces such that the corresponding points in the image space have distance at most $\varepsilon$.

Observation 1**.**

Let $P_{1},P_{2}$ be two polygonal curves parameterized over $[0,m]$ and $[0,n]$, respectively. A graph mapping realizing $\vec{\delta}_{G}(P_{1},P_{2})\leq\varepsilon$ can be characterized as an $x$-monotone path in $[0,m]\times[0,n]$ from the left boundary to the right boundary that is $y$-monotone (either increasing or decreasing) in each column of the free space. A graph mapping realizing $\vec{\delta}_{wG}(P_{1},P_{2})\leq\varepsilon$ is characterized by a path in $[0,m]\times[0,n]$ from the left boundary to the right boundary that is vertex-$x$-monotone, i.e., it is monotone in the traversal of the vertices on the $x$-axis.

This observation implies relationships between graph distance and (weak) Fréchet distance that are summarized in Lemma 2. In this lemma we use the non-standard variant of (weak) Fréchet distance that does not require the homeomorphism to be orientation preserving, but allows to choose an orientation. 444For the weak Fréchet distance one can drop the requirement that the reparameterizations $\alpha,\beta$ keep the endpoints fixed, also called boundary restriction [13]. Our graph distance does this naturally by choosing where to map. This variant of the Fréchet distance for curves can be computed by running the standard algorithm twice, i.e. searching for a path from bottom-left to top-right corner, as well as from top-left to bottom-right corner in the free space. Alternatively, we could enforce an orientation for the graph distance, e.g., using directions on the graphs.

Lemma 2**.**

Let $P_{1},P_{2}$ be paths embedded as polygonal curves. Then {ceqn}

[TABLE]

where $\delta_{wF*}$ denotes the weak Fréchet distance without boundary restriction.

If $P_{1},P_{2}$ are single edges embedded as polygonal paths, equality holds for both the weak and the strong distances.

Proof.

The last inequality holds because a path in the free space realizing the Fréchet distance is a monotone path in $x$ and $y$ from the lower left to upper right corner, hence also realizes both undirected graph distances. For the first inequality, observe that two paths in the free space realizing the two directed weak graph distances can be combined to a path from the left to the right boundary realizing the weak Fréchet distance without boundary restriction.

If $P_{1},P_{2}$ are single edges embedded as polygonal paths, this is essentially the same as a parameterization, hence both distances are equal (now with boundary restriction). ∎

Note that if we require mapping endpoints to themselves then the weak graph distance is also lower bounded by the weak Fréchet distance with boundary restriction. A 1D-example of two paths where $\delta_{wF}$ is strictly smaller than $\delta_{wG}$ when enforcing to map endpoints is the following: $P_{1}=(0,2,0,2)$ and $P_{2}=(0,1,2)$. Here, $\delta_{wF}(P_{1},P_{2})=0<1=\delta_{wG}(P_{1},P_{2})$.

Intuitively, and confirmed by the above lemma, our graph distance measures are at least as hard to compute as the Fréchet distance. It is known that the Fréchet distance cannot be computed in less than subquadratic time unless the strong exponential time hypothesis fail [12]. Hence we do not expect to compute our graph distance measures more efficiently than quadratic time.

3 Algorithms and Hardness for Embedded Graphs

Throughout this paper, let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two straight-line embedded graphs, and let $n_{1}=|V_{1}|$, $m_{1}=|E_{1}|$, $n_{2}=|V_{2}|$ and $m_{2}=|E_{2}|$.

First, we consider the decision variants for the different graph distances defined in Definition 2. Given $G_{1}$ and $G_{2}$ and a value $\varepsilon>0$, the decision problem for the graph distances is to determine whether $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ (resp., $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$). Equivalently, this amounts to determining whether there exists a graph mapping from $G_{1}$ to $G_{2}$ realizing $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ (resp., $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$). Note that the undirected distances can be decided by answering two directed distance decision problems. As we show in Section 3.3, the value of $\varepsilon$ can be optimized by parametric search.

In Section 3.1 we describe a general algorithmic approach for solving the decision problems by computing valid $\varepsilon$-placements for vertices. We show that for general embedded graphs the decision problems for the strong and weak directed graph distances are NP-hard, see Section 3.2. However, we prove in Section 3.3 that our algorithmic approach yields polynomial-time algorithms for the strong graph distance if $G_{1}$ is a tree, and for the weak graph distance if $G_{1}$ is a tree or if both are plane graphs. In the latter scenario ($G_{1}$ and $G_{2}$ plane graphs), deciding if $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ remains NP-hard, see Section 4.1.

3.1 Algorithmic Approach

Recall, that a (directional) graph mapping that realizes a given distance $\varepsilon$ maps each vertex of $G_{1}$ to a point in $G_{2}$ and each edge of $G_{1}$ to a simple path in $G_{2}$ within this distance. In order to determine whether such a graph mapping exists, we define the notion of $\varepsilon$-placements of vertices and edges; see Figures 3 and 4 (a).

Definition 3** ($\varepsilon$-Placement).**

An $\varepsilon$-placement of a vertex $v$ is a maximally connected part of $G_{2}$ restricted to the $\varepsilon$-ball $B_{\varepsilon}(v)$ around $v$. An $\varepsilon$-placement of an edge $e=\{u,v\}\in E_{1}$ is a path $P$ in $G_{2}$ connecting placements of $u$ and $v$ such that $\delta_{F}(e,P)\leq\varepsilon$. In that case, we say that $C_{u}$ and $C_{v}$ are reachable from each other. An $\varepsilon$-placement of $G_{1}$ is a graph mapping $s\colon G_{1}\rightarrow G_{2}$ such that $s$ maps each edge $e$ of $G_{1}$ to an $\varepsilon$-placement.

A weak $\varepsilon$-placement of an edge $e=\{u,v\}$ is a path $P$ in $G_{2}$ connecting placements of $u$ and $v$ such that $\delta_{wF}(e,P)\leq\varepsilon$. A weak $\varepsilon$-placement of $G_{1}$ is a graph mapping $s\colon G_{1}\rightarrow G_{2}$ such that $s$ maps each edge $e$ of $G_{1}$ to a weak $\varepsilon$-placement.

Note that an $\varepsilon$-placement of a vertex $v$ consists of edges and portions of edges of $G_{2}$, depending whether $B_{\varepsilon}(v)$ contains both, one or zero endpoint(s) of the edge, see Figure 4. Also note that each vertex has $O(m_{2})$ $\varepsilon$-placements, since an $\varepsilon$-placement is defined as a connected part of $G_{2}$ of maximal size inside $B_{\varepsilon}(v)$. Furthermore, we consider two graph mappings $s_{1}$ and $s_{2}$ from $G_{1}$ to $G_{2}$ to be equivalent in terms of the directed (weak) graph distance if for each vertex $v\in V_{1}$, $s_{1}(v)$ and $s_{2}(v)$ are points on the same $\varepsilon$-placement of $v$.

General Decision Algorithm.

Our algorithm consists of the following four steps, which we describe in more detail below. We assume $\varepsilon$ is fixed and use the term placement for an $\varepsilon$-placement.

Observe that each connected component of $G_{1}$ needs to be mapped to a connected component of $G_{2}$, and each connected component of $G_{1}$ can be mapped independently of the other components of $G_{1}$. Hence we can first determine the connected components of both graphs, and then consider mappings between connected components only. In the following we present an algorithm for determining if a mapping from $G_{1}$ to $G_{2}$, that realizes a given distance $\varepsilon$, exists, where both $G_{1}$ and $G_{2}$ are connected graphs.

1. Compute vertex placements.

We iterate over all vertices $v\in V_{1}$ and compute all their placements. Each vertex has $O(m_{2})$ placements, so the total number of vertex-placements is $O(n_{1}\cdot m_{2})$, and they can be computed in $O(n_{1}\cdot m_{2})$ time using standard algorithms for computing connected components.

2. Compute reachability information of vertex placements.

Next, we iterate over all edges $e=\{u,v\}\in E_{1}$ to determine all placements of its vertices that allow a placement of the edge. That is, we search for all pairs of vertex-placements $C_{u},C_{v}$ that are reachable from each other according to Definition 3.

For the weak graph distance, we need to find all pairs of placements of $u$ and placements of $v$ that can reach one another using paths contained in the $\varepsilon$-tube $T_{\varepsilon}(e)$ around $e$, i.e., the set of all points with distance $\leq\varepsilon$ to a point on $e$, see Figure 3 (c). If we restrict $G_{2}$ to its intersection with the $\varepsilon$-tube, all placements in the same connected component are mutually reachable. Thus, each edge is processed in time linear in the size of $G_{2}$ using linear space per edge: For each connected component a pair of lists containing the placements of $u$ and $v$ in that component, respectively, is computed. So, all reachability information can be computed in $O(m_{1}\cdot m_{2})$ time and space. Note that the weak Fréchet distance between a straight line edge $e\in E_{1}$ and a simple path $s(e)$ in $G_{2}$ is the maximum of the Hausdorff distance between $e$ and $s(e)$ and the distances of the endpoints of $e$ and $s(e)$.

For the strong graph distance, existence of a path inside the $\varepsilon$-tube is not sufficient to describe the connectivity between placements. We must ensure that the Fréchet distance between $e$ and $P$ is at most $\varepsilon$, i.e., a continuous and monotone map $s$ must exist from $e$ to $P$ such that $\delta_{F}(t,s(t))\leq\varepsilon$ for all $t\in e$. This can be decided in $O(|P|)$ time using the original dynamic programming algorithm for computing the Fréchet distance [9]. In order to determine whether such a path $P$ exists, every placement of $u$ stores a list of all placements of $v$ that are reachable. The connectivity information can be computed by running a graph exploration, starting from each placement, which prunes a branch if the search leaves the $\varepsilon$-tube or backtracking on $e$ is required to map it. This method runs a search for every placement of the start vertex and thus needs $O(m_{2}^{2})$ time per edge of $G_{1}$. Since the connectivity is explicitly stored as pairs of placements that are mutually reachable, it also needs $O(m_{2}^{2})$ space per edge. Hence, in total over all edges, $O(m_{1}\cdot m_{2}^{2})$ time and space are needed. Summing up, we have:

Lemma 3**.**

To run step 1 and step 2 of Algorithm 1, we need $O(m_{1}\cdot m_{2})$ time and space for the weak graph distance and $O(m_{1}\cdot m_{2}^{2})$ time and space for the strong graph distance.

3. Prune invalid placements.

Now, after having processed all vertices and edges, it still needs to be decided whether $G_{1}$ as a whole can be mapped to $G_{2}$. To this end, we delete invalid placements of vertices.

Definition 4** (Valid Placement).**

An $\varepsilon$-placement $C_{v}$ of a vertex $v$ is (weakly) valid if for every neighbor $u$ of $v$ there exists an $\varepsilon$-placement $C_{u}$ of $u$ such that $C_{v}$ and $C_{u}$ are connected by a (weak) $\varepsilon$-placement of the edge $\{u,v\}$. Otherwise, $C_{v}$ is (weakly) invalid.

See Figure 4 for an illustration of (in)valid placements. As shown in the Figure, deleting an invalid placement possibly sets former valid placements to be invalid. Thus, we need to process all placements recursively until all invalid placements are deleted and no new invalid placements occur. Note that the ordering of processing the placements does not affect the final result. To decide which placements of vertices $u$ and $v$ incident to an edge $e$ are valid, we use the reachability information computed in Step 2.

Initially there are $O(n_{1}\cdot m_{2})$ vertex-placements, each of which may be deleted once. For the weak graph distance, connectivity is stored using connected components inside the $\varepsilon$-tube surrounding an edge $\{u,v\}$. On deleting a placement $C_{v}$ of $v$, it is removed from the list containing placements of $v$. If a component no longer contains placements of $v$ (i.e. its list becomes empty), then all placements of $u$ in that component become invalid. A placement $C_{v}$ is deleted at most once and upon deletion it must be removed from one list for every edge incident to $v$. Thus, the time for pruning $C_{v}$ is $O(\deg(v))$. Since the sum of all degrees is $2m_{1}$, all invalid placements can be pruned in $O(m_{1}\cdot m_{2})$ time. For the strong graph distance, every placement has a list of placements to which it is connected. On deleting $C_{v}$, it must be removed from the lists of all placements $C_{u}$ to which $C_{v}$ is connected. Each vertex has $O(m_{2})$ placements which have to be removed from a list for each neighbor of $v$. Thus, pruning a placement runs in $O(\deg(v)\cdot m_{2})$ time and pruning all invalid placements in $O(m_{1}\cdot m_{2}^{2})$ time.

Lemma 4**.**

Pruning all invalid placements takes $O(m_{1}\cdot m_{2})$ time for the weak graph distance and $O(m_{1}\cdot m_{2}^{2})$ time for the strong graph distance.

Note that after the pruning step all remaining vertex placements are (weakly) valid. However, the existence of a (weakly) valid placement for each vertex is not a sufficient criterion for $\vec{\delta}_{G}(G_{1},G_{2})$ ($\vec{\delta}_{wG}(G_{1},G_{2})$) in general, see Figure 7.

4. Decide if there exists a placement for the whole graph $G_{1}$.

After pruning all invalid placements, we want to decide if the remaining valid vertex-placements allow a placement of the whole graph $G_{1}$. The complexity of this step depends on the graph and the distance measure: for plane graphs we show that we can concatenate weakly valid placements of two adjacent faces (Lemma 7), whereas this is not possible for the directed strong graph distance in this setting (Theorem 5) or for general graphs for both distances (Theorem 2). Although deciding the directed (weak) graph distance is NP-hard for general graphs, there are two settings which may occur after running steps 1-3 of Algorithm 1, making step 4 of the algorithm trivial. Clearly $\vec{\delta}_{G}(G_{1},G_{2})>\varepsilon$ ($\vec{\delta}_{wG}(G_{1},G_{2})>\varepsilon$) if there is a vertex that has no (weakly) valid $\varepsilon$-placement. Furthermore, we have the following:

Lemma 5**.**

If, after running steps 1-3 of Algorithm 1, each internal vertex (degree at least two) has exactly one valid $\varepsilon$-placement (resp., weakly valid $\varepsilon$-placement) and each vertex of degree one has at least one valid $\varepsilon$-placement (resp., weakly valid $\varepsilon$-placement), then $G_{1}$ has an $\varepsilon$-placement (resp., weak $\varepsilon$-placement). Thus, $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ (resp., $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$).

Lemma 3, Lemma 4 and Lemma 5 imply the following Theorem.

Theorem 1**.**

If there is a vertex that has no valid $\varepsilon$-placement or if each vertex has exactly one valid $\varepsilon$-placement after running steps 1-3 of Algorithm 1, the directed strong graph distance can be decided in $O(m_{1}\cdot m_{2}^{2})$ time and space. Analogously, if there is a vertex that has no weakly valid $\varepsilon$-placement or if each vertex has exactly one weakly valid $\varepsilon$-placement after running steps 1-3 of Algorithm 1, the directed weak graph distance can be decided in $O(m_{1}\cdot m_{2})$ time and space.

Proof.

Map each internal vertex $v$ to a point $s(v)$ on of its unique (weakly) valid placement $C_{v}$. Consider an edge $e=\{u,v\}\in E_{1}$. In the previous step, at least one (weak) placement $P_{e}$ of $e$ was discovered that connects points $p_{0}$ and $p_{k}$ on $C_{u}$ and $C_{v}$, respectively, since otherwise $C_{u}$ and $C_{v}$ would be invalid. If $p_{k}\neq s(v)$, $P_{e}$ can be adapted by shortening it and/or concatenating a path on $C_{v}$ (i.e. inside $B_{\varepsilon}(v)$) without causing its (weak) Fréchet distance to $e$ to become $>\varepsilon$. Adapt $P_{e}$ to a path $P^{\prime}_{e}$ that has endpoints $p^{\prime}_{0}=s(u)$ and $p^{\prime}_{k}=s(v)$ and define $s(e)=P^{\prime}_{e}$. Now, $s$ is a graph mapping from $G_{1}$ to $G_{2}$ and each edge is mapped to a path with (weak) Fréchet distance at most $\varepsilon$, so $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ (or $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$). Recall, that each vertex $w$ of degree one is either connected to an internal vertex $i$, or $G_{1}$ consists of only one edge $\{w,x\}$. The first case is already covered, since the unique valid (weak) placement $C_{i}$ for $i$ is reachable from any valid (weak) placement of $w$. The latter case follows because every vertex placement for $w$ is valid, i.e., for each placement $C_{w}$ for $w$ there is a (weakly) reachable placement $C_{x}$ for $x$ and any combination of two reachable placements $C_{w}$ and $C_{x}$ yields a valid (weak) placement of $G_{1}$. ∎

3.2 NP-Hardness for the General Case

Notwithstanding the special cases in Theorem 1, deciding the (weak) graph distance is not tractable for general graphs.

Theorem 2**.**

Deciding whether $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ and deciding whether $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$ for two graphs $G_{1}$ and $G_{2}$ embedded in $\mathbb{R}^{2}$ is NP-hard.

Proof.

We show NP-hardness with a reduction from binary constraint satisfaction problem (CSP), which is defined as follows:

Problem 1**.**

\proc

*Binary Constraint Satisfaction Problem (CSP)

Instance: A set of variables $X=\{x_{1},\ldots,x_{n}\}$, variable domains

$D=\{D_{1},\ldots,D_{n}\}$ and a set of constraints $C=\{C_{1},\ldots,C_{m}\}$, where each constraint $C$ has two variables $x_{i}$, $x_{j}$ and a relation $R_{C}\subseteq D_{i}\times D_{j}$.

Question: Can each variable $x_{i}$ be assigned a value $d_{i}\in D_{i}$ such that for each constraint $C$ on variables $x_{i}$, $x_{j}$, their values $(d_{i},d_{j})$ satisfy $R_{C}$?*

Consider an instance $\langle X,D,C\rangle$. Set $\varepsilon=1$. Every variable $x_{i}$ is represented by a vertex $v_{i}$ in $G_{1}$ and for each constraint $C_{k}$ on variables $x_{i}$, $x_{j}$, $G_{1}$ has an edge $\{v_{i},v_{j}\}$ in $G_{1}$. We embed $G_{1}$ such that all adjacent $\varepsilon$-balls are separated by at least $2\varepsilon$ and no $\varepsilon$-tube of an edge overlaps an $\varepsilon$-ball that does not belong to one of the edge’s endpoints. This can for example be realized by placing all vertices on a sufficiently large circle.

Every value $d_{i,a}\in D_{i}$ is represented by a vertex $u_{i,a}$ in $G_{2}$ that is inside the $\varepsilon$-ball of $v_{i}$. For each pair of values $d_{i,a}\in D_{i}$, $d_{j,b}\in D_{j}$ allowed by a constraint on $x_{i}$ and $x_{j}$, $G_{2}$ has an edge $\{u_{i,a},u_{j,b}\}$. This way, every vertex $u_{i,a}$ of $G_{2}$ defines exactly one $\varepsilon$-placement of the corresponding vertex $v_{i}$ in $G_{1}$. Figure 5 illustrates the construction.

A solution to the CSP consists of selecting a value $d_{i}\in D_{i}$ for each variable $x_{i}$, such that if there is a constraint on $v_{i}$ and $v_{j}$, the pair $\{d_{i},d_{j}\}$ satisfies the constraint. This is equivalent to selecting a placement $d_{i}$ of each vertex $v_{i}$, such that if $G_{1}$ has an edge $\{v_{i},v_{j}\}$, then $G_{2}$ has an edge connecting $u_{i}$, $u_{j}$ representing $d_{i}$ and $d_{j}$, respectively. The graph distance problem has the weaker requirement that there exists a path $P_{i,j}$ between $u_{i}$ and $u_{j}$, such that $\delta_{F}(\{v_{i},v_{j}\},P_{i,j})\leq\varepsilon$. However, the construction is such that only paths consisting of a single edge are permitted for the strong distance, since $\varepsilon$-balls must be sufficiently separated and nonoverlapping with $\varepsilon$-tubes. So $G_{2}$ must have an edge $\{u_{i},u_{j}\}$ if $G_{1}$ has an edge $\{v_{i},v_{j}\}$. So, $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ if and only if $\langle X,D,C\rangle$ is a satisfiable binary CSP.

For the weak graph distance, edges in $G_{1}$ can be mapped to paths consisting of multiple edges in $G_{2}$. In this case, there may be weak placements of $G_{1}$ that do not represent a solution to the constraint satisfaction instance. To remedy this, we insert a vertex in the middle of each edge of $G_{1}$. The vertex is placed such that its $\varepsilon$-ball is separated from the $\varepsilon$-balls of the original endpoints of the edge by at least $2\varepsilon$, so each of the new edges is mapped to part of a single edge in $G_{2}$. In this construction $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$ if and only if $\langle X,D,C\rangle$ is a satisfiable binary CSP. ∎

3.3 Efficient Algorithms for Plane Graphs and Trees

Here, we show that that Algorithm 1 yields polynomial-time algorithms for deciding the strong graph distance if $G_{1}$ is a tree (Theorem 4), and the weak graph distance if $G_{1}$ is a tree or if both are plane graphs (Theorem 3). More precisely, we show that the existence of at least one (weakly) valid placement for each vertex is a sufficient condition for $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ or $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$.

Lemma 6**.**

If $G_{1}$ is a tree and every vertex of $G_{1}$ has at least one (weakly) valid $\varepsilon$-placement after running steps 1-3 of Algorithm 1, then $G_{1}$ has a (weak) $\varepsilon$-placement. Thus, $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ (or $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$).

Proof.

We view $G_{1}$ as a rooted tree, selecting an arbitrary vertex as the root. We map all vertices of $G_{1}$ from the root outwards. First, map the root to an arbitrary (weakly) valid placement. When processing a vertex $v$: Map $v$ to an arbitrary (weakly) valid placement that is reachable from the placement its parent $p$ is mapped to. Recall, that a (weak) $\varepsilon$-placement of a vertex is (weakly) valid if there is an $\varepsilon$-placement for every incident edge. Since $p$ was mapped to a (weakly) valid placement and there is an edge $\{p,v\}$ in $G_{1}$, there must be at least one such placement of $v$ by definition of a (weakly) valid placement. Since all edges in $G_{1}$ are tree edges, this ensures that every edge is mapped correctly, that is to a path with (weak) Fréchet distance at most $\varepsilon$. ∎

Lemma 7**.**

If $G_{1}$ and $G_{2}$ are plane graphs and every vertex of $G_{1}$ has at least one weakly valid $\varepsilon$-placement after running steps 1-3 of Algorithm 1, then $G_{1}$ has a weak $\varepsilon$-placement. Thus, $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$.

Proof.

A tree-substructure of $G_{1}$ is a tree $T=(V_{T},E_{T})$ induced by the vertex set $V_{T}\subset V_{1}$ with a root vertex $r\in V_{T}$, such that for all vertices $v\in V_{T}$, $v\neq r$, $v$ is not an endpoint of an edge $e\in E_{1}\backslash E_{T}$ and such that $T$ is maximal, in the sense that when adding one additional vertex, $T$ contains a cycle. We first remove all tree-substructures of $G_{1}$ and map these as in the proof of Lemma 6. Next, we consider all faces of the remainder of $G_{1}$ and show how to iteratively map them.

Consider a cycle $C$ bounding a face $F$ and let $e_{1}$ and $e_{2}$ be two edges of $C$ incident to a vertex $v$. Let $b$ be the line segment of the bisector of $e_{1}$ and $e_{2}$ inside $B_{\varepsilon}(v)$. We define the outermost placement of $v$ as the placement which intersects $b$ at maximum distance to the endpoint of $b$ inside $F$, see Figure 6 (a). Furthermore, we define an outermost path in $G_{2}$ of an edge $e=\{u,v\}$ of $G_{1}$ as the path $P_{out}$ with maximum distance to $F$ connecting the outermost placements of $u$ and $v$. That is, no subpath $Q$ of $P_{out}$ can be replaced by a path $R$ such that $\delta_{H}(R,B)\leq\delta_{H}(Q,B)$, where $\delta_{H}$ is the Hausdorff distance and $B$ is the boundary of the tube $T_{\varepsilon}(e)$ which lies inside the face $F$. Note that if an edge is shorter than $2\varepsilon$, and hence the $\varepsilon$-balls around the vertices overlap, then so possibly do the placements. In particular, in this case the outer placements may overlap, in which case the edge placements degenerate, see Figure 6 (a). Finally, we define an outer placement $O$ of $C$ in $G_{2}$ as the concatenation of all outermost paths of edges of $C$.

Note that if $C$ is sufficiently convex the outer placement is simply the cycle that bounds $H$. See Figure 6 (b) for an example, where the red outer placement bounds the outer face of $G_{2}$ restricted to red and pink vertices and edges. The outer placement of $C$ is a weak $\varepsilon$-placement of $C$.

Now, consider two cycles $C_{1}$ and $C_{2}$ bounding adjacent faces of $G_{1}$, which share a single (possibly degenerate) path $P$ between vertices $u$ and $v$. Let $O_{1}$ and $O_{2}$ be the outer placements of $C_{1}$ and $C_{2}$, respectively. By definition of an outermost placement, $O_{1}$ and $O_{2}$ must intersect inside the intersection of the $\varepsilon$-tubes of $C_{1}$ and $C_{2}$. Let $o_{1}$ and $o_{2}$ of $O_{1}$ and $O_{2}$ be the parts between the intersections of $O_{1}$ and $O_{2}$ containing the respective images of $P$. Again, by definition of an outermost placement, it holds that $o_{1}$ is completely inside $O_{2}$ and $o_{2}$ is completely inside $O_{1}$.

This is illustrated in Figure 6 (c). By planarity there must be a vertex at the intersections of $O_{1}$ and $O_{2}$. Thus, we can construct a mapping $O^{\prime}_{2}$ of $C_{2}$ that consists of $o_{1}$ and $O_{2}\setminus o_{2}$. This is a weak $\varepsilon$-placement of $C_{2}$ for which the image of the shared path $P$ is identical to its image in $O_{1}$. Thus, we can merge $O_{1}$ and $O^{\prime}_{2}$ to obtain a weak $\varepsilon$-placement of these two adjacent cycles. Note that the mapping of $C_{1}$ is not modified in this construction. Additionally, the image of the cycle bounding the outer face is its outer placement. The same argument can be applied iteratively when $C_{1}$ and $C_{2}$ share multiple paths.

If there are two cycles $C_{1}$ and $C_{2}$ which are connected by a path $P$ such that one endpoint $u$ of $P$ lies on $C_{1}$, the other endpoint $v$ of $P$ lies on $C_{2}$ and all other vertices of $P$ are no vertices of $C_{1}$ or $C_{2}$, we can still construct a common placement for $C_{1}$, $C_{2}$ and $P$: Let $C_{u}$, $C_{v}$ be the outermost placements of $u$ and $v$, respectively and let $D_{v}$ be a vaild placement of $v$ which is connected by a path $Q$ in $G_{2}$ to $C_{u}$ such that $\delta_{wF}(Q,P)\leq\varepsilon$. Such a placement $D_{v}$ must exist as $C_{u}$ is a valid placement. If $D_{v}$ = $C_{v}$ we have found a common valid placement for $C_{1}$, $C_{2}$ and $P$. If $D_{v}\neq C_{v}$, by definition of an outermost placement, the path $Q$ must intersect the outermost placement $O$ of $C_{2}$ inside the intersection of the tubes $T_{\varepsilon}(P)$ and $T_{\varepsilon}(C_{2})$. As $G_{2}$ is plane, there is a vertex $w$ at the intersection and the resulting path $R=Q_{C_{u}\rightarrow w}+O_{w\rightarrow C_{v}}$ with $\delta_{wF}(R,P)\leq\varepsilon$ connects $C_{u}$ and $C_{v}$.

Now, we iteratively map the cycles bounding faces of $G_{1}$ until $G_{1}$ is completely mapped. Let $\langle F_{1},F_{2},\ldots,F_{k}\rangle$ be an ordering of the faces of $G_{1}$ such that each $F_{i}$, for $i\geq 2$ is on the outer face of the subgraph $\G_{i-1}:=C_{1}\cup C_{2}\cup\ldots\cup C_{i-1}$ of $G_{1}$, where $C_{j}$ is the cycle bounding face $F_{j}$. Thus, let $F_{1}$ be an arbitrary face of $G_{1}$ and subsequently choose faces adjacent to what has already been mapped. Hence when adding a cycle $C_{i}$, we have already mapped $\G_{i-1}$ such that the cycle bounding its outer face is mapped to its outer placement. Thus, we can treat $\G_{i-1}$ as a cycle, ignoring the part of it inside this cycle, and merge its mapping with $C_{i}$ using the procedure described above. This leaves the mapping of $\G_{i-1}$ unchanged, hence this is still a weak $\varepsilon$-placement of $\G_{i-1}$. However, the mapping of $C_{i}$ is now modified to be identical to that of $\G_{i-1}$ in the parts where they overlap. Thus, we can merge these mappings to obtain a weak $\varepsilon$-placement of $\G_{i}$. After mapping $F_{k}$ we have completely mapped $G_{1}$. ∎

Lemma 3 and Lemma 4 together with Lemma 6 and Lemma 7 directly imply the following theorems. Note, that $m_{1}=O(n_{1})$ for plane graphs and trees, in particular.

Theorem 3** (Decision Algorithm for Weak Graph Distance).**

Let $\varepsilon>0$. If $G_{1}$ is a tree, or if $G_{1}$ and $G_{2}$ are plane graphs, then Algorithm 1 decides whether $\vec{\delta}_{wG}(G_{1},G_{2})\leq\varepsilon$ in $O(n_{1}\cdot m_{2})$ time and space.

Theorem 4** (Decision Algorithm for Graph Distance).**

Let $\varepsilon>0$. If $G_{1}$ is a tree, then Algorithm 1 decides whether $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ in $O(n_{1}\cdot m_{2}^{2})$ time and space.

Computing the Distance

To compute the graph distance, we proceed as for computing the Fréchet distance between two curves: We search over a set of critical values and employ the decision algorithm in each step. The following types of critical values can occur:

A new vertex-placement emerges: An edge in $G_{2}$ is at distance $\varepsilon$ from a vertex in $G_{1}$. 2. 2.

Two vertex-placements merge: The vertex in $G_{2}$ where they connect is at distance $\varepsilon$ from a vertex in $G_{1}$. 3. 3.

The (weak) Fréchet distance between a path and an edge is $\varepsilon$: these are described in [9]. There are exponentially many paths in $G_{2}$, but each value the Fréchet distance may attain is defined by either a vertex and an edge, or two vertices and an edge.

There are $O(n_{1}\cdot m_{2})$ critical values of the first two types, and $O(m_{1}\cdot n_{2}^{2})$ of type three. Parametric search can be used to find the distance as described in [9], using the decision algorithms from Theorems 3 and 4. This leads to a running time of $O(n_{1}\cdot m_{2}\cdot\log(n_{1}+n_{2}))$ for computing the weak graph distance if $G_{1}$ is a tree or both are plane graphs. And the total running time for computing the graph distance if $G_{1}$ is a tree is $O(n_{1}\cdot m_{2}^{2}\cdot\log(n_{1}+n_{2}))$.

4 Hardness Results and Algorithms for Plane Graphs

Lemma 7 does not hold for plane graphs and the directed strong graph distance because in general outer placements of cycles cannot be combined to a placement of $G_{1}$ as shown in the proof of Lemma 7. See Figure 7 for a counterexample. In fact we show that deciding the directed strong graph distance for plane graphs is NP-hard.

4.1 NP-Hardness for the Strong Distance for Plane Graphs

Theorem 5**.**

For plane graphs $G_{1}$ and $G_{2}$, deciding whether $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$ is NP-hard.

Proof.

We prove the NP-hardness by a reduction from Monotone-Planar-3-Sat. In this 3-Sat variant, the associated graph with edges between variables and clauses is planar and each clause contains only positive or only negative literals. The overall idea is to construct two graphs $G_{1}$ and $G_{2}$ based on a Monotone-Planar-3-Sat instance $A$, such that $A$ is satisfiable if and only if $\vec{\delta}_{G}(G_{1},G_{2})\leq\varepsilon$. That is, we construct subgraphs of $G_{1}$ and $G_{2}$, where some edges of $G_{2}$ are labeled True or False in a way, such that only certain combinations of True and False values can be realized by a placement of $G_{1}$ to $G_{2}$. To realize other combinations, backtracking, at least along one edge of $G_{1}$, is necessary. But this is not allowed for the strong graph distance. In the following, we describe the construction of the gadgets (subgraphs) for the variables and the clauses of a Monotone-Planar-3-Sat instance. Additionally, we need a gadget to split a variable if it is contained in several clauses and a gadget which connects the variable gadgets with the clause gadgets. Furthermore, we prove for each gadget which True and False combinations can be realized and which combinations are not possible. All constructed edges are straight line edges. The graph $G_{1}$ is shown in blue color and $G_{2}$ is shown in red color in the sketches used to illustrate the ideas of the proof. We denote $T_{\varepsilon}(e)$ as the $\varepsilon$-tube around the edge $e$. All vertices of the graph can be either mapped arbitrarily within a given $\varepsilon$-surrounding and with a given minimal distance from each other or must lie at the intersection of two lines. Thus, we can ensure that the construction uses rational coordinates only and can be computed in polynomial time.

Following a reparameterization of an edge or a path (embedded curve) as described in Definition 2, we denote by the term ”walking along an edge or a path”. In this sense backtracking means that the reparametization is not injective and thus, if backtracking is necessary to stay within $\varepsilon$-distance to another path or edge one can conclude that the Fréchet distance distance between these paths or edges is greater than $\varepsilon$ as in this case it is required that the reparameterization is injective.

Furthermore, we call a path labeled True (False) shortly a True (False) signal.

In the following, we give a detailed description of the construction and the properties of the gadgets. The Variable gadget: For each variable of $A$, we add a vertex $v$ and two edges, $e_{1}$ and $e_{2}$, of $G_{1}$ incident to a vertex $v$, where the angle between $e_{1}$ and $e_{2}$ is between $90^{\circ}$ and $120^{\circ}$. We add a vertex $w_{1}$ ($w_{2}$) of $G_{2}$ on the intersection of the outer boundaries of $T_{\varepsilon}(e_{2})$ ($T_{\varepsilon}(e_{1})$) and a line through $e_{1}$ ($e_{2}$). Furthermore, we add a vertex $w_{3}$ of $G_{2}$ at the intersection of the boundaries of $T_{\varepsilon}(e_{1})$ and $T_{\varepsilon}(e_{2})$. See the upper left sketch in Figure 8 for an illustration. We connect $w_{1}$ and $w_{2}$ with $w_{3}$ and draw an edge from $w_{1}$ and $w_{2}$ inside the $\varepsilon$-tubes around $e_{1}$ and $e_{2}$, with labels True. Analogously, we embed two edges from $w_{3}$ with the label False. For the Variable gadget a True-True combination is not possible: There are two placements $p_{1}$ and $p_{2}$ of the vertex $v$. Assume we choose $p_{1}$. Note that one can map $e_{1}$ to a path containing the edge of $G_{2}$ with the True labeling inside $T_{\varepsilon}(e_{1})$. Now, we want to map $e_{2}$ to a path $P$ starting at some point of $p_{1}$, where $P$ contains the edge of $G_{2}$ with the True labeling inside $T_{\varepsilon}(e_{2})$. In this case, one has to walk along $e_{2}$ up to $q$ (the point on $e_{2}$ with distance $\varepsilon$ to $w_{3}$) while walking simultaneously to $w_{3}$ on $P$. But then, when walking along $P$ up to $w_{1}$, one must walk back along $e_{2}$ up to $v$ as any point along the interior of $e_{1}$ has distance greater than $\varepsilon$ to $w_{1}$. Thus, $\delta_{F}(e_{2},P)>\varepsilon$. It is easy to see that for any other combination of labels there is a placement $p$ of $v$, such that $e_{1}$ and $e_{2}$ can be mapped to a path $P_{1}$ ($P_{2}$) starting at a point $a$ of $p$ with $\delta_{F}(e_{2},P_{1})\leq\varepsilon$ ($\delta_{F}(e_{2},P_{2})\leq\varepsilon$).

A Permute gadget is the same as the Variable gadget, but with a different labeling, see Figure 8. The construction of the Split gadget is similar to the Variable gadget. Additionally, we add a third edge $e_{3}$ of $G_{1}$ and edges of $G_{2}$ from $w_{2}$ and $w_{3}$ inside the $\varepsilon$-tube around $e_{3}$. For the labeling, see Figure 8.

The same argument as for the Variable gadget is used to proof the following statements:

• •

A False signal can never be converted to a True signal in the Split gadget.

• •

A False signal can never be converted to a True signal in the Permute gadget.

Furthermore, a True signal can, but need not to be converted to a False signal in the Permute gadget.

We construct the Wire gadget used to connect all the other gadgets by drawing two edges $e_{1}$ and $e_{2}$ of $G_{1}$ incident to a vertex $v$ (with arbitrary angle) and two vertices $w_{1}$ and $w_{2}$ of $G_{2}$ inside $B_{\varepsilon}(v)$ with non intersecting incident edges inside $T_{\varepsilon}(e_{1})$ and $T_{\varepsilon}(e_{2})$. Obviously, it is not possible to convert a True signal to a False signal or vice versa, here.

For the Clause gadget, we first introduce a NAE-Clause gadget. Here it is required that the three values in each clause are not all equal to each other. We start the construction of the NAE-Clause gadget by drawing three edges $e_{1}$, $e_{2}$ and $e_{3}$ incident to a vertex $v$ with a pairwise $120^{\circ}$ angle. We draw three vertices $w_{1}$, $w_{2}$ and $w_{3}$ on the intersections of $T_{\varepsilon}(e_{1})$, $T_{\varepsilon}(e_{2})$ and $T_{\varepsilon}(e_{3})$ with a maximum distance to $v$. Furthermore, we draw two edges of $G_{2}$ inside the tubes for each vertex and label them as shown in Figure 8. Let $q_{1}$ be the point on $e_{1}$ with distance $\varepsilon$ to $w_{1}$ and $w_{2}$. Here, it is not enough to simply connect $w_{1}$ and $w_{2}$ with one edge as shown in the bottom left sketch of Figure 8. To force backtracking along $e_{1}$ for a combination of labels which we want to exclude, we have to ensure that a path from $w_{1}$ to $w_{2}$ leaves $B_{\varepsilon}(q_{1})$ but stays, once entered, inside $B_{\varepsilon}(v)$. For the other pairs, $(w_{1},w_{3})$ and $(w_{2},w_{3})$ we do the same. A possible drawing of these paths maintaining the planarity of $G_{2}$ is shown in Figure 8. The three placements of $v$ are connected by the vertices $w_{1}$, $w_{2}$ and $w_{3}$ and it holds that there is no placement of $v$ such that an all-False or an all-True labeling can be realized: Suppose we map $v$ to $s(v)$ as shown in the Figure 8. Then, edges $e_{2}$ and $e_{3}$ can be mapped to paths through edges labeled True. But we cannot map $e_{1}$ to such a path $P$: When $P$ reaches vertex $w_{1}$, any corresponding reparameterization of $e_{1}$ realizing $\delta_{F}(e_{1},P)\leq\varepsilon$ must have reached $q_{1}$ as $q_{1}$ is the only point with distance at most $\varepsilon$ to $w_{1}$ on $e_{2}$. As $P$ leaves $B_{\varepsilon}(q_{1})$ between $w_{1}$ and $w_{2}$ and any point on $e_{1}$ with distance at most $\varepsilon$ to the part of $P$ outside $B_{\varepsilon}(q_{1})$ lies between $v$ and $q_{1}$ it follows that $\delta_{F}(e_{1},P)>\varepsilon$. For symmetric reasons it follows that any other all-equal labeling cannot be realized. However, there is a placement of $v$, such that all three edges $e_{1}$, $e_{2}$ and $e_{3}$ can be mapped to a path in $G_{2}$ with Fréchet distance at most $\varepsilon$, for each configuration where not all three signals have the same value.

Monotone-Planar-NAE-3-Sat is in $P$, but we can use the NAE-Clause gadget as core of our Clause gadget referring to the NP-complete version Monotone-Planar-3-Sat: We obtain the Clause gadget by connecting each NAE-Clause gadget with three Permute gadgets, as shown in Figure 8.

Figure 9 partially shows the constructed graphs for a given Monotone-Planar-3-Sat $A$ consisting of the subgraphs (gadgets) described above.

Now, given a Monotone-Planar-3-SAT instance $A$, one can construct the graphs $G_{1}$ and $G_{2}$ with the gadgets described above. Note that all gadgets are plane subgraphs. By placing them next to each other with no overlap, we can ensure that $G_{1}$ and $G_{2}$ are plane graphs.

A valid placement of the whole graph $G_{1}$ induces a solution of $A$: In the corresponding gadget for each positive NAE-clauses, at least one of the outgoing edges of $G_{1}$ must be mapped to a path through an edge labeled True. By construction, this label cannot be converted to False in any of the gadgets and therefore the corresponding variable $v$ gets the value True. In this case, $v$ cannot set any of the negative clauses True because the other outgoing edge must be mapped to a path through the edge of $G_{2}$ labeled False and this signal can never be switched to True. The same holds for the case of negative NAE-clauses.

Conversely, given a solution $S$ of the Monotone-Planar-3-SAT instance $A$, it is easy to construct a placement of $G_{1}$. In the variable gadget of variable $x_{1}$, we chose placement $p_{1}$ and map $e_{1}$ to a path through the edge of $G_{2}$, labeled True and map $e_{2}$ to a path through the edge labeled False if $x_{1}$ is positive in $S$. If $x_{1}$ is negative, we chose placement $p_{2}$ and map $e_{1}$ and $e_{2}$ accordingly. All edges of the other gadget now can be mapped to $G_{2}$ in a signal preserving manner (True stays True, False stays False). If there exists a clause $C$ in $A$, such that all three variables of $C$ are positive (negative) in $S$, we change one signal in the Permute gadget from True to False. Thus, we have found a placement for the whole graph $G_{1}$. ∎

The following stronger result follows from the observation that the characteristics of the subgraphs we constructed in the proof of Theorem 5 still hold for a slightly larger $\varepsilon$ value.

Theorem 6**.**

It is NP-hard to approximate $\vec{\delta}_{G}(G_{1},G_{2})$ within a $1.10566$ factor.

Proof.

We give a detailed proof for the NAE-Clause gadget and note that a similar argument holds for the other gadgets. See Figure 10 for an illustration of the arguments and the calculations below.

Let us fix $\varepsilon=1$. As described in the proof of Theorem 5, we connect the vertices $w_{1}$ and $w_{2}$ by a path which leaves $B_{1}(q_{1})$, but stays inside $B_{1}(v_{1})$. (Introducing a spike which leaves $B_{1}(q_{1})$ and returns to $B_{1}(q_{1})$, see Figure 8). We draw the spike such that its peak is arbitrarily close to the intersection of a straight line through the edge $e_{1}$ and the $1$-circle around $v$. When enlarging $\varepsilon$, the point $q_{1}$ moves along $e_{1}$ toward $v$. We need to compute the smallest value $\delta_{min}$, such that $B_{1}(v)$ is completely contained in $B_{1+\delta_{min}}(q_{1})$. For any value $\delta<\delta_{min}$, there exists a drawing of the spikes, such that the characteristics of the NAE-Clause gadget still hold, e.g., there is no placement of $v$ allowing an all-equal-labeling.

Note that $\delta_{min}$ equals the distance from $q_{1}$ to $v$, when $q_{1}$ is at distance $1+\delta_{min}$ to $w_{1}$. Let $q^{\prime}$ be the position of $q_{1}$ for $\delta=0$ and let $d$ be the distance between $q^{\prime}$ and $q_{1}$. Then we have $\tan(30^{\circ})=\frac{\delta_{min}+d}{1}=\delta_{min}+d$. Furthermore, we have $d=\sqrt{(1+\delta_{min})^{2}-1}$ and therefore $\delta_{min}=\tan(30^{\circ})-\sqrt{(1+\delta_{min})^{2}-1}$, which solves to $\delta_{min}=\frac{1}{4}-\frac{1}{4\sqrt{3}}\approx 0.10566$. The factor by which $\varepsilon$ can be multiplied is greater than $1+\delta_{min}$ for all other gadgets. Thus, $\delta_{min}$ is the critical value for the whole construction and the theorem follows. ∎

4.2 Deciding the Strong Graph Distance in Exponential Time

A brute-force method to decide the directed strong graph distance is to iterate over all possible combinations of valid vertex placements, which takes $O(m_{1}\cdot m_{2}^{n_{1}})$. time. Another approach is to decompose $G_{1}$ into faces and merge the substructures bottom-up.

First, we remove all tree-like substructures of $G_{1}$ and map these as described in the proof of Lemma 6. Next, we decompose the remainder of $G_{1}$ into chordless cycles, where a chord is a maximal path in $G_{1}$ incident to two faces, see Figure 11.

We map the parts of $G_{1}$ from bottom up, deciding in each step if we can map two adjacent cycles and all the nested substructures of the cycles simultaneously. To do so, we start with storing all combinations of placements of endpoints of a chord -which separates two faces (chordless cycles)- allowing us to map the two faces simultaneously. We prune all placements which are not part of any valid combination. In the following steps, for each placement $C_{u}$ of an endpoint $u$ of a chord and each valid combination $c$ of nested chords computed in the previous step, we run one graph exploration. For each placement $C_{v}$ of the other endpoint $v$ of the chord, which allow to map both cycles simultaneously, we store a new combination consisting of $C_{u}$, $c$ and $C_{v}$. We prune all placements of $u$ where we cannot reach a valid placement of $v$ by using any of the previous computed combinations. Furthermore, we prune those placements of $v$ which are never reached by any graph exploration. If the list of placements gets empty for one vertex, we can conclude, that the graph distance is greater than $\varepsilon$. Conversely, if we find a valid combination of placements of the endpoints of the chord in the last step, we can conclude that we can map the whole graph $G_{1}$ as we guarantee in each step that all substructures can be mapped, too.

Theorem 7**.**

For plane graphs, the strong graph distance can be decided in $O(Fm_{2}^{2F-1})$ time and $O(m_{2}^{2F-1})$ space, where $F$ is the number of faces of $G_{1}$.

Proof.

Each graph exploration takes $O(m_{2})$ time and in each node we have to run $O(m_{2}k)$ explorations, where $k$ is the number of valid combinations of endpoint placements from previously investigated chords. As the tree has a depth of $\log(F)$, we have {ceqn}

[TABLE]

as the total running time of the graph explorations. We can upper bound this term as follows: {ceqn}

[TABLE]

where the second equality uses {ceqn}

[TABLE]

for $a\in\mathbb{R}$. In the first step, we have to store $O(m_{2}^{2})$ combinations for each two faces we want to map simultaneously. Let $k$ be the number of combinations in the previous step. Then we have to store up to $k^{2}m_{2}^{2}$ combinations in the next step. This results in storing $O(m_{2}^{2F-1})$ combinations in the root node of the decomposition. ∎

Thus, this method is superior to the brute-force method if $2F-1\leq n_{1}$.

4.3 Approximation for Plane Graphs

For plane graphs, Algorithm 1 yields an approximation depending on the angle between the edges for deciding the strong graph distance. The decision is based on the existence of valid placements. Therefore, the runtime is the same as stated in Theorem 4.

Theorem 8**.**

Let $G_{1}:=(V_{1},E_{1})$ and $G_{2}:=(V_{2},E_{2})$ be plane graphs. Assume that for all adjacent vertices $v_{1}$, $v_{2}\in V_{1}$, $B_{\varepsilon}(v_{1})$ and $B_{\varepsilon}(v_{2})$ are disjoint. Let $\alpha_{v}$ be the smallest angle between two edges of $G_{1}$ incident to vertex $v$ with $deg(v)\geq 3$, and let $\alpha:=\frac{1}{2}\min_{v\in V_{1}}(\alpha_{v})$. If there exists at least one valid $\varepsilon$-placement for each vertex of $G_{1}$, then $\vec{\delta}_{G}(G_{1},G_{2})\leq\frac{1}{\sin(\alpha)}\varepsilon$.

Proof.

Let $\alpha$ be the smallest angle between two edges incident to a vertex $v$ with degree at least three and let $C_{1},C_{2},\dots,C_{j}$ be the valid placements of $v$ for a given distance value $\varepsilon$. Furthermore, let $V_{C_{i}}$ be the set of vertices of $C_{i}$. It can be easily shown that for a larger distance value of $\hat{\varepsilon}\geq\frac{1}{\sin(\alpha)}\varepsilon$ there exists vertices $v_{1},v_{2},\dots,v_{k}$, embedded inside $B_{\hat{\varepsilon}}$, such that the subgraph $C=(V^{\prime},E^{\prime})$, where $V^{\prime}=\bigcup\limits_{i=1}^{j}V_{C_{i}}\cup\{v_{1}\}\cup\{v_{2}\}\cup\dots\cup\{v_{k}\}$ and $E^{\prime}=\{\{uw\}\in E_{2}|u\in V^{\prime},w\in V^{\prime}\}$ is connected, see Figure 12 (a). Note that this property is not true if $B_{\varepsilon}(v_{1})\cap B_{\varepsilon}(v_{2})\neq\emptyset$ for two adjacent vertices $v_{1}$, $v_{2}\in V_{1}$: Figure 12 (b) shows an example with $\alpha=45^{\circ}$ where the two placements do not merge inside $B_{\frac{\varepsilon}{\sin(45^{\circ})}}(v)$. However, with the condition $B_{\varepsilon}(v_{1})\cap B_{\varepsilon}(v_{2})=\emptyset$, there is only one valid $\frac{1}{\sin(\alpha)}\varepsilon$-placement $C$ for each vertex with degree at least three. Furthermore, every valid $\varepsilon$ placement is a valid $\frac{1}{\sin(\alpha)}\varepsilon$-placement. Now, a path $P$ of $G_{1}$ starting at a vertex $v$ with $deg(v)\geq 3$ and ending at a vertex $w$ with $deg(w)\neq 2$, with vertices of degree two in the interior of $P$, can be mapped as described in the proof of Lemma 6 . For two paths which start and/or end at a common vertex $v$, $v$ is mapped to the same placement as there is only one valid $\frac{1}{\sin(\alpha)}\varepsilon$-placement of $v$. This ensures that each edge of $G_{1}$ is mapped correctly. ∎

5 Experiments on Road Networks

In the last decade several algorithms have been developed for reconstructing maps from the trajectories of entities moving on the network [4, 5]. This naturally asks to assess the quality of such reconstruction algorithms. Recently, Duran et al [17] compared several of these algorithms on hiking data, and found that inconsistencies often arise due to noise and low sampling of the input data, for example unmerged parallel roads or the addition of short off-roads.

When assessing the quality of a network reconstruction from trajectory data, several aspects have to be taken into account. Two important aspects are the geometric and topological error of the reconstruction. Another important aspect is the coverage, i.e., how much of the network is reconstructed from the data. We believe our measures to be well suited for assessing the geometric error while still maintaining connectivity information.

We have used the weak graph distance for measuring the distance between different reconstructions and a ground truth, as well as a simplification of part of the road network of Chicago. Figure 13 (a) shows two reconstructed road map graphs $R$ (red) and $B$ (blue), overlayed on the underlying ground truth road map $G$ from OpenStreetMap. The reconstruction $R$ in red resulted from Ahmed et al.’s algorithm [6], whereas the reconstruction $B$ in blue from Davies et al.’s [16] algorithm. Our directed graph distance from $B$ to $G$ is 25 meters, and from $R$ to $G$ it is 90 meters. This reflects the local geometric error of the reconstructions (note that it does not evaluate the difference in coverage). Figure 13 (b) shows an example where the topology of $R$ and $G$ differs (blue circle), affecting for instance navigation significantly. Our measure captures this difference. Although the reconstruction approximates the geometry well, our measure computes a directed distance of 200 m from $G$ (restricted to the part covered by $R$) to $R$.

Figure 14 shows a partial road network of Chicago at different resolutions. Both maps show vertices as blue dots, connected by straight-line edges. Our approach yields a distance of 22 meters between the graphs, which corresponds to the geometric error of the lower resolution map in comparison to the higher resolution map. The data is extracted from OpenSteetMap using the OSMnx Python library.

6 Conclusion

We developed new distances for comparing straight-line embedded graphs and presented efficient algorithms for computing these distances for several variants of the problem, as well as proving NP-hardness for other variants. Our distance measures are natural generalizations of the Fréchet distance and the weak Fréchet distance to graphs, without requiring the graphs to be homeomorphic. Although graphs are more complicated objects than curves, the runtimes of our algorithms are comparable to those for computing the Fréchet distance between polygonal curves. A large-scale comparison of our approach with existing graph similarity measures is left for future work.