Gromov-Hausdorff Distance Between Metric Graphs

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1 Groov-Hausdorff Distance Between Metric Graphs Jiwon Choi St Mark s School January, 019 Abstract In this paper we study the Groov-Hausdorff distance between two etric graphs We copute the precise value of the Groov-Hausdorff distance between two path graphs Moreover, we copute the precise value of the Groov-Hausdorff distance between a cycle graph and a tree Given a graph X, we consider a graph Y that results fro adding an edge to X without changing the nuber of vertices We copute the precise value of the Groov-Hausdorff distance between X and Y 1 Introduction Groov-Hausdorff distance, a concept that was first introduced by Groov in his study of convergence of anifolds [], is a very iportant concept in odern differential geoetry because this notion allows us to evaluate the distance between two distinct etric spaces Suppose we have a etric space The standard notion of distance allows us to deterine the distance between two subspaces of a etric space However, when we are considering the distance between two subspaces that intersect, this definition of distance does not give us enough inforation to distinguish between the two subspaces This definition of distance was later iproved by Hausdorff Hausdorff proposed a way of finding the distance between two subspaces of the etric space Out of all the Hausdorff distances that result fro different isoetric ebeddings and different etric spaces, the infiu of all of the is the Groov-Hausdorff distance This notion allows us find the distance between two etric spaces, thus allowing us to copare the two etric spaces It is often challenging to copute the exact value of Groov-Hausdorff distances since the process involves finding the infiu of all possible Hausdorff distances It is relatively siple to find the upper bound for the Groov- Hausdorff distance Because the Groov-Hausdorff distance is the infiu of all the Hausdorff distances resulting fro different ethods of isoetric ebeddings, we only need to exhibit one specific isoetric ebedding such that the condition is satisfied in order to find the upper bound Nevertheless, it is 1

2 ore challenging to get the exact lower bound for Groov-Hausdorff distance because the condition has to told true for all possible isoetric ebeddings In this paper, we copute the exact value of the Groov-Hausdorff distance between specific graphs The ain difficulty in this paper is finding the lower bound of the Groov-Hausdorff distance We start out with Lea 1, which gives the lower bound for the Hausdorff distance between two etric graphs Fro Lea 1, we find out that the Hausdorff distance can be controlled by the distances between points within the two etric graphs that are being considered While finding the lower bound for the Groov-Hausdorff distance between specific graphs, we can use these controls to obtain the optial lower bound We copute the exact value of the Groov-Hausdorff distance between path graphs, which are graphs that can be drawn so that all of their vertices and edges lie on a single straight line Path graphs are subgraphs of any ore coplicated graphs Therefore, the result we get for path graphs prove to be significant when we later copute the Groov-Hausdorff distance between ore coplicated graphs We consider other graphs such as cycle graphs and trees Finding the Groov-Hausdorff distance between two graphs has various applications In the biostatistics field, these results ay be helpful when coparing large data sets in high diensions H Lee et al in their paper on brain networks [3] discuss possible applications in the biostatistics field by introducing the Groov-Hausdorff distance as an already well-established ethod in shape analysis for coparing shapes of networks Moreover, they [3] argue that the concept can also be used for coparing the shapes of neural networks in brains The Groov-Hausdorff distance between graphs has further applications in coputer science In particular, it can be used as a proising ethod for shape atching and coparison Such applications ay help iprove algoriths for face recognition, pattern recognition, or atching of articulated objects [4] In Section, basic definitions are discussed More specifically, the definitions of the Hausdorff distance and the Groov-Hausdorff distance are discussed In Section 3, the proof of the fundaental lea is given Section 4 and Section 5 are devoted to coputing the exact value of the Groov-Hausdorff distance between graphs using the lea In Section 6, future directions are discussed Basic definitions Definition 1 Let X be an arbitrary set A function d : X X R { } is a etric on X if the following conditions are satisfied for all x, y, z X: (1) Identity: d(x, y) = 0 x = y, () Coutativity: d(x, y) = d(y, x), (3) Triangle Inequality: d(x, z) d(x, y) + d(y, z) Eleents of X are called points of the etric space; d(x, y) refers to the distance between points x and y The Euclidean distance in R is an exaple of a etric

3 Definition Given a etric space X, a ap of f : X Z is called an isoetric ebedding if for all x 1, x X: d X (x 1, x ) = d Z (f(x 1 ), f(x )) The etric on Z is denoted as d Z An isoetric ebedding of X into etric space Z is distance preserving Using isoetric ebedding, we can copute the distance between two different spaces by ebedding the into a etric space Z Definition 3 A graph G is an ordered pair G = (V, E) consisting of a set V of vertices, nodes or points together with a set E of edges, which are -eleent subsets of V Reark 1 All graphs entioned in this paper are finite graphs Definition 4 Given a point x and a graph Y ebedded into a etric space Z, the distance between point x and graph Y is defined as d Z (x, Y ) = inf y Y d Z(x, y) Definition 5 Given two graphs X, Y isoetrically ebedded into a etric space Z, for all x X, y Y, the Hausdorff distance between X and Y is d H (X, Y ) = ax(sup x X d Z (x, Y ), sup d Z (X, y)) y Y In a etric space Z, there exists a shortest distance fro point x X to Y Siilarly, there exists a shortest distance fro point y Y to X Out of all these distances, the Hausdorff distance between X and Y is the longest distance Reark We will say that vertices x and y correspond to each other if d(x, y) = d H X, Y Definition 6 The Groov-Hausdorff distance between two graphs X and Y is d GH (X, Y ) = inf d H(i(X), j(y )) i:x Z,j:Y Z The Groov-Hausdorff distance between X and Y is defined to be the infiu of all nubers d H (i(x), j(y )) for all etric spaces Z and all isoetric ebeddings i : X Z and j : Y Z There are infinitely any isoetric ebeddings of X and Y into any etric space Z and different Hausdorff distances resulting fro the Out of all possible Hausdorff distances, the Groov-Hausdorff distance between X and Y is the infiu Reark 3 When proving the lower bound for the Groov-Hausdorff distance, the lower bound ust hold true for all possible Hausdorff distances When proving the upper bound for the Groov-Hausdorff distance, showing one specific isoetric ebedding that exhibits a Hausdorff distance bounded by the upper bound 3

4 Definition 7 Take two graph X, Y with each n and vertices Let us denote the vertices of X as x i for all 1 i n and the vertices of Y as y j for all 1 j Then, isoetrically ebed graphs X, Y into a etric space Z We say that x i corresponds to y j if d Z (x i, y j ) = d Z (x i, Y ) In the rest of this section, we will define specific graphs that will play an iportant role in the paper later Definition 8 We say that graph P is a path graph of length 1 if it consists of vertices and 1 edges Denote the vertices as v i for all 1 i All vertices v i other than v 1 and v are connected to v i 1 and v i+1 We call v 1 and v endpoints, and they are only connected to v and v 1 respectively Figure 1 is a path graph with 6 vertices and 5 edges Figure 1: P 6 Definition 9 A graph C n is a cycle graph if it consists of n 3 vertices v 1, v, v n and n edges, and there is an edge between v and v +1 for all 1 n 1 and an edge between v 1 and v n Figure is a cycle graph with 5 vertices and 5 edges Figure : C 5 Definition 10 A graph X is a tree if it is a set of connected edges containing no cycle graphs as a subgraph In other words, a tree graph is a connected acyclic graph Figure 3 shows exaples of a tree graph Figure 3: Tree graphs 4

5 3 Lower Bound for the Hausdorff Distance In this section, we will prove a fundaental lea that will help prove the subsequent theores in this paper We use basic definitions to prove this lea, which gives us the lower bound for the Hausdorff distance between two graphs X and Y Notably, we use the definition of correspondence and the definition of isoetric ebedding Lea 1 Take two graphs X, Y and isoetrically ebed the into a etric space Z If x i X corresponds to y i Y and x j X corresponds to y j Y, then d H (X, Y ) d X (x i,x j) d Y (y i,y j) Proof Because X and Y are isoetrically ebedded into a etric space Z, d X (x i, x j ) d Z (x i, y i ) + d Y (y i, y j ) + d Z (x j, y j ) Because x i corresponds to y i and x j corresponds to y j, Moreover, Therefore, we get d X (x i, x j ) d Z (x i, Y ) + d Y (y i, y j ) + d Z (x j, Y ) d X (x i, x j ) d H (X, Y ) + d Y (y i, y j ) + d H (X, Y ) d X (x i, x j ) d Y (y i, y j ) d H (X, Y ), and d X (x i,x j) d Y (y i,y j) is the lower bound for d H (X, Y ) Reark 4 In order to find the best possible lower bound for d H (X, Y ), we want to axiize d X (x i, x j ) and iniize d Y (y i, y j ) 4 Groov-Hausdorff Distance Between Two Path Graphs We consider the Groov-Hausdorff distance between two path graphs To find the best possible lower bound, we know fro Lea 1 that we ust axiize d X (x i, x j ) and iniize d Y (y i, y j ) In order to do so, we consider the endpoints of the two path graphs Finding the Groov-Hausdorff distance between path graphs is significant because path graphs are subgraphs of any ore coplicated graphs We copute the precise value of the Groov-Hausdorff distance between two path graphs We prove that d GH (P, P n ) n by contradiction, and we prove that d GH (P, P n ) n by exhibiting a special isoetric ebedding of X, Y into a etric space Z 5

6 Theore Given two graphs P and P n, the Groov-Hausdorff distance between P and P n is exactly n Proof Assue n First, let us prove that d GH (P, P n ) n By the definition of the Groov-Hausdorff distance between two spaces X, Y, in order to prove d GH (P, P n ) n, we need to prove that d H (P, P n ) n holds for any isoetric ebedding of P, P n into Z By the definition of Hausdorff distance, we need to prove that either d Z (x, P n ) n or d Z (P, y) n for soe point x P and soe point y P n In order to prove this stateent, we want to show that sup x P d Z (P n, x) n To axiize the lower bound, we consider the endpoints x 1 and x In other words, we want to show that either d Z (P n, x 1 ) n or d Z (P n, x ) n We can prove this by contradiction Assue that d Z (P n, x 1 ) < n and d Z (P n, x ) < n Then, for soe i, j such that 1 i, j n, d Z (x 1, y i ) < n and d Z (x, y j ) < n Because P and P n are path graphs, each edge is of length 1, thus d Y (y i, y j ) = i j and d X (x 1, x ) = 1 Since d Z (x 1, y i ) < n and d Z (x, y j ) < n, d Z (x 1, y i ) + d Y (y i, y j ) + d Z (x, y j ) < n + j i Because isoetric ebedding is distance-preserving, and d X (x 1, x ) d Z (x 1, y i ) + d Y (y i, y j ) + d Z (x, y j ), 1 < n + j i However, this is a contradiction Thus, either d Z (P n, x 1 ) n or d Z (P n, x ) n, and we can conclude that d GH (P, P n ) n Second, let us prove that d GH (P, P n ) n In order to prove this, we want to show that there exists an isoetric ebedding of P, P n into a etric space such that the d H (P, P n ) n Consider the isoetric ebedding shown in Figure 4 We first build a pyraid with edges and vertices as shown below Construct the pyraid so that the horizontal edges are each of length 1, and the diagonal edges are each of length 1 Figure 4: Isoetric Ebedding for d GH (P, P n ) n 6

7 Consider the bases of the triangles in each layer as path graphs The first layer exhibits P, the second layer exhibits P 3, and so on In this isoetric ebedding, d H (P, P n ) n Thus, since d GH (P, P n ) n and d GH (P, P n ) n, d GH (P, P n ) = n 5 Groov-Hausdorff Distance Between Graphs that Contain Path Graphs as Subgraphs We consider the Groov-Hausdorff distance between a cycle graph and a tree Burago [1] clais in his book that the Groov-Hausdorff distance between two subspaces of two etric spaces is equal to the Groov-Hausdorff distance between the two etric spaces Since path graphs are subgraphs of a cycle graph, we divide up the cycle graph into ultiple path graphs and use Lea 1 to find the distance between those path graphs and the path graphs found in the tree We prove the lower bound for the Hausdorff distance by contradiction, and the upper bound is given by exhibiting a specific isoetric ebedding of the two graphs into a etric space To find the best upper bound possible, we want to show an isoetric ebedding that iniizes the Hausdorff distance Intuitively, we want to divide up the vertices of the cycle graph as evenly as possible to correspond to each vertex of the tree Theore 3 If C is a cycle graph with vertices and X is a tree where n vertices with > n, then d GH (C, X) = n 1 Proof First, we will prove that d GH (C, X) n 1 Given all isoetric ebeddings of C, X into a etric space Z, we can find an isoetric ebedding where there are at least n vertices of C that correspond to one vertex of X, using the pigeon hole arguent One resulting Hausdorff distance is greater than or equal to the Hausdorff distance between a path graph of length n and a path graph of length 1 Since n vertices correspond to a single vertex in X, the longest possible distance between two vertices that correspond to the sae vertex is greater than n This condition holds for any isoetric ebedding of C and X Therefore, if x i and x j are the two vertices that are farthest apart that correspond to the sae vertex y i, then d X (x i, x j ) > n Using Lea 1, d H (C, X) 1, n and thus n 1 d H (C, X), for all possible isoetric ebeddings of C and X into a etric space Therefore, d GH (C, X) n 1 7

8 Second, we will prove that d GH (C, X) n 1 In order to prove this stateent, we need to exhibit an isoetric ebedding of C and X into etric space Z that satisfies the condition Consider the following isoetric ebedding of C and X into a etric space Z Let p = n Call the vertices of C v i where 1 i Then, take the vertices v 1, v 1+p,, v 1+kp where k is an integer such that 1 + kp, and let the each correspond to a single vertex of X Make the correspondence so that each vertex of C corresponds to a different vertex of X Then, take the reaining vertices and construct the isoetric ebedding such that two adjacent vertices of C correspond to vertices that are no longer than p apart For all vertices of X, at least one vertex of C should correspond to a single vertex of X Moreover, no ore than p vertices of C ust correspond to the sae vertex of X This isoetric ebedding of C and X gives us a Hausdorff distance less than or equal to n 1 Therefore, d GH (C, X) n 1 Figure 6 shows the steps of how to construct such an isoetric ebedding of C 8 and X The dotted red lines represent correspondence Figure 5: Isoetric Ebedding of C 8 and X Previously, we had used Lea 1 to prove Theore 3 by finding the distance between path graphs that are subgraphs of the cycle graph and the tree Now we consider the Groov-Hausdorff distance between two graphs X and Y with the sae nuber of vertices and different nuber of edges Here, we use a different approach because the length of the longest path graphs for both X and Y are equal We prove the lower bound by contradiction To prove the upper bound, we exhibit a specific isoetric ebedding of X and Y into a etric space where the vertices of X and Y have a one-to-one correspondence Theore 4 Consider a graph X with vertices v 1, v,, v Add an edge between v i, v j where 1 i, j and d X (v i, v j ) = to create graph Y Then, d GH (X, Y ) = 1 8

9 To prove Theore 4, we will prove the lower bound of d GH using contradiction by Lea 1 To prove the upper bound, we will show an isoetric ebedding that exhibits d H = d X (v i,v j) 1 Proof First, we will prove that d GH (X, Y ) 1 In order to prove this, we need to prove that d H (X, Y ) 1 for all possible isoetric ebeddings of X and Y Assue that d H (X, Y ) < 1 Using Lea 1, we get d H (X, Y ) > d X(v i, v j ) d X (w i, w j ) However, d X (v i, v j ) = and d Y (w i, w j ) = 1 Then, we get d H (X, Y ) > 1, and this is a contradiction Therefore, the Hausdorff distance between X and Y is always greater than or equal to 1 for all possible isoetric ebeddings of X and Y Thus, d GH (X, Y ) 1 Second, we will prove that d GH (X, Y ) 1 Consider an isoetric ebedding of X and Y where all vertices v i, v,, v of X each respectively correspond to w 1, w,, w of Y, and vice versa In this isoetric ebedding, there is a two-way one-to-one correspondence between all the vertices of X and Y Figure 5 shows an exaple of such an isoetric ebedding The red dotted lines represent the two-way correspondence between the vertices In this isoetric ebedding, d H (X, Y ) 1 Figure 6: Isoetric Ebedding of X and Y where = 7 Because d GH (X, Y ) 1 and d GH(X, Y ) 1, d GH(X, Y ) = 1 6 Conclusion and Future Directions One possibility for future work is a deeper investigation into the Groov- Hausdorff distance between two graphs that have the sae nuber of vertices 9

10 and a different nuber of edges For exaple, let us call the original graph G and the new graph with additional edges but the sae nuber of vertices G Unlike in Theore 4, we can add ore than one edge to create G For these graphs, can we also show that the one-to-one correspondence between the vertices of the two graphs is the only isoetric ebedding that works 7 Acknowledgeents I a grateful for the MIT PRIMES Progra for the opportunity to work on atheatical research with such an aazing entor, Ao Sun I want to thank y entor, Ao Sun, fro MIT for offering great guidance and support throughout the year I also thank Tanya Khovanova for the guidance she offered for writing quality research papers References [1] D Burago, Y Burago, S Ivanov A course in etric geoetry Graduate Studies in Matheatics, 33 Aerican Matheatical Society, Providence, RI, 001 [] M Groov Metric structures for Rieannian and non-rieannian spaces Modern Birkhuser Classics Birkhuser Boston, Inc, Boston, MA, 007 [3] H Lee, M Chung, H Kang, B Ki, D Lee Coputing the Shape of Brain Networks using Graph Filtration and Groov-Hausdorff Metric University of Wisconsin, Madison, WI 53705, USA, 011 [4] F Meoli On the use of Groov-Hausdorff Distances for Shape Coparison Eurographics Syposiu on Point-Based Graphics, Stanford, CA, USA,

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