Journal of Algorithms 40, 24–36 (2001)doi:10.1006/jagm.2000.1118, available online at http://www.idealibrary.com on

Improved Bandwidth Approximation for Trees andChordal Graphs1

Anupam Gupta2

Computer Science Division, University of California, Berkeley, California 94720E-mail: anupamg@bell-labs.com

Received March 14, 2000; published online June 9, 2001

A linear arrangement of an n-vertex graph G = �V� E� is a one–one mapping f ofthe vertex set V onto the set �n� = �0� 1� � � � � n− 1�. The bandwidth of this lineararrangement is the maximum difference between the images of the endpoints of anyedge in E�G�. When the input graph G is a tree, the best known approximationalgorithms for the minimum bandwidth linear arrangement (which are based onthe principle of volume respecting embeddings) output a linear arrangement whichhas bandwidth within O�log3 n� of the optimal bandwidth. In this paper, we presenta simple randomized O�log2�5 n�-approximation algorithm for bandwidth minimiza-tion on trees. We also show that a close variant of this algorithm gives a similarperformance guarantee for chordal graphs. © 2001 Academic Press

1. INTRODUCTION

The bandwidth minimization problem is the following: given an undi-rected graph G = �V�E�, find a one–one mapping of the verticesf V → �n� whose bandwidth, which is defined to be

Bw�G� f � = max�i� j�∈E

�f �i� − f �j���

is minimum. This problem is equivalent to the matrix bandwidth minimiza-tion problem which, given a square symmetric matrix M , seeks to find apermutation matrix P such that PMPT has all its nonzero entries in a bandof minimum width about the diagonal. This not only reduces the space

1Portions of this work appeared in the Proceedings of the 11th Annual ACM-SIAMSymposium on Discrete Algorithms [13].

2Supported by NSF Grants CCR-9505448 and CCR-9820951.

24

0196-6774/01 $35.00Copyright © 2001 by Academic PressAll rights of reproduction in any form reserved.

bandwidth approximation for trees 25

needed to store the matrices, but also can help improve the performanceof matrix operations such as LU and Cholesky factorizations, making this ofmuch importance in many engineering applications. Other applications ofthe bandwidth minimization problem are given in [8, 16].

In 1976, this problem was shown to be NP-hard for general graphs byPapadimitriou [24]. Subsequent work strengthened the hardness result totrees with maximum degree 3 and to caterpillars of hair-length at mostthree [10, 23], making this one of the few problems known to be hard evenwhen the input graphs are trees of a very simple form. Furthermore, theproblem is also hard on trees for any fixed level of the fixed parameterhierarchy [4], which suggests that the dynamic programming algorithm [14]that decides whether the bandwidth is at most k and runs in O�nk� timecannot be made substantially faster. Furthermore, it has also been shownthat the Bandwidth Minimization problem is hard to even approximate ontrees. In fact, it is NP-hard to approximate it to within any constant evenwhen the input graph is a caterpillar of maximum degree three [27].

On the positive side, exact algorithms were known for caterpillars of hairlength at most two [2], interval graphs [17], and some other special classesof graphs. Further, approximation algorithms were known only for specialclasses of trees and for asteroidal-triple free graphs [15, 16, 18] until 1998,when Feige developed a O�log4�5 n�-approximation algorithm for generalgraphs [8], and independently, Blum et al. [3] obtained O�√n/B∗ log n�-approximation algorithms, where B∗ is the optimal bandwidth of the inputgraph. Both these algorithms are based on the idea of obtaining a “nice”embedding of the input graph into Euclidean space and then projectingdown onto a random line, though they differ in the way these embeddingsare obtained and the properties they possess.

The embeddings used in [8] were called volume respecting embeddings.Subsequent improvements on the bandwidth minimization problem havebeen achieved by displaying better volume respecting embeddings: Feige [9]gave a better analysis of his volume respecting embeddings to improve theapproximation guarantee to O�log3�5 n√log log n� for general graphs, andindependently, Rao [26] obtained improved volume respecting embeddingsfor planar graphs and Euclidean graphs, which gave even better guaranteesof O�log3 n� and O�log3 n√log log n� respectively for the bandwidth mini-mization problem on those graphs. This O�log3 n� guarantee is also the bestknown result for trees and outerplanar graphs, which are a fortiori planar.

The algorithms presented in this paper try to circumvent the step of get-ting an explicit volume-respecting embedding and are much simpler by com-parison. The algorithm for trees assigns carefully chosen random lengths tothe edges of a graph associated with the input and then places the verticesin order of their resulting distance from some arbitrarily chosen vertex.We show that this algorithm outputs an ordering that approximates the

26 anupam gupta

minimum bandwidth of the tree T to within a factor of O�log2 n√κ�T ��,where κ�T � is the caterpillar dimension of the tree T 3. Since it can be shownthat the caterpillar dimension of any n-vertex tree is at most log n, the per-formance of our algorithm is always within O�log2�5 n� of optimal.

The algorithm for chordal graphs uses a result of Brandstadt et al. [5],which shows that distances in any unweighted chordal graph G can beapproximated to within a constant by some unweighted tree T �G�. Thisimplies that the bandwidth of a linear arrangement of G is within someconstant of the bandwidth of the same arrangement for T �G�. Using thealgorithm for trees to approximate the bandwidth of T �G� then gives usthe algorithm for chordal graphs.

We note in passing that better approximation algorithms are knownfor some special classes of trees: deterministic O�log n�-approximationsare known for caterpillars [16] and some generalizations of height-balanced trees [15]. Though our general analysis only gives an O�log2 n�-approximation guarantee for caterpillars, we can, by a more specializedargument, show that our algorithm is in fact an O�log n�-approximationalgorithm when the input is a caterpillar. For the case of chordal graphs, a2k-approximation algorithm is also known for the case when there are atmost k leaves in the clique tree of the input chordal graph [19].

The rest of the paper is organized as follows: in Section 2, we fix somenotation and definitions. A crucial concentration bound is presented inSection 3. Then in Section 4, we describe the approximation algorithm andgive its analysis. The proof of the O�log n�-approximation guarantee forcaterpillars is presented in Section 5. Finally, in Section 6, we present thealgorithm for chordal graphs.

2. SOME DEFINITIONS

We will consider trees T = �V�E� with n vertices and l leaves, which weroot at an arbitrary vertex r. This imposes an ancestor–descendent relation-ship on the vertex set of T . We shall also assume that a vertex is its ownancestor. Finally, let d�u� v� be the number of edges in the unique pathbetween u and v in T .

The caterpillar dimension [21, 22] of a rooted tree T , henceforth denotedby κ�T �, is defined thus: For a tree with a single vertex, κ�T � = 0. Else,κ�T � ≤ k + 1 if there exist paths P1� P2� � � � � Pt beginning at the rootand pairwise edge-disjoint such that each component Tj of T − E�P1�−

3This quantity, formally defined in Section 2, has been previously used in [12, 21, 22] tocapture the “complexity” of trees, being, for example, 2 for caterpillars and log n for thecomplete binary tree.

bandwidth approximation for trees 27

E�P2� − · · · −E�Pt� has κ�Tj� ≤ k, where T −E�P1� −E�P2� − · · · −E�Pt�denotes the tree T with the edges of the Pi’s removed, and the componentsTj are rooted at the unique vertex lying on some Pi. The collection ofedge-disjoint paths in the above recursive definition form a partition of Eand are called the caterpillar decomposition of T . It is simple to see thatthe unique path between any two vertices of T intersects at most 2κ�T �of these paths. It can also be shown that κ�T � is at most log l and that adecomposition with the minimum value of κ�T � can be computed in poly-nomial time by dynamic programming (see, e.g., [22]). Furthermore, if timeis at a premium, computing a (possibly suboptimal) decomposition of valuelog n is possible in linear time.

We assign the vertices of the tree to paths in the caterpillar decomposi-tion in the following manner: a vertex v belongs to the path P if the edgeconnecting v to its parent belongs to P . The root vertex r is arbitrarilyassigned to one of the paths of its children. This also allows us to imposean ordering on the children of a vertex v: the child w which lies on thesame path as v is defined to be its leftmost child, and its other children arearbitrarily ordered after it.

A graph G is chordal [6, 7], if it has no chordless cycles of length morethan 3. See, e.g., [11] for other definitions and many properties of chordalgraphs.

Given an unweighted graph G = �V�E�, the ith power of G is the graphGi = �V�Ei�, where �x� y� ∈ Ei iff there is a path of at most i edges betweenx and y in G.

The tree volume Tvol�S� of a k-point metric space �S�µ� is the productof the lengths of the edges of a minimum spanning tree of S (considered asa graph with the weight of edge �i� j� being µ�i� j�) [8]. Hence, if T is anyspanning tree of S, the product of its edge lengths of T is at least Tvol�S�.Now given a graph G = �V�E� where d�·� ·� is the shortest-path distancebetween its vertices, the tree volume of a subset of the vertices S ⊆ V isjust the tree volume of the submetric �S� d�S×S�.

The local density D of a graph G is defined to be maxv∈V maxd��N�v� d��/2d�, where N�v� d� is the set of vertices in G at distance at most d fromthe vertex v. It is easy to see that this is a lower bound on the bandwidthof G.

3. A CONCENTRATION BOUND

In the following discussion, we will need an upper bound on the chancethat the sum of some well-spread-out real valued random variables lie insome unit interval on the real line. Let us define the concentration c�X� of

28 anupam gupta

a real-valued random variable X to be the following:

c�X� = supx∈�

Pr�X ∈ �x� x+ 1���

The following result due to Leader and Radcliffe tells us how the concen-tration of a sum of independent random variables behaves in terms of theconcentrations of the individual variables.

Theorem 3.1 [20]. Let Xi be a set of t independent random variablessuch that the concentration of Xi is at most 1/di, where di ∈ �. Then theconcentration of

∑ti=1Xi is at most the fraction of elements of the poset

M�d1� � � � � dt� which lie in its largest antichain.Here, the poset M�d1� d2� � � � � dt� is the partially ordered set with the

base elements from !ti=1�di� and the partial order given by e � e′ iff ei ≤ e′ifor all 1 ≤ i ≤ t. To estimate the quantity mentioned, we can use thefollowing result:

Theorem 3.2 [1]. The number of elements in the largest antichain inM�d1� � � � � dt� is

W = $( ∏t

i=1 di

�∑ti=1�d2i − 1��1/2

)� (1)

As a check, note that if di = 2 for all i, the above theorems imply that thewidth of the Boolean lattice is $�2n/√n� and that the chance of the sym-metric random walk with unit step lengths being at any fixed unit intervalafter n steps is O�1/√n�, which is optimal to within constants.

Before we proceed to the algorithm, let us record a simple but usefulcorollary of the above results.

Corollary 3.3. Let Xi be independent random variables, where Xi takesa value uniformly from the interval �di� 2di� ⊆ �, where di ∈ �>0. Then thereexists a constant c > 0 (independent of the di’s) such that for any unit openinterval I ⊆ �,

Pr

[t∑i=1

Xi ∈ I]≤ c

�∑i d2i �1/2

≤ c√t

�∑ti=1 di�

�

Proof. Immediately follows from Theorems 3.1 and 3.2 and the fact thatthe concentration of the random variables Xi is 1/di.

In passing, we may mention that if the values di are bounded by ',then the result of Corollary 3.3 will hold even if we choose Xi from theset �di� di�1 + 1

'�� di�1 + 2

'�� � � � � 2di� instead of the continuous interval

�di� 2di�, though with a possibly larger constant. This follows from thefact that the concentration of the random variables in the discrete caseis bounded by twice the concentration in the continuous case.

bandwidth approximation for trees 29

4. THE APPROXIMATION ALGORITHM FOR TREES

Let us consider the following simple algorithm for producing a lineararrangement of a rooted tree: let φ�r� = 0 and φ�v� = d�r� v�. Thoughthis is not a one–one map, we can make it so by arranging the set of ver-tices falling on a particular position in some arbitrary (or random) fashion.Unfortunately, this is a poor algorithm for bandwidth, since there are sim-ple examples where the bandwidth obtained is off by a polynomial factorfrom the local density D (which is within a polylogarithmic factor of theoptimal bandwidth B∗, a consequence of the results of Feige). One suchexample is given in Fig. 1, where the degrees of vertices a and all the bi isabout

√n, a is connected to each bi by a path of length about

√n, and the

children of the bi are the leaves. In this case, the output bandwidth will be+�n�, while the local density is O�√n�.

A simple twist to the algorithm is to give independent random lengths(say, from the interval �1� 2�) to the edges to get a weighted tree (withdistance function d′) and embed as above. Unfortunately, a simple analysiscan indicate that this performs very poorly on the same example givenabove. Indeed, all the n leaves will fall in an interval of length O�n1/4� onthe real line with constant probability, and hence the bandwidth will be+�n3/4�, again off by a polynomial factor from the local density.

Persevering with this general approach, we again choose lengths for theedges in a random fashion, but instead of choosing the length of each edgeindependently, we fix a caterpillar decomposition of the tree and for anypath P in the decomposition, we choose a length in �1� 2� and assign thislength to all edges lying on P . The main result of this paper is that this algo-rithm outputs an O�log2�5 n�-approximation for the minimum bandwidthproblem.

FIG. 1. A bad example for the simplest algorithms.

30 anupam gupta

4.1. The Algorithm

Let T = �V�E� be an unweighted undirected tree rooted at r. The algo-rithm Random-Lengths outputs a linear arrangement of T , i.e., a mappingf V → �n� in O�n log n� time, given a caterpillar decomposition:

Algorithm Random-Lengths

1. For each path Pi in caterpillar decomposition, choose a rate Riindependently and uniformly in �1� 2�. For each edge in Pi, let its lengthbe Ri. Let the distance function using these edge lengths be denoted by d′,and let φ�v� = d′�r� v�.

2. The map φ V → � induces a linear order on the vertices in V .The map f is the natural conversion of φ into a map from V to �n�; thus,f �i� = j if ��v ∈ V � φ�v� < φ�i��� = j. (Note that this will be a one–onemap with probability 1.)

4.2. The Analysis for Trees

Theorem 4.1. Random-Lengths is an O�log2 n√κ�T ��-approximationalgorithm for the bandwidth minimization problem on trees.

Proof. The proof closely follows that given by Feige [8], the basic struc-ture of which is sketched below.

1. Let an integer interval be an interval �a� a+ 1�, where a is an inte-ger. For any set S ⊆ V with �S� = k, S is called bad if φ�S� falls in someinteger interval of unit length. We show that the chance that S is bad is atmost .k−1/Tvol�S�, for some ..

2. By the linearity of expectation, the expected number of bad sets ofsize k is thus at most .k−1 ∑�S�=k�1/Tvol�S��. By Markov’s inequality, thetotal number of bad sets is not more than twice this value with probabilityat least a half.

3. By Theorem 7 of [8],∑S 1/Tvol�S� ≤ n�D log n�k−1. Substituting

this into the previous expression, we get that the number of bad sets is atmost 2n�D. log n�k−1 with probability at least a half. Let us condition onthis event happening for the rest of proof.

4. Now if the bandwidth of f is B, there is an edge e such that �B− 1�points lie between its endpoints. If the length of each edge is at most M ,the pigeon-hole principle ensures that one of the (at most) M + 1 integerintervals the edge e spans has at least �B − 1�/�M + 1� points in it. Thisimplies that we have at least

(�B−1�/�M+1�k

)bad sets.

5. Choosing k = log n and simplifying, we get that B ≤ O�D ×M. log2 n�. Since D is a lower bound for the optimal bandwidth, thisproves that we have an O�M. log2 n�-approximation.

bandwidth approximation for trees 31

In the previous results of Rao [26], both M and . were O�√log n�.Lemma 4.1 now shows that our algorithm satisfies the assertion in Step 1with . = O�√κ�T ��. Since the length of any edge is at most M = 2in our algorithm, we will get the claimed approximation guarantee ofO�log2 n√κ�T ��.

Lemma 4.1. For any set S of k points, the probability that all the points ofS fall in some integer interval is bounded above by O�√κ�T ��k−1/Tvol�S�.Proof. Look at the set S of k vertices of the tree T . Note that no two

vertices of T which are related to each other can fall into I, and hence wecan assume (w.l.o.g.) that S has no ancestor–descendent pairs in it.

Another important observation is the following: any two vertices of Smust be at almost the same distance (measured according to d) from theirleast common ancestor or else S will never be bad. More formally, forany pair u� v ∈ S with least common ancestor r ′, it must be the case that13 < d�v� r ′�/d�u� r ′� < 3. This follows from the fact that edges are stretchedby at most a factor of 2. A simple calculation shows that if the set S doesnot satisfy this property, then the vertices of S cannot all lie in the sameunit interval. We can thus (again, without loss of generality) assume that Ssatisfies this property.

Let us now give an ordering on the vertices of S. Consider the in-ordertraversal of the tree T (where the ordering on the children of any node isinduced by the caterpillar decomposition and is defined in Section 2), andlet S = �v1� v2� � � � � vk�, where vi is visited in this traversal before vi+1. Nowlet us choose the random rates for the paths in a delayed fashion based onthis in-order traversal. The rate for a path is chosen when a vertex belongingto it is visited for the first time. Finally, we choose rates for paths in thisway until the position of v1 is fixed: we now fix I to be the interval �a� a+ 1�into which v1 falls, where a ∈ �.

We claim that, conditional on the rates chosen till the time at whichvi is visited and the event that φ�vi� ∈ I, the probability that vi+1 alsofalls into I is at most 4c

√κ�T �/d�vi� vi+1�, where c is the constant in

Corollary 3.3. This implies that the chance of S being bad is at most!k−1i=1 4c

√κ�T �/d�vi� vi+1�, which is O�√κ�T ��k−1/Tvol�S�.

To prove the claim, consider the time at which vi is visited. Let x bethe least common ancestor of vi and vi+1. Clearly, x is distinct from boththese vertices, since they both lie in S and thus are unrelated. Hence ithas children y and z which are ancestors of vi and vi+1, respectively, andy lies to the left of z. This implies that x and z lie on different pathsin the caterpillar decomposition, and the path containing z, as well as allthe paths that hang off it, have not been seen yet. In particular, none ofthe paths in the caterpillar decomposition that intersect the path betweenx and vi+1 have been seen. Note that the total contribution of the paths

32 anupam gupta

is d�x� vi+1� which, by the above observation, is at least d�x� vi�/3 and thusat least d�vi� vi+1�/4.

Let Xj be the contribution (according to d′) of the jth path between xand vi+1. This Xj is uniformly distributed in the range �dj� 2dj�, where djis the length of that portion of the path according to the distance functiond. Note that

∑dj = d�x� vi+1�. Now the position of vi+1, conditional on

the events up to when vi is visited, is a sum of at most κ�T � of theseindependent random variables Xj . By Lemma 3.3, the chance that thissum lies in some fixed unit interval, and hence vi+1, lies in I is at mostc√κ�T �/d�x� vi+1� ≤ 4c

√κ�T �/d�vi� vi+1�.

4.3. Random Projections

Though the Random-Lengths algorithm above was described withoutany reference to random projections and volume-respecting embeddings, itturns out to be closely related to the notion of volume-respecting embed-dings. In this section, we give another view of this algorithm in the languageof volume-respecting embeddings.

In Feige’s algorithm for bandwidth, we first take a volume-respectingembedding ψ V → �L of the input graph G = �V�E�, and then “project”this embedding onto the real line by taking an L-dimensional vector X =�X1�X2� � � � �XL�, where each Xi is an independent random variable dis-tributed according to the N�0� 1� distribution and setting φ�x� = ψ�x� ·X.The analysis then proceeds as sketched in the proof of Theorem 4.1.

Unfortunately, for step 1 of this analysis to go through, it is necessary thatthe volumes of all the �log n�-sets be preserved, i.e., for any set S ⊆ V ofk = log n vertices, the �k− 1�-dimensional Euclidean volume of the simplexformed by the images of these vertices be “close” to its tree volume Tvol�S�.(See [8] for formal definitions and details.) In fact, if the embedding is η-volume-respecting, then we can prove that the chance of any set S of sizek being bad is at most ηk−1/Tvol�S�.

Let us now consider the following√log n-distance-preserving embedding

for trees [21, 22]: associate with a path Pi in the caterpillar decomposition ofthe tree the unit vector ui pointing in the ith direction. Now, for any vertexv, let P�v� be the path from v to the root r and l�v� i� be the length of theintersection P�v� ∩ Pi. The mapping now sets ψ�v� = ∑

i l�v� i�ui, wherethe sum is a vector sum. However, this embedding for trees is not volume-respecting. In general, the volume of any set which contains more than onepoint from a single path in the caterpillar decomposition is not preserved.As an example, consider the path of length n. This graph is embedded by ψalong the vector u1 and hence the �k − 1�-dimensional Euclidean volumeof the images of any set of k ≥ 3 points is 0. If the volume of a set S is notpreserved, the analysis breaks down, since bounding the chance of being

bandwidth approximation for trees 33

bad after “projecting” ψ down to φ by .k−1/Tvol�S� yields a very largevalue of ., defeating our purpose.

However, if we were to change the projection scheme slightly so thateach Xi was chosen independently and uniformly from the interval �1� 2�,we would get the algorithm Random-Lengths. No two vertices from thesame path in the caterpillar decomposition would now lie in the same unitinterval, and hence a k-set whose �k − 1�-volume is zero would never bebad. Furthermore, it is not difficult to show that the volumes of the remain-ing sets, which contain only a single point from each caterpillar path, areindeed preserved to within

√log n. However, since we changed the projec-

tion scheme, we cannot use previously used arguments to show that thechance of any such set being bad under the projection is low, but have toargue this afresh. This fact can be proved by techniques essentially identicalto those used in Lemma 4.1.

5. A BETTER ANALYSIS FOR CATERPILLARS

In this section, we will show that the algorithm given above has a betterperformance guarantee when the input tree is a caterpillar.

Theorem 5.1. Given a caterpillar as input, the bandwidth of the outputproduced by the algorithm Random-Lengths is an O�log n�-approximationto the optimal bandwidth.

Proof. A caterpillar is a path P (called the spine) with paths �Lij�j(called hairs) attached to the vertex i ∈ P and has caterpillar dimensionof at most 2. For the sake of analysis, let us imagine adding paths of length2n to either end of the spine to get a new caterpillar which we shall embed.Note that the local density D′ of this modified caterpillar is almost the sameas the local density D of the input graph. (In fact, D′ ≤ D + 2.) Hence ifwe can show that B ≤ O�D′ log n�, we will be done. We will assume that the(extended) spine P is given a unit rate and is isometrically embedded alongthe real line. We shall indicate in a moment why this assumption does notaffect the analysis by more than a constant factor.

Again, it suffices to bound the number of vertices that fall in some unitinterval. Let us focus on the unit interval I = �i� i + 1�. Look at the hairL attached to vertex j. If �L� < 1

2 �i − j�, then no vertex from that haircan ever fall into i or else it will have at most one vertex in I. In eithercase, the (worst-case) number of vertices from L falling into I is at mostnL = 2

∑v∈L 1/d�vi� v� for some suitably chosen constant c, where the vi

is the vertex of P lying at position i on the real line. Thus the total num-ber of vertices in I can be bounded by

∑L nL = 2

∑v∈V 1/d�vi� v�. It is

not difficult to see that the latter sum is bounded above by O�D′ log n�,

34 anupam gupta

since there are at most 2D′d vertices at distance d, and thus the sum canbe bounded above by 2D′ ∑

i 1/i. (See, e.g., [8, Theorem 7].)Finally, we can address the assumption about the rate of the spine. Since

the rate rP for the spine P could be no more than 2 in general, we couldimagine scaling all distances down by a factor of rP and then performingthe above analysis. The only modification required in this argument is that,due to the scaling step, there could be two vertices (instead of a singlevertex) from a hair lying in a unit interval. Hence the assumption changesthe calculations by a factor of at most 2, completing the proof.

6. AN ALGORITHM FOR CHORDAL GRAPHS

In this section, we give a O�log2�5 n�-approximation algorithm for chordalgraphs. For this, we use the following result of Brandstadt et al. [5] whichsays that distances in any chordal graph can be well approximated by a tree.

Theorem 6.1 [5]. For any unweighted chordal graph G, there exists a treeT ⊆ G2, such that dT �u� v� ≤ 3dG�u� v� for all u� v ∈ V �G�. Further, thistree can be constructed in linear time.

The algorithm for chordal graphs is now the following: given the inputchordal graph G, we construct the tree T by the procedure implicit in theproof of Theorem 6.1. We now simply run the Random-Lengths algorithmon T and output the resulting linear arrangement f .

Theorem 6.2. The above algorithm is a O�log2�5 n� approximation algo-rithm for the bandwidth minimization problem on chordal graphs.

Proof. It is easy to see that for the input graph G, the tree created issuch that T ⊆ G2 ⊆ T 6. Indeed, by Theorem 6.1, T ⊆ G2. Now considerany edge �u� v� ∈ E�G2�. Then dG�u� v� ≤ 2, and hence dT �u� v� ≤ 6, andso �u� v� ∈ E�T 6�.

We now use the fact that raising a graph to the kth power can onlyincrease its bandwidth by a factor of k. This implies that Opt�G2� ≤2Opt�G�. Clearly, Opt�T � ≤ Opt�G2�, and hence Opt�T � ≤ 2Opt�G�.Furthermore, note that Bw�G� f � ≤ Bw�G2� f � ≤ Bw�T 6� f � ≤ 6Bw�T� f �.Combining these two facts with Theorem 4.1 gives us that the bandwidthBw�G� f � is within O�log2�5 n� of Opt�T �, and hence of Opt�G�.

In fact, the same technique can be used for any graph in which dis-tances can be approximated well by an (unweighted) tree. Unfortunately,it is known that not all graphs have this property. For example, any treeapproximating the n-cycle incurs +�n� distortion [25], and other techniquesare required for broader classes of graphs.

bandwidth approximation for trees 35

ACKNOWLEDGMENTS

Many thanks to David Aldous, Jaikumar Radhakrishnan, and AlexRussell for discussions on the concentration bounds. Thanks also toAshwin Nayak and Alistair Sinclair for several helpful discussions.

REFERENCES

1. I. Anderson, A variance method in combinatorial number theory, Glasgow Math. J. 10(1969), 126–129.

2. S. F. Assmann, G. W. Peck, M. M. Sysło, and J. Zak, The bandwidth of caterpillars withhairs of length 1 and 2, SIAM J. Algebraic Discrete Methods 2 (1981), 387–393.

3. A. Blum, G. Konjevod, R. Ravi, and S. Vempala, Semi-definite relaxations for minimumbandwidth and other vertex-ordering problems, in “Proceedings of the 30th Annual ACMSymposium on Theory of Computing,” pp. 100–105, 1998.

4. H. L. Bodlaender, M. R. Fellows, and M. T. Hallett. Beyond NP-completeness for prob-lems of bounded width: Hardness for the W hierarchy (extended abstract), in “Proceedingsof the 26th Annual ACM Symposium on Theory of Computing,” pp. 449–458, 1994.

5. A. Brandstadt, V. Chepoi, and F. Dragan, Distance approximating trees for chordal anddually chordal graphs, J. Algorithms 30 (1999), 166–184.

6. P. Buneman, A characterisation of rigid circuit graphs, Discrete Math. 9 (1974), 205–212.7. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76.8. U. Feige, Approximating the bandwidth via volume respecting embeddings, in “Proceed-

ings of the 30th Annual ACM Symposium on Theory of Computing,” pp. 90–99, 1998.9. U. Feige, Improved analysis of the volume distortion of the random subsets embedding,

Manuscript, October 28, 1998.10. M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for

bandwidth minimization, SIAM J. Appl. Math. 34 (1978), 477–495.11. M. C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs,” Academic Press,

New York, 1980.12. A. Gupta, Embedding trees into low dimensional Euclidean spaces, Discrete Comput.

Geom. 24 (2000), 105–116.13. A. Gupta, Improved bandwidth approximation for trees, in “Proceedings of the 11th

Annual ACM-SIAM Symposium on Discrete Algorithms,” pp. 788–793, 2000.14. E. M. Gurari and I. Hal Sudborough, Improved dynamic programming algorithms for

bandwidth minimization and the mincut linear arrangement problem, J. Algorithms 5(1984), 531–546.

15. J. Haralambides and F. Makedon, Approximation algorithms for the bandwidth minimiza-tion problem for a large class of trees, Theory Comput. System 30 (1997), 67–90.

16. J. Haralambides, F. Makedon, and B. Monien, Bandwidth minimization: an approximationalgorithm for caterpillars, Math. Systems Theory 24 (1991), 169–177.

17. D. J. Kleitman and R. V. Vohra, Computing the bandwidth of interval graphs, SIAMJ. Discrete Math. 3 (1990), 373–375.

18. T. Kloks, D. Kratsch, and H. Muller, Approximating the bandwidth for asteroidal triple-free graphs, J. Algorithms 32 (1999), 41–57.

19. D. Kratsch and L. Stewart, Approximating bandwidth by mixing layouts of interval graphsin “Proceedings of the 16th Annual Symposium on Theoretical Aspects of ComputerScience (STACS ’99)” (C. Meinel and S. Tison, Eds.), Lecture Notes in Computer Science,Vol. 1563 pp. 248–258, Springer-Verlag, Berlin/New York, 1999.

36 anupam gupta

20. I. Leader and A. J. Radcliffe, Littlewood-Offord inequalities for random variables, SIAMJ. Discrete Math. 7 (1994), 90–101.

21. N. Linial, A. Magen, and M. Saks, Trees and Euclidean metrics, in “Proceedings ofthe 30th Annual ACM Symposium on Theory of Computing,” pp. 169–177, 1998.

22. J. Matousek, On embedding trees into uniformly convex Banach spaces, Israel J. Math. 114(1999), 221–237. [Czech version in “Lipschitz Distance of Metric Spaces,” C.Sc. degreethesis, Charles University, 1990].

23. B. Monien, The bandwidth minimization problem for caterpillars with hair length 3 isNP-complete, SIAM J. Algebraic Discrete Methods 7 (1986), 505–512.

24. C. H. Papadimitriou, The NP-completeness of the bandwidth minimization problem,Computing 16 (1976), 263–270.

25. Y. Rabinovich and R. Raz, Lower bounds on the distortion of embedding finite metricspaces in graphs, Discrete Comput. Geom. 19 (1998), 79–94.

26. S. B. Rao, Small distortion and volume preserving embeddings for planar and Euclideanmetrics, in “15th Annual ACM Symposium on Computational Geometry,” pp. 300–306,1999.

27. W. Unger, The complexity of the approximation of the bandwidth problem, in“Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Sci-ence,” pp. 82–91, 1998.