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CCCG 2023, Montreal, QC, Canada, July 31 – August 4, 2023
Graph Mover’s Distance: An Efficiently Computable Distance Measure for
Geometric Graphs
Sushovan Majhi
*
Abstract
Many applications in pattern recognition represent pat-
terns as a geometric graph. The geometric graph dis-
tance (GGD) has recently been studied in [13] as a
meaningful measure of similarity between two geometric
graphs. Since computing the GGD is known to be NP–
hard, the distance measure proves an impractical choice
for applications. As a computationally tractable alter-
native, we propose in this paper the Graph Mover’s Dis-
tance (GMD), which has been formulated as an instance
of the earth mover’s distance. The computation of the
GMD between two geometric graphs with at most n
vertices takes only O(n3)-time. Alongside studying the
metric properties of the GMD, we investigate the stabil-
ity of the GGD and GMD. The GMD also demonstrates
extremely promising empirical evidence at recognizing
letter drawings from the LETTER dataset [18].
1 Introduction
Graphs have been a widely accepted object for pro-
viding structural representation of patterns involving
relational properties. While hierarchical patterns are
commonly reduced to a string [7] or a tree represen-
tation [6], non-hierarchical patterns generally require a
graph representation. The problem of pattern recogni-
tion in such a representation then requires quantifying
(dis-)similarity between a query graph and a model or
prototype graph. Defining a relevant distance measure
for a class of graphs has been studied for almost five
decades now and has a myriad of applications including
chemical structure matching [21], fingerprint matching
[16], face identification [11], and symbol recognition [12].
Depending on the class of graphs of interest and the
area of application, several methods have been pro-
posed. Graph isomorphisms [5] or subgraph isomor-
phisms can be considered.
These, however, cannot cope with (sometimes minor)
local and structural deformations of the two graphs. To
address this issue, several alternative distance measures
have been studied. We particularly mention edit dis-
tance [20, 9] and inexact matching distance [3].
*
School of Information, University of California, Berkeley,
USA, smajhi@berkeley.edu
Although these distance measures have been battle-
proven for attributed graphs (i.e., combinatorial graphs
with finite label sets), the formulations seem inadequate
in providing meaningful similarity measures for geomet-
ric graphs.
A geometric graph belongs to a special class of at-
tributed graphs having an embedding into a Euclidean
space Rd, where the vertex labels are inferred from the
Euclidean locations of the vertices and the edge labels
are the Euclidean lengths of the edges.
In the last decade, there has been a gain in practical
applications involving comparison of geometric graphs,
such as road-network or map comparison [1], detection
of chemical structures using their spatial bonding ge-
ometry, etc. In addition, large datasets like [18] are be-
ing curated by pattern recognition and machine learning
communities.
1.1 Related Work and Our Contribution
We are inspired by the recently developed geometric
graph distance (GGD) in [4, 13]. Although the GGD
succeeds to be a relevant distance measure for geomet-
ric graphs, its computation, unfortunately, is known to
be NP-hard. Our motivation stems from applications
that demand an efficiently computable measure of sim-
ilarity for geometric graphs. The formulation of our
graph mover’s distance is based on the theoretical un-
derpinning of the GGD. The GMD provides a mean-
ingful yet computationally efficient similarity measure
between two geometric graphs.
In Section 2, we revisit the definition of the (GGD)
to investigate its stability under Hausdorff perturbation.
Section 3 is devoted to the study of the GMD. The GMD
has been shown to render a pseudo-metric on the class
of (ordered) geometric graphs. Finally, we apply the
GMD to classify letter drawings in Section 4. Our ex-
periment involves matching each of 2250 test drawings,
modeled as geometric graphs, to 15 prototype letters
from the English alphabet. For the drawings with LOW
distortion, the correct letter has been found among the
top 3 matches at a rate of 98.93%, where the benchmark
accuracy is 99.6% obtained using a k-nearest neighbor
classifier (k-NN) with the graph edit distance [3].
35th Canadian Conference on Computational Geometry, 2022
2 Geometric Graph Distance (GGD)
We first formally define a geometric graph. Throughout
the paper, the dimension of the ambient Euclidean space
is denoted by d≥1. We also assume that the cost
coefficients CVand CEare positive constants.
Definition 2.1 (Geometric Graph) Ageomet-
ric graph of Rdis a (finite) combinatorial graph
G= (VG, EG)with vertex set VG⊂Rd, and the
Euclidean straight-line segments {ab |(a, b)∈EG}
intersect (possibly) at their endpoints.
We denote the set of all geometric graphs of Rdby
G(Rd). Two geometric graphs G= (VG, E G) and
H= (VH, EH) are said to be equal, written G=H,
if and only if VG=VHand EG=EH. We make no
distinction between a geometric graph G= (VG, EG)
and its geometric realization as a subset of Rd; an edge
(u, v)∈EGcan be identified as the line-segment uv in
Rd, and its length by the Euclidean length |uv|.
Following the style of [13], we first revisit the def-
inition of GGD. The definition uses the notion of an
inexact matching. In order to denote a deleted vertex
and a deleted edge, we introduce the dummy vertex ϵV
and the dummy edge ϵE, respectively.
Definition 2.2 (Inexact Matching) Let G, H ∈
G(Rd)be two geometric graphs. A relation π⊆(VG∪
{ϵV})×(VH∪ {ϵV})is called an (inexact) matching
if for any u∈VG(resp. v∈VH) there is exactly
one v∈VH∪ {ϵV}(resp. u∈VG∪ {ϵV}) such that
(u, v)∈π.
The set of all matchings between graphs G, H is de-
noted by Π(G, H). Intuitively, a matching πis a relation
that covers the vertex sets VG, V Hexactly once. As a
result, when restricted to VG(resp. VH), a matching
πcan be expressed as a map π:VG→VH∪ {ϵV}
(resp. π−1:VH→VG∪ {ϵV}). In other words, when
(u, v)∈πand u=ϵV(resp. v=ϵV), it is justified to
write π(u) = v(resp. π−1(v) = u). It is evident from
the definition that the induced map
π:{u∈VG|π(u)=ϵV}→{v∈VH|π−1(v)=ϵV}
is a bijection. For edges e= (u1, u2)∈EGand
f= (v1, v2)∈EH, we introduce the short-hand π(e) :=
(π(u1), π(u2)) and π−1(f) := (π−1(v1), π−1(v2)).
Another perspective of πis to view it as a match-
ing between portions of Gand H, (possibly) after ap-
plying some edits on the two graphs. For example,
π(u) = ϵV(resp. π−1(v) = ϵV) encodes deletion of the
vertex ufrom G(resp. vfrom H), whereas π(e) = ϵE
(resp. π−1(f) = ϵE) encodes deletion of the edge efrom
G(resp. ffrom H). Once the above deletion opera-
tions have been performed on the graphs, the resulting
subgraphs of Gand Hbecome isomorphic, which are
finally matched by translating the remaining vertices u
to π(u). Now, the cost of the matching πis defined as
the total cost for all of these operations:
Definition 2.3 (Cost of a Matching) Let G, H ∈
G(Rd)be geometric graphs and π∈Π(G, H)an inex-
act matching. The cost of π, is Cost(π) =
X
u∈VG
π(u)=ϵV
CV|u−π(u)|
| {z }
vertex translations
+X
e∈EG
π(e)=ϵE
CE|e|−|π(e)|
| {z }
edge translations
+
X
e∈EG
π(e)=ϵE
CE|e|
| {z }
edge deletions
+X
f∈EH
π−1(f)=ϵE
CE|f|
| {z }
edge deletions
.(1)
Definition 2.4 (GGD)For geometric graphs G, H ∈
G(Rd), their geometric graph distance, GGD(G, H), is
GGD(G, H)def
= min
π∈Π(G,H)Cost(π).
2.1 Stability of GGD
A distance measure is said to be stable if it does not
change much if the inputs are perturbed only slightly.
Usually, the change is expected to be bounded above
by the amount of perturbation inflicted on the inputs.
The perturbation is measured under a suitable choice of
metric. In the context of geometric graphs, it is natural
to wonder if the GGD is stable under the Hausdorff
distance between two graphs. To our disappointment,
we can easily see for the graphs shown in Fig. 1 that
the GGD is positive, whereas the Hausdorff distance
between their realizations is zero. So, the Hausdorff
distance between the graphs can not bound their GGD
from above.
v1v2
u1u2u3
H
G
Figure 1: The graphs G(top) and H(bottom) are em-
bedded in the real line; the distance between consec-
utive ticks is 1 unit. The Hausdorff distance between
Gand His zero, however GGD(G, H) = CV+CEis
non-zero. The optimal matching is given by π(u1) = v1,
π(u2) = v2, and π(u3) = ϵV.
One might think that the GGD is stable when the
Hausdorff distance only between the vertices is consid-
ered. However, the graphs in Fig. 2 indicate otherwise.
Under strong requirements, however, it is not difficult
to prove the following result on the stability of GGD
under the Hausdorff distance.
CCCG 2023, Montreal, QC, Canada, July 31 – August 4, 2023
0 1 2 3
0
1
2
3
u1
u3
u2
0 1 2 3
0
1
2
3
v3
v1
v2
Figure 2: For the graphs G, H ∈ G(R2), the Haus-
dorff distance between the vertex sets is zero, however
GGD(G, H)=4CEis non-zero. The optimal matching
is given by π(u1) = v1,π(u3) = v3,π(u2) = ϵV, and
π−1(v2) = ϵV.
Theorem 1 (Hausdorff Stability of GGD) Let
G, H ∈ G(Rd)be geometric graphs with a graph
isomorphism π:VG→VH. If δ > 0is such that
|u−π(u)| ≤ δfor all u∈VG, then
GGD(G, H)≤CV|VG|δ.
Proof. The given graph isomorphism πis a bijective
mapping between the vertices of Gand H. So, π∈
Π(G, H), i.e., it defines an inexact matching. Since πis
a graph isomorphism, it does not delete any vertex or
edge. More formally, for all u∈VGand v∈VH, we
have π(u)=ϵVand π−1(v)=ϵV, respectively. Also,
for all e∈EGand f∈EH, we have π(e)=ϵEand
π−1(f)=ϵE, respectively. From (1), the cost
Cost(π) = X
u∈VG
CV|u−π(u)| ≤ CV|VG|δ.
So, GGD(G, H)≤Cost(π)≤CV|VG|δ.□
3 Graph Mover’s Distance (GMD)
We define the Graph Mover’s Distance for two ordered
geometric graphs. A geometric graph is called ordered if
its vertices are ordered or indexed. In that case, we de-
note the vertex set as a (finite) sequence VG={ui}m
i=1.
Let us denote by GO(Rd) the set of all ordered geomet-
ric graphs of Rd. The formulation of the GMD uses the
framework known as the earth mover’s distance (EMD).
3.1 Earth Mover’s Distance (EMD)
The EMD is a well-studied distance measure between
weighted point sets, with many successful applications
in a variety of domains; for example, see [8, 10, 17, 19].
The idea of the EMD was first conceived by Monge [14]
in 1781, in the context of transportation theory. The
name “earth mover’s distance” was coined only recently,
and is well-justified due to the following analogy. The
first weighted point set can be thought of as piles of
earth (dirt) lying on the point sites, with the weight
of a site indicating the amount of earth; whereas, the
other point set as pits of volumes given by the corre-
sponding weights. Given that the total amount of earth
in the piles equals the total volume of the pits, the EMD
computes the least (cumulative) cost needed to fill all
the pits with earth. Here, a unit of cost corresponds to
moving a unit of earth by a unit of “ground distance”
between the pile and the pit.
The EMD can be cast as a transportation problem
on a bipartite graph, which has several efficient imple-
mentations, e.g., the network simplex algorithm [2, 15].
Let the weighted point sets P={(pi, wpi)}m
i=1 and
Q={(qj, wqj)}n
j=1 be a set of suppliers and a set of
consumers, respectively. The weight wpidenotes the to-
tal supply of the supplier pi, and wqjthe total demand
of the consumer qj. The matrix [di,j ] is the matrix of
ground distances, where di,j denotes the cost of trans-
porting a unit of supply from pito qj. We also assume
the feasibility condition that the total supply equals the
total demand:
m
X
i=1
wpi=
n
X
j=1
wqj.(2)
Aflow of supply is given by a matrix [fi,j] with fi,j
denoting the units of supply transported from pito qj.
We want to find a flow that minimizes the overall cost
m
X
i=1
n
X
j=1
fi,j di,j
subject to:
fi,j ≥0 for any i= 1, . . . , m and j= 1, . . . , n (3)
n
X
j=1
fi,j =wifor any i= 1, . . . , m (4)
m
X
i=1
fi,j =wjfor any j= 1, . . . , n, (5)
Constraint (3) ensures a flow of units from Pto Q, and
not vice versa; constraint (4) dictates that a supplier
must send all its supply—not more or less; constraint
(5) guarantees that the demand of every consumer is
exactly fulfilled.
The earth mover’s distance (EMD) is then defined by
the cost of the optimal flow. A solution always exists,
provided condition (2) is satisfied. The weights and the
ground distances can be chosen to be any non-negative
numbers. However, we choose them appropriately in
order to solve our graph matching problem.
3.2 Defining the GMD
Let G, H ∈ GO(Rd) be two ordered geometric graphs
of Rdwith VG={ui}m
i=1 and VH={vj}n
j=1. For
35th Canadian Conference on Computational Geometry, 2022
1
1
1
2
u11
u21
u31
u42
v1
1
v2
1
v3
3
Figure 3: The bipartite network used by the GMD is
shown for two ordered graphs G, H with vertex sets
VG={u1, u2, u3}and VH={v1, v2}, respectively. The
dummy nodes u4for Gand v3for H, respectively, have
been shown in gray. Below each node, the corresponding
weights are shown. A particular flow has been depicted
here. The gray edges do not transport anything. A
red edge has a non-zero flow with the transported units
shown on them.
each i= 1, . . . , m, let EG
idenote the (row) m–vector
containing the lengths of (ordered) edges incident to the
vertex uiof G. More precisely, the
kth element of EG
i=(|eG
i,k|,if eG
i,k := (ui, uk)∈EG
0,otherwise.
Similarly, for each j= 1, . . . , n, we define EH
jto be the
(row) n–vector with the
kth element of EH
j=(|eH
j,k|,if eH
j,k := (vj, vk)∈EH
0,otherwise.
In order to formulate the desired instance of the EMD,
we take the point sets to be P={ui}m+1
i=1 and Q=
{vj}n+1
j=1 . Here, um+1 and vn+1 have been taken to be a
dummy supplier and dummy consumer, respectively, to
incorporate vertex deletion into our GMD framework.
The weights on the sites are defined as follows:
wui= 1 for i= 1 ...,m and wum+1 =n .
And,
wvj= 1 for j= 1 ...,n and wvm+1 =m .
We note that the feasibility condition (2) is satisfied:
m+nis the total weight for both Pand Q. An instance
of the transportation problem is depicted in Fig. 3.
Finally, the ground distance from uito vjis defined
by:
di,j =
CV|ui−vj|+CE∥EG
iDm×p−EH
jDn×p∥1,
if 1 ≤i≤m, 1≤j≤n
CE∥EH
j∥1,if i=m+ 1 and 1 ≤j≤n
CE∥EG
i∥1,if 1 ≤i≤mand j=n+ 1
0,otherwise.
Here, p= min{m, n}, the 1–norm of a row vector is
denoted by ∥·∥1, and Ddenotes a diagonal matrix with
the all diagonal entries being 1.
0 1 2
0
1
2u4u1
u5
u2u3
0 1 2
0
1
2v3
v1
v5
v2
v4
G H
Figure 4: For the geometric graph G, H ∈ GO(R2), the
GMD is zero. The optimal flow is given by the matching
π(u1) = v2,π(u2) = v1,π(u3) = v4,π(u4) = v3, and
π(u5) = v5.
3.3 Metric Properties
We can see that the GMD induces a pseudo-metric on
the space of ordered geometric graphs GO(Rd). Non-
negativity, symmetry, and triangle inequality follow
from those of the cost matrix [di,j ] defined in the GMD.
In addition, we note that G=H(as ordered graphs)
implies that di,j = 0 whenever i=j. The trivial flow,
where each uisends its full supply to vi, has a zero cost.
So, GMD(G, H) = 0. The GMD does not, however,
satisfy the separability condition on GO(Rd).
For the graphs G, H shown in Fig. 4, we have
GMD(G, H) = 0. We note that G, H have the follow-
ing adjacency length matrices [EG
i]iand [EH
j]j, respec-
tively:
0 0 0 2 √2
0 0 2 0 √2
0 2 0 0 0
2 0 0 0 0
√2√2 0 0 0
and
0 0 2 0 √2
0 0 0 2 √2
2 0 0 0 0
0 2 0 0 0
√2√2 0 0 0
.
It can be easily checked that the flow that transports
a unit of supply from u17→ v2,u27→ v1,u37→ v4,
u47→ v3,u57→ v5, and five units from u67→ v6has total
cost zero. So, GMD(G, H ) = 0. However, the graphs G
and Hare not the same geometric graph. The fact that
GGD(G, H)= 0 implies the GGD is not stable under
the GMD.
One can easily find even simpler configurations for
two distinct geometric graphs with a zero GMD—if the
graphs are allowed to have multiple connected compo-
nents.
We conclude this section by stating a stability result
for the GMD under the Hausdorff distance. We omit
the proof, since it uses a similar argument presented in
Theorem 1.
CCCG 2023, Montreal, QC, Canada, July 31 – August 4, 2023
Theorem 2 (Hausdorff Stability of GMD) Let
G, H ∈ GO(Rd)be ordered geometric graphs with a
bijection π:VG→VHsuch that eG
i,j =eH
π(i),π(j)for
all i, j . If δ > 0is such that |ui−π(ui)| ≤ δfor all
ui∈VG, then
GMD(G, H)≤CV|VG|δ.
3.4 Computing the GMD
As pointed out earlier, the GMD can be computed as an
instance of transportation problem—using, for example,
the network simplex algorithm. If the graphs have at
most nvertices, computing the ground cost matrix [di,j]
takes O(n3)-time. Since the bipartite network has O(n)
vertices and O(n2) edges, the simplex algorithm runs
with a time complexity of O(n3), with a pretty good
constant. Overall, the time complexity of the GMD is
O(n3).
4 Experimental Results
We have implemented the GMD in Python, using net-
work simplex algorithm from the networkx package. We
ran a pattern retrieval experiment on letter drawings
from the IAM Graph Database [18]. The repository pro-
vides an extensive collection of graphs, both geometric
and labeled.
In particular, we performed our experiment on the
LETTER database from the repository. The graphs in
the database represent distorted letter drawings. The
database considers only 15 uppercase letters from the
English alphabet: A,E,F,H,I,K,L,M,N,T,V,W,X,Y,
and Z. For each letter, a prototype line drawing has been
manually constructed. On the prototypes, distortions
are applied with three different level of strengths: LOW,
MED, and HIGH, in order to produce 2250 letter graphs
for each level. Each test letter drawing is a graph with
straight-line edges; each node is labeled with its two-
dimensional coordinates. Since some of the graphs in
the dataset were not embedded, we had to compute the
intersections of the intersecting edges and label them
as nodes. The preprocessing guaranteed that all the
considered graphs were geometric; a prototype and a
distorted graph are shown in Fig. 5.
We devised a classifier for these letter drawings us-
ing the GMD. For this application, we chose CV= 4.5
and CE= 1. For a test letter, we computed its GMD
from the 15 prototypes, then sorted the prototypes in
an increasing order of their distance to the test graph.
We then check if the letter generating the test graph is
among the first kprototypes. For each level of distor-
tion and various values of k, we present the rate at which
the correct letter has been found in the first kmodels.
The summary of the empirical results have been shown
0 1 2 3
0
1
2
3
u3
u4
u1
u2
u5
0 1 2 3
0
1
2
3
v1
v2
v3
v4
v5
v6
v7
Figure 5: The prototype geometric graph of the letter
Ais shown on the left. On the right, a (MED) distorted
letter Ais shown.
correct letter in first kmodels (%)
Distortion k= 1 k= 3 k= 5
LOW 96.66% 98.93% 99.37%
MED 66.66% 85.37% 91.15%
HIGH 73.73% 90.48% 95.51%
Table 1: Empirical result on the LETTER dataset
in Table 1. Although the graph edit distance based k-
NN classifier still outperforms the GMD by a very small
margin, our results has been extremely satisfactory.
One possible reason why the GMD might fail to cor-
rectly classify some of the graphs is that lacks the sep-
arability property as a metric.
5 Discussions
We have successfully introduced an efficiently com-
putable and meaningful similarity measure for geomet-
ric graphs. However, the GMD lacks some of the de-
sirable properties, like separability and stability. The
currently presented stability results for the GGD and
GMD have a factor that depends on the size of the in-
put graphs. The question remains if the distance mea-
sures are in fact stable under much weaker conditions,
possibly with constant factors on the right side. It will
also be interesting to study the exact class of geometric
graphs for which the GMD is, in fact, a metric.
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