Sp ectral Partitioni ng Works:
Planar graphs and nite element meshes
Preliminary Draft
y
Daniel A. Spielman Shang-Hua Teng
February 13, 1996
Abstract
Sp ectral partitioning metho ds use the Fiedler vector|the eigenvector of the second-
smallest eigenvalue of the Laplacian matrix|to nd a small separator of a graph. These
metho ds are imp ortant comp onents of many scienti c numerical algorithms and have b een
demonstrated by exp erimenttowork extremely well. In this pap er, we show that sp ectral
partitioning metho ds work well on b ounded-degree planar graphs and nite element meshes|
the classes of graphs to which they are usually applied. While naive sp ectral bisection do es
not necessarily work, we prove that sp ectral partitioning techniques can b e used to pro duce
p
n for b ounded-degree planar separators whose ratio of vertices removed to edges cut is O
1=d
graphs and two-dimensional meshes and O n for well-shap ed d-dimensional meshes. The
heart of our analysis is an upp er b ound on the second-smallest eigenvalues of the Laplacian
matrices of these graphs.
1. Intro duction
Sp ectral partitioning has b ecome one of the most successful heuristics for partitioning graphs
and matrices. It is used in many scienti c numerical applications, such as mapping nite el-
ement calculations on parallel machines [Sim91, Wil90], solving sparse linear systems [PSW92 ],
and partitioning for domain decomp osition [CR87, CS93 ]. It is also used in VLSI circuit de-
sign and simulation [CSZ93 , HK92, AK95]. Substantial exp erimental work has demonstrated
that sp ectral metho ds nd go o d partitions of the graphs and matrices that arise in many ap-
plications [BS92, HL92 , HL93 , PSL90 , Sim91, Wil90]. However, the quality of the partition that
Computer Science Division, U.C. Berkeley, CA 94720, [email protected]. Supp orted in part byan
NSF p ostdo c. After July 1996, Department of Mathematics, M.I.T.
y
Department of Computer Science, University of Minnesota, Minneap olis, MN 55455, [email protected]. Sup-
p orted in part by an NSF CAREER award CCR-9502540 and an Intel research grant. Part of the work was done
while visiting XeroxPARC. 1
these metho ds should pro duce has so far eluded precise analysis. In this pap er, we will prove that
sp ectral partitioning metho ds give go o d separators for the graphs to which they are usually applied.
The size of the separator pro duced by sp ectral metho ds can b e related to the Fiedler value|the
second smallest eigenvalue of the Laplacian|of the adjacency structure to which they are applied.
2=d
,we By showing that well-shap ed meshes in d dimensions have Fiedler value at most O 1=n
1 1=d
. show that sp ectral metho ds can b e used to nd bisectors of these graphs of size at most O n
While a small Fiedler value do es not immediately imply that there is a cut along the Fiedler vector
that is a balanced separator, it do es mean that there is a cut whose ratio of vertices separated to
1=d
. By removing the vertices separated by this cut, computing a Fiedler vector of edges cut is O n
1 1=d
edges. In particular, the new graph, and iterating as necessary, one can nd a bisector of O n
we prove that b ounded-degree planar graphs have Fiedler value at most O 1=n , which implies that
p
sp ectral techniques can b e used to nd bisectors of size at most O n in these graphs. These
b ounds are the b est p ossible for well-shap ed meshes and planar graphs.
1.1. History
The sp ectral metho d of graph partitioning was b orn in the works of Donath and Ho man [DH72,
DH73 ] who rst suggested using the eigenvectors of adjacency matrices of graphs to nd partitions.
Fiedler [Fie73, Fie75a , Fie75b] asso ciated the second-smallest eigenvalue of the Laplacian of a graph
with its connectivity and suggested partitioning by splitting vertices according to their value in the
corresp onding eigenvector. Thus, we call this eigenvalue the Fied ler value and a corresp onding
vector a Fied ler vector.
A few years later, Barnes and Ho man [Bar82 , BH84] used linear programming in combination
with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar vein, Bop-
pana [Bop87 ] analyzed eigenvector techniques in conjunction with convex programming. However,
the use of linear and convex programming made these techniques impractical for most applications.
By recognizing a relation b etween the Fiedler value and the Cheeger constant [Che70] of con-
tinuous manifolds, Alon [Alo86] and Sinclair and Jerrum [SJ89 ] demonstrated that if the Fiedler
value of a graph is small, then directly partitioning the graph according to the values of vertices
in the eigenvector will pro duce a cut with a go o d ratio of cut edges to separated vertices see
also [AM85, Fil91, DS91 , Mih89, Moh89 ] . Around the same time, improvements in algorithms
for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation
of eigenvectors practical [PSS82 , Sim91]. In the next few years, a wealth of exp erimental work
demonstrated that sp ectral partitioning metho ds work well on graphs that usually arise in prac-
tice [BS92, HL92 , PSL90 , Sim91, Wil90]. Sp ectral partitioning b ecame a standard to ol for mesh
partitioning in many areas [HL93 ]. Still, researchers were unable to prove that sp ectral partition-
ing techniques would work well on the graphs encountered in practice. This failure is partially
explained by results of Guattery and Miller [GM95] demonstrating that naive applications of sp ec-
tral partitioning, such as sp ectral bisection, will fail miserably on some graphs that could con-
ceivably arise in practice. By b ounding the Fiedler values of the graphs of interest in scienti c
applications|b ounded-degree planar graphs and well-shap ed meshes|we are able to show that 2
sp ectral partitioning metho ds will successfully nd go o d partitions of these graphs.
In a related line of research, algorithms were develop ed along with pro ofs that they will always
nd small separators in various families of graphs. The seminal work in this area was that of
Lipton and Tarjan [LT79 ], who constructed a linear-time algorithm that pro duces a 1=3-separator
p
of 8n no des in any n-no de planar graph. Their result improved a theorem of Ungar [Ung51 ] which
p
demonstrated that every planar graph has a separator of size O n log n . Gilb ert, Hutchinson, and
Tarjan [GHT84] extended these results to show that every graph of genus at most g has a separator
p
of size O gn . Another generalization was obtained by Alon, Seymour, and Thomas [AST90],
p
3=2
who showed that graphs that do not haveanh-clique minor have separators of O h n no des.
3=2
Plotkin, Rao, and Smith [PRS94] reduced the dep endency on h from h to h. Using geometric
techniques, Miller, Teng, Thurston, and Vavasis [MT90, MTTV96a , MTTV96b , MTV91 , MV91,
Ten91 ] extended the planar separator theorem to graphs emb edded in higher dimensions and showed
d 1 1=d
has a 1= d + 2 -separator of size O n . Using multicommo dity that every well-shap ed mesh in R
ow, Leighton and Rao [LR88 ] designed a partitioning metho d guaranteed to return a cut whose
ratio of cut size to vertices separated is within logarithmic factors of optimal. While sp ectral
metho ds have b een favored in practice, they lacked a pro of of e ectiveness.
1.2. Outline of pap er
In Section 2, weintro duce the concept of a graph partition, review some facts from linear algebra
that we require, and describ e the class of sp ectral partitioning metho ds.
In Section 3, we prove the embedding lemma, which relates the quality of geometric emb eddings
of a graph with its Fiedler value. We then show using the main result of Section 4 that every
planar graph has a \nice" emb edding as a collection of spherical caps on the surface of a unit
sphere in three dimensions. By applying the emb edding lemma to this emb edding, we prove that
the Fiedler value of every b ounded-degree planar graph is O 1=n .
In Section 4, we show that, for almost every arrangement of spherical caps on the unit sphere in
d
R , there is a sphere-preserving map that transforms the caps so that the center of the sphere is the
centroid of their centers. It is this fact that enables us to nd nice emb eddings of planar graphs.
In Sections 5 and 6, we extend our sp ectral planar separator theorem to the class of overlap
graphs of k -ply neighb orho o d systems emb edded in any xed dimension. This extension enables us
1=d
to show that the sp ectral metho d nds cuts of ratio O 1=n for k -nearest neighb or graphs and
well-shap ed nite element meshes.
In Section 7, we present an elementary pro of that from anyvector p erp endicular to the all-ones
vector with small Rayleigh quotient, one can obtain a cut of small ratio.
In Section 8, we extend the results of Guattery and Miller to show that our results are essentially
the b est p ossible given currentcharacterizations of well-shap ed meshes. We present natural families
of graphs for which Fiedler vectors can b e used to nd cuts of go o d ratio, but not go o d balance.
We discuss why these graphs exist and why they might not app ear in practice. 3
2. Intro duction to Sp ectral Partitioning
In this section, we de ne the sp ectral partitioning metho d and intro duce the terminology that we
will use throughout the pap er.
2.1. Graph Partitioning
Throughout this pap er, G = V; E will b e a connected, undirected graph on n vertices.
A partition of a graph G is a division of its vertices into two disjoint subsets, A and A. Without
loss of generality,we can assume that jAj A . Let E A; A b e the set of edges with one endp oint
in A and the other in A. The cut size of the partition A; A is simply jE A; A j. The cut ratio,or
simply the ratio of the cut, denoted A; A , is equal to the ratio of the size of the cut to the size
of A, namely,
jE A; A j
A; A = :
min jAj; jAj
The isop erimetric numb er of a graph, which measures how go o d a ratio cut one can hop e to nd,
is de ned to b e
E A; A
G = min :
jAjn=2
jAj
In Section 7, we describ e a relation b etween the isop erimetric numb er of a graph and its Fiedler
value.
A partition is a bisection of G if A and A di er in size by at most 1. For in the range
0 < 1=2, a partition is called a -separator if min jAj; jAj n.We use the word cut to refer
to a partition separating anynumber of vertices and reserve the word separator for partitions that
are -separators for some >0. Given an algorithm that can nd cuts of ratio in G and its
subgraphs, we can nd a bisector of G of size O n see Lemma 22 .
2.2. Laplacians and Fiedler Vectors
The adjacency matrix, A G , of a graph G is the n n matrix whose i; j -th entry is 1 if i; j 2 E
and 0 otherwise. The diagonal entries are de ned to b e 0. Let D b e the n n diagonal matrix with
entries D = d , where d is the degree of the ith vertex of G. The Laplacian, L G , of the graph
i;i i i
G is de ned to b e L G =D A.
Let M be an n n matrix. An n-dimensional vector ~x is an eigenvector of M if there is a scalar
such that M~x = ~x. is the eigenvalue of M corresp onding to the eigenvector ~x.IfMis a real
symmetric matrix, then all of its n eigenvalues are real. The only matrices we consider in this pap er
will b e the Laplacians of graphs. Notice that the all-ones vector is an eigenvector of any Laplacian
matrix and that its asso ciated eigenvalue is 0. Because Laplacian matrices are p ositive semide nite,
all the other eigenvalues must b e non-negative. We will fo cus on the second smallest eigenvalue,
, of the Laplacian and an asso ciated eigenvector ~u. Fiedler called this eigenvalue the \algebraic
2
connectivity of a graph", so we will call it the Fied ler value and an asso ciated eigenvector a Fied ler
vector. 4
The following prop erties of Fiedler values and vectors play an imp ortant role in this pap er:
The Fiedler value of a graph is greater than zero if and only if the graph is connected.
A Fiedler vector ~u = u ; :::; u satis es
1 n
n
X
u =0;
i
i=1
b ecause all-ones vector is an eigenvector of the Laplacian and the eigenvectors of a symmetric
matrix are orthogonal.
The Fiedler value, ,of G satis es
2
T
~x L G ~x
; = min
2
T
~x? 1;1;:::;1
~x ~x
with the minimum o ccurring only when ~x is a Fiedler vector.
n
For anyvector ~x 2 R ,wehave
X
T 2
~x L G ~x= x x :
i j
i;j 2E
Let M b e a symmetric n n matrix and ~x be an n-dimensional vector. Then, the Rayleigh
quotient of ~x with resp ect to M is
T
~x M~x
:
T
~x ~x
For pro ofs of these statements and many other fascinating facts ab out the eigenvalues and
eigenvectors of graphs consult one of [Str88, Moh88 , CDS90 , Big93].
The Fiedler value, , of a graph is closely linked to its isop erimetric numb er. If G is a graph
2
on more than three no des, then one can show [AM85, Moh89 , SJ88 ]
q
2
G 2d :
2 2
2
In Section 7, we will fo cus on the second inequality.
2.3. Sp ectral Partitioning Metho ds
Let ~u = u; :::; u b e a Fiedler vector of the Laplacian of a graph G. The idea of sp ectral
1 n
partitioning is to nd a splitting value s and partition the vertices of G into the set of i such that
u >sand the set such that u s. We call such a partition a Fied ler cut. There are several
i i
p opular choices for the splitting value s:
bisection: s is the median of fu ; :::; u g.
1 n 5
ratio cut: s is the value that gives the b est ratio cut.
sign cut: s is equal to 0.
gap cut: s is a value in the largest gap in the sorted list of Fiedler vector comp onents.
Other variations have b een prop osed.
In this pap er, we will analyze the sp ectral metho d that uses the splitting value that achieves
the b est ratio cut. We will show that, for b ounded-degree planar graphs and well-shap ed meshes, it
always nds a go o d ratio cut. In fact, it is not necessary to use a Fiedler vector; an approximation
will suce. If a vector ~x that is orthorgonal to the all-ones vector has a small Rayleigh quotient
with resp ect to the Laplacian of G, then ~x can b e used to nd a go o d ratio cut of G.
Guattery and Miller [GM95] showed that there exist b ounded-degree planar graphs on n vertices
with constant-size separators for which sp ectral bisection and sp ectral sign cuts give separators that
cut n=3 edges.
p
We will show that planar graphs have Fiedler cuts of ratio O 1= n . By Lemma 22, our result
p
implies that a bisector of size O n can b e found by rep eatly nding Fiedler cuts. In Section 8,
we extend the results of Guattery and Miller to show that this rep eated application of Fiedler cuts
is necessary,even for some quite natural graphs. We will show that, for any constant in the range
0 < 1=2, there are natural families of well-shap ed two-dimensional meshes that have no Fiedler
cut of small ratio that is also a -separator. We discuss why these graphs exist as well as why they
might fail to arise in practice. Our explanation is not entirely satisfactory and the problem remains
of nding a b etter characterization of the graphs that do arise in practice as well as those for which
p
there is a Fiedler cut that pro duces a -separator of size O n .
3. The eigenvalues of planar graphs
In this section, we will prove that the Fiedler value of every b ounded-degree planar graph is O 1=n .
Our pro of establishes and exploits a connection b etween the Fiedler value and geometric emb eddings
of graphs. We obtain the eigenvalue b ound by demonstrating that every planar graph has a \nice"
emb edding in Euclidean space.
p
A b ound of O 1= n can b e placed on the Fiedler value of any planar graph by combining the
planar separator theorem of Lipton and Tarjan [LT79 ] with the fact that =2 G . Bounds of
2
O 1=n on the Fiedler values of planar graphs were previously known for graphs such as regular grids
[PSL90 ], quasi-uniform graphs [GK95 ], and b ounded-degree trees. Bounds on the Fiedler values of
regular grids and quasi-uniform graphs essentially follow from the fact that the diameters of these
graphs are large see [Chu89] . Bounds on trees can b e obtained from the fact that every b ounded-
degree tree has a -separator of size 1 for some constant in the range 0 < <1=2 that dep ends
only on the degree. However, in order to estimate the Fiedler value of general b ounded-degree
planar graphs and well-shap ed meshes, we need di erent techniques.
p
T
x x.We relate We denote the standard l norm of a vector ~x in Euclidean space by k~xk =
2
the qualityofanemb edding of a graph in Euclidean space with its Fiedler value by the following
lemma: 6
Lemma 1 emb edding lemma . Let G = V; E beagraph. Then , the Fied ler value of G,is
2
given by
P
2
k~v ~v k
i j
i;j 2E
= min ;
P
2
n
2
k~v k
i
i=1
n
where the minimum is taken over vectors f~v ;:::;~v gR such that
1 n
n
X
~
~v = 0;
i
i=1
~
where 0 denotes the al l-zeroes vector.
n m
Remark 2. While we state this lemma for vectors in R , it applies equal ly wel l for vectors in R
for any m 1.
Pro of: Because the all-ones vector is the eigenvector of L G corresp onding to the eigenvalue 0,
can b e characterized by
2
P
2
x x
i j
i;j 2E
; = min
P
2
n
2
x
i
i=1
P
n
where the minimum is taken over real x 's such that x = 0. The minimum is achieved precisely
i i
i=1
when x ;:::;x is an eigenvector.
1 n
The emb edding lemmanow follows from a comp onent-wise application of this fact. Write ~v as
i
P
n
~
v ;:::;v . Then, for all f~v ;:::;~v g such that ~v = 0, wehave
i;1 i;n 1 n i
i=1
P P P
n
2 2
k~v ~v k v v
i j i;k j;k
i;j 2E i;j 2E
k =1
=
P P P
n n n
2
2
k~v k v
i
i=1 i=1 k =1 i;k
P P
n
2
v v
i;k j;k
i;j 2E
k =1
= :
P P
n n
2
v
k =1 i=1 i;k
But, for each k ,
P
2
v v
i;k j;k
i;j 2E
;
P
2
n
2
v
i=1 i;k
so
P P
n
2
v v
i;k j;k
i;j 2E
k =1
P P
2
n n
2
v
k =1 i=1 i;k
P P
this follows from the fact that x = y min x =y , for x ;y > 0 . 2
i i i i i i i
i i
Our metho d of nding a go o d geometric emb edding of a planar graph is similar to the way
in which Miller, Teng, Thurston, and Vavasis [MTTV96a ] directly nd go o d separators of planar
graphs.
We rst nd an emb edding of the graph on the plane by using the \kissing disk" emb edding of
Ko eb e, Andreev, and Thurston [Ko e36, And70a , And70b , Thu88 ]: 7
Theorem 3 Ko eb e-Andreev-Thurston . Let G be a planar graph with vertex set V =
f1;:::;ng and edge set E . Then, there exists a set of disks fD ;:::;D g in the plane with disjoint
1 n
interiors such that D touches D if and only if i; j 2 E .
i j
Suchanemb edding is called a kissing disk emb edding of G.
The analogue of a disk on the sphere is a cap.Acap is given by the intersection of a half-space
with the sphere, and its b oundary is a circle. We de ne kissing caps analogously with kissing disks.
Following [MTTV96a ], we use stereographic pro jection to map the kissing disk emb edding of the
graph on the plane to a kissing cap emb edding on the sphere See Section 4 for more information
on stereographic pro jection . In Theorem 9, we will show that we can nd a sphere preserving map
map that sends the centroid also known as the center of gravity or center of mass of the centers
of the caps to the center of the sphere. Using this theorem, we can b ound the eigenvalues of planar
graphs:
Theorem 4. Let G be a planar graph on n nodes of degree at most d. Then, the Fied ler value of
G is at most
8d
:
n
p
Accordingly, G has a Fied ler cut of ratio O 1= n , and one can iterate Fied ler cuts to nd a
p
bisector of size O n .
Pro of: By Theorem 3 and Theorem 9, there is a representation of G by kissing caps on the unit
sphere so that the centroid of the centers of the caps is the center of the sphere. Let ~v ;:::;~v be
1 n
P
n
~
the centers of these caps. Make the center of the sphere the origin, so that ~v = 0.
i
i=1
Let r ;:::;r b e the radii of the caps. If cap i kisses cap j , then the edge from ~v to ~v will have
1 n i j
2 2 2
length at most r + r . As this is at most 2 r + r , we can divide the contribution of this edge
i j
i j
between the two caps. That is, we write
n
X X
2 2
k~v ~v k 2d r :
i j
i
i=1
i;j 2E
But, b ecause the caps do not overlap,
n
X
2
r 4:
i
i=1
Moreover, k~v k = 1 b ecause the vectors are on the unit sphere.
i
Applying the emb edding lemma, we nd that the Fiedler value of G is at most
P
2
k~v ~v k
8d
i j
i;j 2E
P :
n
2
n kv~ k
i
i=1
Given the b ound on the Fiedler value, the ratio achievable by a Fiedler cut follows immediately
from Theorem 21 and the corresp onding bisector size follows Lemma 22. 2 8
Remark 5. One can prove a slightly weaker result by using a result of Mil ler, Teng, Thurston,
and Vavasis [MTTV96a] to nd a circle-preserving map that makes the center of the spherea
centerpoint [Ede87] of the images of particular points in the caps. If the center of the sphereisa
centerpoint, then the centroid must be far away from at leastaconstant fraction of the centers of
the caps. Thus, the numerator of the Rayleigh quotient wil l be the same as in Theorem 4, and the
denominator wil l be n .
Remark 6. The embedding lemma can be viewed as a semi-de nite relaxation of an integer program
for minimum balanced cut. The integer program would be
X
2
x x min
i j
i;j 2E
n
X
s:t: x =0; and
i
i=1
x 2f1g:
i
However, unlike the semi-de nite relaxation of MaxCut usedbyGoemans and Wil liamson [GW94],
we do not know if our relaxation provides a constant factor approximation of the optimum.
4. Sphere-preserving maps
P
n
d 2 d
Let B b e the unit ball in d dimensions: f x ;:::;x j x 1g. Let S denote the sphere
1 d
i=1 i
d d d
de ning the surface of B . This section is concerned with sphere-preserving maps from S to S .
d d
A sphere-preserving map from S to S is a continuous function that sends every sphere of lower
d d d
dimension contained in S to a sphere in S and such that every sphere in S has a pre-image
under the map that is also a sphere. Familiar sphere-preserving maps include rotations and the
map that sends each p oint to its antip o de.
We will make use of a slightly larger family of sphere-preserving maps. We obtain this family
d
by rst considering sphere-preserving maps b etween the sphere and the plane. Let H b e the
d d d
hyp erplane tangenttoS at 1; 0;:::;0 . One can map H to S by stereographic projection:
d d
: H ! S by
d
z = the intersection of S with the line connecting z to 1; 0;:::;0 :
1 d d d d
Similarly, one de nes a map : S ! H that sends a p oint z 2 S to the intersection of H
1
with the line through z and 1; 0;:::;0 . Note that is not well-de ned at 1; 0;:::;0 . To
d 1
x this, we add the p oint 1 to the hyp erplane H , and de ne 1; 0;:::;0 = 1 as well as
1 = 1;0;:::;0 .
d
For any p oint 2 S ,we de ne to b e the stereographic pro jection from the plane p erp en-
d 1
dicular to S at , and we let b e its inverse so, 1 = . One can show that the maps
1
and are sphere-preserving maps see [HCV52] or [MTTV96a] for a pro of .
9
Sphere-preserving maps in the plane include rigid motions of the plane as well as dilations and
other mobius transformations . We will obtain sphere-preserving maps in the sphere by applying
a pro jection onto a plane, then applying a dilation of the plane, and then mapping backby stere-
d a
ographic pro jection. Thus, for 2 S and a 0, we de ne D to b e the map that dilates the
d a
hyp erplane p erp endicular to S at by a factor of a note that D 1 =1 . For example,
a
D : 1;x ;:::;x 7! 1;ax ;:::;ax :
2 d 2 d
1;0;:::;0
As the comp osition of sphere-preserving maps is again a sphere-preserving map, we can now de ne
the sphere-preserving maps that we will use. For any such that k k < 1, de ne f z by
1 k k
1
f z = D z :
=k k
=k k =k k
2
It is routine to verify that f is continuous. We wish to extend the de nition of f to on S ,even
though the resulting maps will not b e continuous. For k k =1,we de ne
if z = , and
f z =
otherwise.
d
We will now examine the e ect of the maps f on arrangements of spherical caps on S . Recall
d d
that a spherical cap on S is a connected region of S whose b oundary is a d 1 -dimensional
sphere. Thus, the image of a cap under a map f is determined by the image of its b oundary
d d
along with a p oint in its interior. For a cap C on S , let p C denote the p ointonS that is
the center of C i.e., the p oint inside C that is equidistant from its b oundary . Wewant to show
d d
that, for any arrangement of caps fC ;:::;C g on S , there is an 2 S so that the centroid of
1 n
fp f C ;:::;p f C g is the origin. But rst, wemust exclude some degenerate cases:
1 n
d
De nition 7. An arrangement of caps fC ;:::;C g in S is wel l-behaved if there is no p oint that
1 n
b elongs to at least half of the caps.
Remark 8. Al l of the arrangements of caps obtainedfrom graphs contained in the other sections
of this paper are wel l-behaved. Otherwise, the inducedgraphs would have cliques on half of their
vertices and no smal l separators.
d
Theorem 9. For any wel l-behaved arrangement of caps fC ;:::;C g in S , thereisan so that
1 n
k k < 1 and
P
n
p f C
i
i=1
~
= 0:
n 10
Pro of: Consider the map from to the centroid of fp f C ;:::;p f C g. Wewantto
1 n
~
show that 0 has a preimage under this map. This would b e easier if the map were continuous, but
it is not continuous for k k =1:as crosses the b oundary of C , p f C jumps from one side
i i
of the sphere to the other.
To x this problem, we construct a slightly mo di ed map that is continuous. Because the set of
caps is well-b ehaved, we can cho ose an >0 so that, for all such that k k1 , most of the
caps ff C ;:::;f C g are entirely contained within the ball of radius 1=2n around =k k.In
1 n
particular, this implies that f do es not map the centroid of the centers of the caps to the origin.
d
For 2 B ,wenow de ne the map
P
n
w C ; f p C
i i
i=1
= ;
n
where the weight function w is given by
2 d ; C = if d ; C 2 , and
w C; =
1 otherwise,
where by d ; C , we mean the greatest distance from to a p oint in the cap C for example, if
2 C , then d ; C =1+k k . Wehavechosen w to b e a continuous function of that go es to
zero as approaches the b oundary of a cap; so, is also a continuous function.
d
From the fact that fC ;:::;C g is well-b ehaved, it is easy to verify that, for 2 S , lies
1 n
~
on the line connecting 0to and is closer to than it is to . By combining this fact with some
~
elementary algebraic top ology,we can use Lemma 10 to show that there is an such that =0.
By our choice of , k k < 1 , so all of the terms w ; C are 1, which implies that f is the
i
map that wewere lo oking for. 2
d d d
Lemma 10. Let : B ! B beacontinuous function so that, for 2 S , lies on the line
d
~
connecting with 0 and is closer to than it is to . Then, there exists an 2 B such that
~
=0.
d
~
Pro of: Assume, bywayofcontradiction, that there is no p oint 2 B such that =0.
n o
d d
~
Now, consider the map b , where b : B 0 ! S by
b z = z=kzk:
d d d
Since b is a continuous map, b is a continuous map of B onto S that is the identityon S .
d d
Then z 7! b z is a map from B onto S that has no xed p oint. This contradicts Brouwer's
d d
Fixed Point Theorem, whichsays that every continuous map from B into B has a xed p oint.
2
d
Wehave shown that, for most collections of balls in H , there is a sphere preserving map from
d d
H to S such that the centroid of the centers of the caps is the origin. Wenow show that one
d d
can nd such a map by p erforming a rigid motion of H followed by a dilation of H followed by
stereographic pro jection. 11
d
De nition 11. An arrangement of balls fD ;:::;D g in H is wel l-behaved if there is no p oint
1 n
that b elongs to at least half of the balls.
d
Theorem 12. Let fD ;:::;D g be a wel l-behav edcol lection of bal ls. Then, thereisapoint x 2 H
1 n
and an a>0 such that the spherepreserving map
g : z 7! a z x
x;a
sends the bal ls to a col lection of caps, the centroid of whose centers is the origin.
d
1
Pro of: [sketch] For an 2 S , consider the map g followed by a rotation of the
; 1 k k
sphere that sends 1; 0;:::;0 to . As we did in the pro of of Theorem 9, we can construct a
d
continuous map from toaweighted centroid of the centers of the caps, which for 2 S sends
~
to a p oint on the line segmentbetween and 0. We can then apply Lemma 10 to prove that there
1
is some map such that the map g sends the centroid of the centers of the caps to the
; 1 k k
origin. 2
5. The Sp ectra of k -Nearest Neighb or Graphs
We extend our sp ectral planar separator theorem to graphs emb edded in three and more dimensions.
We show that Fiedler cuts of small ratio can b e found in -overlap graphs of k -ply neighborhood
systems. One corollary of this extension is that the sp ectral metho d nds small ratio cuts for
k -nearest neighb or graphs and well-shap ed nite element meshes in any xed dimension. In this
section, we analyze intersection graphs and nearest neighb or graphs. Results on overlap graphs and
well-shap ed meshes will b e given in the next section.
In this section and the next, we will use the following notation: We use capital letters to denote
d d 0 d+1
balls in R .IfAis a ball in R , then we will use A to denote its image on the sphere S under
stereographic pro jection. If is a p ositive real and A is a ball of radius r , then A is the ball
0
with the same center as A and radius r . Similarly,if A is a spherical cap of spherical radius r ,
0 0
then A is the spherical cap with the same center as A and radius r . Let V b e the volume of
d
a unit d-dimensional ball. Let A b e the surface volume of a unit d-dimensional ball.
d
5.1. Intersection Graphs
The graphs that we consider are de ned by neighb orho o d systems. A neighborhood system is a set of
closed balls in Euclidean space. A k -ply neighb orho o d system is one in which no p ointiscontained
in the interior of more than k of the balls. Given a neighb orho o d system, = fB ;:::;B g,we
1 n
de ne the intersection graph of to b e the undirected graph with vertex set V = fB ;:::;B g and
1 n
edge set
E = f B ;B :B \B 6=;g :
i j i j 12
For example, the Ko eb e-Andreev-Thurston emb edding theorem says that every planar graph is
isomorphic to the intersection graph of some 1-ply neighb orho o d system in two dimensions.
d
Let P = fp ;:::;p g beapoint set in R .For each p 2 P , let N p b e the set of k p oints
1 n i k i
closest to p in P if there are ties, break them arbitrarily . A k -nearest neighbor graph of P is a
i
graph with vertex set fp ;:::;p g and edge set
1 n
E = f p ;p :p 2N p or p 2N p g:
i j i k j j k i
d
Miller et. al. [MTTV96b] show that every k -nearest neighb or graph in R is a subgraph of an inter-
section graph of a k -ply neighb orho o d system, where is the kissing number in d dimensions|the
d d
d
maximum numb er of nonoverlapping unit balls in R that can b e arranged so that they all toucha
central unit ball [CS88]. Moreover, the maximum degree of a k -nearest neighb or graph is b ounded
by k .
d
5.2. A Sp ectral Bound
d
Theorem 13. Let G be a subgraph of an intersection graph of a k -ply neighborhood system in R
2=d
such that the maximum degreeof G is . Then, the Fied ler value of L G is boundedbyc k=n ,
d
2=d
where c =2 A =V .
d d+1 d
Pro of: Let =fB ; :::; B g b e the k -ply neighb orho o d system of which G is the intersection
1 n
d d
graph. By Theorem 9, there exits a sphere-preserving map :R !S such that the centroid of
the centers of the images of the B 's is the center of the sphere.
i
0 0
Let = fB ; :::; B g b e the images of the balls in under . Then, the balls in also
i n
0 d 0
form a k -ply system. Let r b e the radius of B . Because V r volume B ,
i d
i i i
n n
X X
d 0
V r volume B kA : 1
d d+1
i i
i=1 i=1
By Lemma 1,
P
n
2
2r
i=1 i
L G
2
n
2=d 1 2=d
kA =V n
d+1 d
2
n
!
2=d
2=d
A k
d+1
: 2
V n
d
Note that the second inequality follows from 1 . 2
The next two corollaries follow from Theorem 13, Theorem 21, and Lemma 22.
d
Corollary 14. The Fied ler value of a k -nearest neighbor graph of n points in R is boundedby
1+2=d 2=d 1+1=d 1=d
O k =n . Therefore, G has a Fied ler cut of ratio O k =n , and one can repeatedly
1+1=d 1 1=d
take Fied ler cuts to nd a bisector of size O k n . 13
d
Corollary 15. Let G be a subgraph of an intersection graph of a k -ply neighborhood system in R
1+1=d 1=d
, and one can whose maximum degreeis . Then, G has a Fied ler cut of ratio O =n
1+1=d 1 1=d
. iterate Fied ler cuts to obtain a bisector of size O n
6. The Sp ectra of Well-Shap ed Meshes
One of the main applications of the sp ectral metho d is the partitioning of meshes for parallel
numerical simulations. Many exp erments demonstrate the e ectiveness of this metho d [BS92, HL92,
HL93 , PSL90 , Sim91, Wil90]. In this section, we explain why the sp ectral metho d nds such good
partitions of well-shap ed meshes.
6.1. Well-Shap ed Meshes
Most numerical metho ds work by approximating continuous problems with discrete problems on
nite structures whose solutions can b e eciently computed. The nite structure used is often
called a mesh. Many such metho ds have b een develop ed and applied to imp ortant problems in
mechanics and physics.
Most of these numerical metho ds can b e classi ed as equation based methods e.g., the nite
element, nite di erence, and nite volume metho ds or particle methods e.g., the N-b o dy sim-
ulation metho d . However di erent the particular metho ds may b e, a basic principle is common
to all|accuracy of approximation is ensured by using meshes that satisfy certain numerical and
geometric constraints. Meshes that satisfy these constraints are said to b e wel l-shaped.
To motivate our sp ectral analysis of well-shap ed meshes, we review the conditions required of
nite element and nite di erence meshes. More detailed discussions can b e found in several b o oks
and pap ers for example, see [SF73, Joh92 , BEG94, BE92 , Fri72] . Background material on the
particle metho d can b e found in [BH86, GR87 , HE81 , Zha87 ].
The nite element method approximates a continuous problem by sub dividing the domain a
d
subset of R of the problem into a mesh of p olyhedral elements and then approximates the con-
tinuous function by piecewise p olynomial functions on the elements. A common choice for an
elementisa d-dimensional simplex. Accordingly,ad-dimensional nite element mesh is a d-
dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared
faces [BEG94, BE92 , MT90].
The computation graph asso ciated with each simplicial complex is often its 1-skeleton or the
1-skeleton of its geometric dual as used in the nite volume metho d . In the nite element metho d,
a linear system is de ned over a mesh, with variables representing physical quantities at the no des.
The nonzero structure of the co ecient matrix of such a linear system is exactly the adjacency
structure of the 1-skeleton of the simplicial complex.
To ensure accuracy, in addition to the conditions that a mesh must conform to the b oundaries
of the region and b e ne enough, each individual element of the mesh must b e wel l-shaped. A
common shap e criterion for the nite element metho d is that the angles of each element are not
to o small, or the aspect ratio of each element is b ounded [BA76, BEG94 , Fri72]. Other numerical 14
formulations require slightly di erent conditions. For example, the controlled volume formulation
[Nic92, MTTW95 ] using a Voronoi diagram requires that the radius aspect ratio the ratio of the
circumscrib ed radius to the shortest edge length of an element in the dual Delaunay diagram is
b ounded.
The nite di erence metho d also uses a discrete structure, a nite di erence mesh, to approxi-
mate a continuous problem. Finite di erence meshes are often pro duced by inserting a uniform grid
2 3
from R or R into the domain via a b oundary-matching conformal mapping. Notice that, unlike
a nite element mesh, a nite di erence mesh need not b e a collection of simplices or elements,
so we can not analyze it as we do a triangulation. In general, the derivative of the conformal
transformation must vary gradually with resp ect to the mesh size in order to pro duce go o d results
See, for example [TWM85 ] . This means that the mesh will probably satisfy a density condition
[BB87, MV91].
d
Let G b e an undirected graph and let b e an emb edding of its no des in R .Wesay is an
emb edding of density if the following inequality holds for all vertices v in G: Let u b e the no de
closest to v . Let w b e the no de farthest from v that is connected to v by an edge. Then
jj w v jj
:
jj u v jj
d d
In general, G is an -density graph in R if there exists an emb edding of G in R with density
.
6.2. Mo deling Well-Shap ed Meshes
We will use the overlap graph to mo del well-shap ed meshes Miller et al [MTTV96a] . An overlap
graph is based on a k -ply neighb orho o d system. The neighb orho o d system and a parameter, 1,
de ne an overlap graph: Let 1, and let = fB ;:::;B g be a k -ply neighb orho o d system in
1 n
d
R . The -overlap graph of is the graph with vertex set fB ;:::;B g and edge set
1 n
f B ;B : B \ B 6=; and B \ B 6= ; g;
i j i j i j
where by B ,we mean the ball whose center is the same as the center of B and whose radius is
larger bya multiplicative factor of .
Overlap graphs are go o d mo dels of well-shap ed meshes b ecause eachwell-shap ed mesh in two,
three, or higher dimensions is a bounded-degree subgraph of some overlap graph for suitable choices
of the parameters and k . For example,
d
Let M b e a nite element mesh emb edded in R in whichevery element has asp ect ratio
b ounded by a. Then, there is a constant dep ending only on d and a so that the 1-skeleton
of M is a subgraph of an -overlap graph of a 1-ply neighb orho o d system. Moreover its
maximum degree is b ounded by a constant that also dep ends only on d and a [MTTV96a] .
d
in which the radius asp ect Let M be a Voronoi diagram from a nite volume metho d in R
ratio of its dual Delaunay diagram is b ounded by a. Then there is a constant dep ending
only on d and a so that the dual Delaunay diagram is an -density graph [MTTW95] . 15
d
If G is an -density graph in R , then the maximum degree of G is b ounded by a constant
dep ending only on and d; and, G is a subgraph of an -overlap graph of a 1-ply neighborhood
system [MV91, MTTV96a ] .
The computation/communication graph used in hierarchical N-b o dy simulation metho ds such
as the Barnes-Hut's treeco de metho d [BH86] and the fast-multip ole metho d [GR87] is a
subgraph of an -overlap graph of an O log n -ply neighb orho o d system [Ten96] .
6.3. Spherical Emb eddings of Overlap Graphs
In this section, we show that an -overlap graph is a subgraph of the intersection graph obtained by
pro jecting its neighborhoods onto the sphere and then dilating eachbyan O factor. By cho osing
the prop er pro jection, we are able to use this fact to b ound the eigenvalues of these graphs.
d
Theorem 16. Let 1.Let A and B bebal ls in R such that
A \ B 6= ; and A \ B 6= ; :
0 0
Then, + + A touches + + B .
Our pro of uses two lemmas that handle orthogonal sp ecial cases.
r 2α r A C
S
Figure 1:
d
Lemma 17. Let A and C bebal ls in R equidistant from the origin and having the same radius.
0 0 d+1
Let A and C be their images under stereographic projection onto S .If Atouches C , then
0 0
touches =2 C . =2 A 16
Pro of: Let r b e the radius of A and C . Because A touches C , the centers of A and C are
at distance at most 2 r from each other.
Let S b e the sphere centered at the origin that passes through the centers of A and C . The
geo desic arc b etween the centers of A and C on S has length at most 2 r =2. The p ortion of this
arc that lies in the interior of A has length at least r See Figure 1 for a two dimensional example .
Since stereographic pro jection preserves the relations b etween the intersections of A and C with S ,
0 0
=2 A will touch =2 C . 2
a b
Figure 2: Restriction to the plane through the top of the sphere, the origin, and the centers of A
and B .
d
Lemma 18. Let A and B bebal ls in R so that the center of A, the center of B , and the origin
arecolinear and the origin does not lie on the line segment between the center of A and the center
0 0
of B .IfAis closer to the origin than B and A touches B , then A touches B .
Pro of: We will restrict our attention to the plane through the top of the sphere, the origin, and
the centers of A and B see Figure 2 . Let a denote the interval that is the intersection of A with
the plane. Observe that an interval of the same size as a but lo cated further to the rightonthe
line will have a smaller pro jection on the circle. The lemma follows. 2
d 0 0
Pro of: [of Theorem 16] Let A and B be anytwo balls in R and let A and B b e their images
d+1
under stereographic pro jection on S . Assume that A touches B and B touches A.We will
0 0
show that + + A touches + + B .
Assume, without loss of generality, that A is at least as large as B . Let C b e the disk of the
same distance to the origin as A and congruenttoAthat is closest to B . Then, the centers of C
0
and B are colinear with the origin See Figure 3 . Let C b e the image of C . Since C is closer to
0 0
B than A is, C touches B and B touches A. By Lemma 18, C touches B . 17 A C
B
Figure 3: C is the circle congruent to and equidistant from the center to A that is closest to B .
The distance b etween the centers of A and B is less than + 1 times the radius of A b ecause
we assume that A is at least as large as B . The same holds for the distance b etween the center of C
0
and the center of B . Therefore, +1 A touches +1 C , so Lemma 17 implies that +1 =2A
0 0 0 0 0
touches +1 =2 C . Since A and C have the same spherical radius, C +1 + A .
0 0
Thus, + + A must touch + + B . 2
6.4. A Sp ectral Bound
Wenow show that the Fiedler value of a b ounded degree subgraph of an -overlap graph is small.
d
Theorem 19. If G is a subgraph of an -overlap graph of a k -ply neighborhood system in R and
2 2=d
the maximum degreeofGis , then the Fied ler value of L G is boundedby k=n , where
d
2 2=d 1=d
=2 +1 + = A =V .Accordingly, G has a Fied ler cut of ratio O k=n , and
d d+1 d
1=d 1 1=d
one can iterate Fied ler cuts to obtain a bisector of size O k n .
Pro of: Let = fB ; :::; B g b e the k -ply neighb orho o d system whose intersection graph contains
1 n
d d+1
G. By Theorem 12, there is a stereographic pro jection from R onto a particular sphere S so
that the centroid of the centers of the images of the neighb orho o ds is the center of the sphere.
0 0 0
Let = fB ; :::; B g b e the images of the balls in under . Let r b e the radius of B .
i
i n i
d 0
Because V r v ol ume B , We know that
d
i
n n
X X
d 0
V r v ol ume B kA :
d d+1
i i
i=1 i=1 18
0
By Theorem 16, G is a subgraph graph of the intersection graph of f + + B :1ing.
i
Thus, by Lemma 1,
P
n
2 2
2 + + r
i=1 i
L G
2
n
!
2=d
2=d
A k
d+1
2
2 + + :
V n
d
Given the b ound on the Fiedler value, the ratio achievable by a Fiedler cut follows immediately
from Theorem 21 and the corresp onding bisector size follows Lemma 22. 2
Remark 20. Recently, Agarwal and Pach [AP95] and, independently, Spielman and Teng [ST96]
gave an elementary proof of the sphere separator theorem of Mil ler et al [MTTV96b] on planar
graphs and intersection graphs. However, these proofs do not directly extend to overlap graphs.
The relation between overlap graphs and intersection graphs established by Theorem 16 enables
us to prove the overlap graph separator theorem using the intersection graph separator theorem.
The same reduction also extends the deterministic linear time algorithm for nding a good sphere
separator from intersection graphs to overlap graphs [EMT95].
7. Go o d ratio cuts
Alon [Alo86] and Sinclair and Jerrum [SJ88 ] proved that graphs with small Fiedler eigenvalue have
a go o d ratio cut Alon's theorem actually demonstrates the existence of a small vertex separator .
A corollary of an extension of their work by Mihail [Mih89] demonstrates that one can obtain a
go o d ratio cut from anyvector with small Rayleigh quotient that is p erp endicular to the all-ones's
vector although this is not explicitly stated in her work . In this section, we will present a new
pro of of Mihail's theorem see also [AM85, Fil91, DS91 , Mih89] and [Moh89] for a tighter b ound .
Theorem 21 Mihail . Let G = V; E bea graph on n nodes of maximum degree d, let Q be
n
its Laplacian matrix, and let be its isoperimetric number. For any vector ~x 2 R such that
P
n
x =0,
i
i=1
T 2
~x Q~x
:
T
~x ~x 2d
2
Moreover, thereisansso that the cut fi : x sg ; fi : x >sg has ratio at most = 2d .
i i
Pro of: Assume, without loss of generality, that x x x , and consider the emb edding
1 2 n
of G in R by ~x.For i n=2, at least i edges must cross over x . Similarly, for i n=2, at least
i
n i edges must cross x .We will use this fact to show that the Rayleigh quotientof~xcannot
i
b e to o small.
In our pro of, we will deal with the i n=2 and the i n=2 similarly.To simplify matters, assume
that n is o dd. Toachieve symmetry,we will work instead with the vector ~y , where y = x x ,
i i n+1 =2 19
so that y = 0. In a moment, we will show that
n+1 =2
T T
~x Q~x ~y Q~y
;
T T
~x ~x ~y ~y
so it suces to nd a lower b ound for the Rayleigh quotientof ~y with resp ect to Q. Recall that
P
2
T
x x
~x Q~x
i j
i;j 2E
= :
P
n
2
T
~x ~x x
i
i=1
Since x x = y y , the inequality follows from
i j i j
n n n
X X X
T 2 2 2
~y ~y = x x = x 2x x + nx
i i
n+1 =2 n+1 =2
i n+1 =2
i=1 i=1 i=1
n
X
P
2 2
= x + nx recall x =0
i
i n+1 =2
i=1
n
X
2 T
x = ~x ~x:
i
i=1
So that we can treat the i< n+1 =2 and the i> n+1 =2 indep endently,wewould liketo
eliminate all edges i; j where i< n+1 =2 ~ edges: one from i to n +1 =2 and one from n +1 =2toj. Let E denote this new set of edges. ~ ~ ~ Also, let E b e those edges i; j 2 E such that i; j n +1 =2, and let E b e the others. Because + 2 2 2 y y y y + y y ,we nd j i j i n+1 =2 n+1 =2 P P 2 2 y y y y ~ i j i j i;j 2E i;j 2E P P n n 2 2 y y i i i=1 i=1 P P 2 2 y y y y + ~ ~ i j i j i;j 2E i;j 2E + = : P P n+1 =2 n 2 2 y + y i=1 i i i= n+1 =2 Wenow provealower b ound on the terms involving i n +1 =2. As a similar b ound will hold for the other terms, the combination of the two will prove the theorem. The diculty in proving a b ound on the Rayleigh quotientof ~y is that the numerator is comp osed 2 2 of terms like y y , while the denominator is a sum of y 's. Toovercome this problem, wewould i j i 2 2 2 like to b ound the terms y y by a combination of y and y . Since y and y are usually i j i j i j separated, we can nd a b ound that usually works. For any a> 0, the inequality 2 2 y 1 + a y i j 2 ; y y i j 1+1=a follows from 2 2 2 2 y = y y +y 1+1=a y y + 1+a y : i j j i j i j 20 + We will cho ose a value for a later in the pro of. For now, let d b e the numb er of edges i; j such i that j>i, let d b e the numb er such that j i P P + 2 2 y y d d 1 + a y ~ ~ i j i i i i;j 2E i;j 2E : P P n+1 =2 n=2 2 2 y 1 + 1=a y i=1 i i=1 i We will nowcho ose a value of a so that P + 2 d d 1 + a y ~ i i i i;j 2E =2: P n=2 2 y i=1 i 2 Because the terms y are strictly decreasing, it suces cho ose a so that i P + k d d 1 + a i i i=1 =2; k P + k for all 1 k n=2. Since d is the numb er of edges with an endp oint less than y and k i=1 i P k d is the numb er of edges b oth of whose endp oints are less than y , their di erence is the k i i=1 numb er of edges that cross y , which is at least k . Moreover, since the maximum degree of a no de k P k is d, d d k =2. Thus, a sucient constraintonais that i i=1 k a d k=2 =2 a = d : k If we set a = = d , then 1 + 1=a =d= , and we obtain a lower b ound on the quotientof 2 : 2d 2 Wenow explain how to use go o d ratio cuts to pro duce a bisection of a graph. Lemma 22. Assume that we are given an algorithm that wil l nd a cut of ratio at most k in ev- ery k -node subgraph of G, for some monotonical ly decreasing function . Then repeated application of this algorithm can be used to nd a bisection of G of size at most Z n x dx: x=1 Pro of: The following algorithm see [LT79 , Gil80] will nd the bisection. 0 i. Initially, let D = G, let A and B b e empty sets, and let i =0. i is empty, then return A and B ; otherwise rep eat ii. If D i i i i a Find a cut of ratio at most jD j that divides D into F and F .We assume that i i jF jjF j. 21 i i b If jAj jBj, let A = A [ F ; otherwise, let B = B [ F ; i+1 i+1 c Let D = F , let i = i + 1, and return to step a . We assume that the algorithm terminates after t iterations. To show that this algorithm pro duces i a bisection, we need to prove that, for all i in the range 0 i i i Because jF jjF j, i i i min jAj; jB j + jF j jAj+jBj+jF j+j F j =2=n=2: i Wenow analyze the total cut size. Because the algorithm nds cuts of ratio at most jD j i at the ith iteration, the numb er of edges we cut to separate F is at most i jF j X i i i jD j jF j = jD j j =1 i i jD j jF j+1 X i jD j = i j =jD j i i jD j jF j+1 X j i j =jD j The inequality follows from the fact that is monotonicly decreasing. The total numb er of edges cut by this algorithm is at most 0 1 i i jD j jF j+1 t 1 t 1 X X X i i @ A jD j jF j j i i=0 i=0 j =jD j n X = j j =1 Z n x dx 1 The last inequality follows from the assumption that is monotonically decreasing. 2 1=d Remark 23. If x = x then Z n d 1 1=d x dx = n 1 : d 1 1 p Lipton and Tarjan [LT79] showed that by rep eatedly applying an -separator of size n, one p p can obtain a bisection of size = 1 1 n. Gilb ert [Gil80] extended this result to graphs p with p ositivevertex weights at the exp ense of a 1= 1 2 factor in the bisection b ound. Djidjev and Gilb ert [DG92 ] further generalized this result to graphs with arbitrary weights. Leighton and Rao [LR88 ] showed that one can obtain an O -approximation to a 1/3-separator from an -approximation to a ratio cut. 22 8. When Small Fiedler Cuts Are Unbalanced Recall that a Fiedler cut of a graph is obtained by taking its Fiedler vector x ;:::;x and a 1 n splitting value s and cutting the edges that cross s. In this section, we present examples of planar graphs for which there is no balanced Fiedler cut of go o d ratio. In fact, we show that no eigenvector corresp onding to a small eigenvalue has a splitting value that induces a go o d balanced cut. These graphs are generalizations of graphs constructed by Guattery and Miller [GM95]. The prop erties that we demand of these graphs are easy to achieve, and the reader should b e able to generalize our techniques to construct examples in many di erent contexts. Recently, Guattery and Miller [GM96] have extended our extensions of their work to pro duce b ounded-degree planar graphs in which the ratio achieved byany Fiedler cut is far from the graph's isop erimetric numb er. p All of our example graphs have cuts of ratio O 1= n that separate o n no des. This is necessary. In Section 3, we proved that every planar graph has a splitting value that induces a cut of ratio p p O 1= n . Thus, if there is no unbalanced Fiedler cut of ratio O 1= n , then the Fiedler cut of p ratio O 1= n must b e balanced. This might explain why most graphs that arise in practice have p balanced Fiedler cuts: a fairly regular planar graph should not have a cut of ratio O 1= n that separates few vertices. The graphs that we build in this section will b e constructed from: P , the path graph on k vertices V = f0;:::;k 1g and E = f i 1;i : 1 i< kg , k R , the ring graph on k vertices V = f0;:::;k 1g and E = f i; i + 1 mo d k :1ikg , k W , the Cartesian pro duct of P with R , and i;j i j i i+1 B , the complete binary tree with 2 leaves and a total of 2 1 no des. i Prop osition 24. The Fied ler value of P the path graph of k vertices is equal to k 2 : 4 sin 2k The Fied ler vector ~u is given by ! 2i 1 ~u = cos : i 2k The s-th eigenvalue of P is equal to k ! s 1 2 4 sin : 2k Its eigenvector ~u is given by ! 2i 1 s 1 : ~u = cos i 2n 23 Prop osition 25. The Fied ler value of R is equal to k 2 4 sin : k A Fied ler vector ~u is given by 2i ~u = cos i k Prop osition 26. Let G be the Cartesian product of graphs G and G . Then the eigenvalues of G 1 2 areequal to al l the possible sums of eigenvalues of G and G . Therefore, the Fied ler value of G is 1 2 the smal ler of the Fied ler values of G and G . 1 2 Wenow consider graphs S obtained by joining one copyofW with two copies of P , which k ;a;b a;b k 0