0018-9162/99/$10.00 © 1999 IEEE68 Computer

Clustering is a discovery process in data min-ing.1 It groups a set of data in a way thatmaximizes the similarity within clusters andminimizes the similarity between two dif-ferent clusters.1,2 These discovered clusters

can help explain the characteristics of the underlyingdata distribution and serve as the foundation for otherdata mining and analysis techniques. Clustering is use-ful in characterizing customer groups based on pur-chasing patterns, categorizing Web documents,3

grouping genes and proteins that have similar func-tionality,4 grouping spatial locations prone to earth-quakes based on seismological data, and so on.

Most existing clustering algorithms find clustersthat fit some static model. Although effective in somecases, these algorithms can break down—that is, clus-ter the data incorrectly—if the user doesn’t selectappropriate static-model parameters. Or sometimesthe model cannot adequately capture the clusters’characteristics. Most of these algorithms break downwhen the data contains clusters of diverse shapes, den-sities, and sizes.

Existing algorithms use a static model of the clustersand do not use information about the nature of indi-vidual clusters as they are merged. Furthermore, oneset of schemes (the CURE algorithm and relatedschemes) ignores the information about the aggregateinterconnectivity of items in two clusters. The otherset of schemes (the Rock algorithm, group averagingmethod, and related schemes) ignores informationabout the closeness of two clusters as defined by thesimilarity of the closest items across two clusters. (Formore information, see the “Limitations of TraditionalClustering Algorithms” sidebar.)

By only considering either interconnectivity or close-ness, these algorithms can easily select and merge the

wrong pair of clusters. For instance, an algorithm thatfocuses only on the closeness of two clusters will incor-rectly merge the clusters in Figure 1a over those inFigure 1b. Similarly, an algorithm that focuses only oninterconnectivity will, in Figure 2, incorrectly mergethe dark-blue with the red cluster rather than the greenone. Here, we assume that the aggregate interconnec-tivity between the items in the dark-blue and red clus-ters is greater than that of the dark-blue and greenclusters. However, the border points of the dark-bluecluster are much closer to those of the green clusterthan those of the red cluster.

CHAMELEON: CLUSTERING USING DYNAMIC MODELING

Chameleon is a new agglomerative hierarchical clus-tering algorithm that overcomes the limitations ofexisting clustering algorithms. Figure 3 (on page 70)provides an overview of the overall approach used byChameleon to find the clusters in a data set.

The Chameleon algorithm’s key feature is that itaccounts for both interconnectivity and closeness inidentifying the most similar pair of clusters. It thusavoids the limitations discussed earlier. Furthermore,Chameleon uses a novel approach to model the degreeof interconnectivity and closeness between each pair ofclusters. This approach considers the internal charac-teristics of the clusters themselves. Thus, it does notdepend on a static, user-supplied model and can auto-matically adapt to the internal characteristics of themerged clusters.

Chameleon operates on a sparse graph in whichnodes represent data items, and weighted edges rep-resent similarities among the data items. This sparse-graph representation allows Chameleon to scale tolarge data sets and to successfully use data sets that

Many advanced algorithms have difficulty dealing with highly variableclusters that do not follow a preconceived model. By basing its selectionson both interconnectivity and closeness, the Chameleon algorithm yieldsaccurate results for these highly variable clusters.

Chameleon:HierarchicalClustering UsingDynamic Modeling

GeorgeKarypisEui-Hong(Sam) HanVipin KumarUniversity ofMinnesota

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are available only in similarity space and not in met-ric spaces.5 Data sets in a metric space have a fixednumber of attributes for each data item, whereas datasets in a similarity space only provide similaritiesbetween data items.

Chameleon finds the clusters in the data set by usinga two-phase algorithm. During the first phase,Chameleon uses a graph-partitioning algorithm tocluster the data items into several relatively small sub-clusters. During the second phase, it uses an algorithmto find the genuine clusters by repeatedly combiningthese subclusters.

Modeling the dataGiven a matrix whose entries represent the similar-

ity between data items, many methods can be used tofind a graph representation.2,8 In fact, modeling dataitems as a graph is very common in many hierarchicalclustering algorithms.2,8,9 Chameleon’s sparse-graphrepresentation of the items is based on the commonlyused k-nearest-neighbor graph approach. Each vertexof the k-nearest-neighbor graph represents a data item.An edge exists between two vertices v and u if u isamong the k most similar points of v, or v is amongthe k most similar points of u.

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Partition-based clustering techniquessuch as K-Means2 and Clarans6 attemptto break a data set into K clusters suchthat the partition optimizes a given crite-rion. These algorithms assume that clus-ters are hyper-ellipsoidal and of similarsizes. They can’t find clusters that vary insize, as shown in Figure A1, or concaveshapes, as shown in Figure A2.

DBScan7 (Density-Based Spatial Clus-tering of Applications with Noise), a well-known spatial clustering algorithm, canfind clusters of arbitrary shapes. DBScandefines a cluster to be a maximum set ofdensity-connected points, which meansthat every core point in a cluster musthave at least a minimum number of points(MinPts) within a given radius (Eps).DBScan assumes that all points withingenuine clusters can be reached from oneanother by traversing a path of density-connected points and points across dif-ferent clusters cannot. DBScan can findarbitrarily shaped clusters if the clusterdensity can be determined beforehand and

the cluster density is uniform.Hierarchical clustering algorithms pro-

duce a nested sequence of clusters with asingle, all-inclusive cluster at the top andsingle-point clusters at the bottom.Agglomerative hierarchical algorithms2

start with each data point as a separatecluster. Each step of the algorithm involvesmerging two clusters that are the mostsimilar. After each merger, the total num-

ber of clusters decreases by one. Users canrepeat these steps until they obtain thedesired number of clusters or the distancebetween the two closest clusters goesabove a certain threshold. The many vari-ations of agglomerative hierarchical algo-rithms2 primarily differ in how theyupdate the similarity between existing andmerged clusters.

In some hierarchical methods, each clus-

(1) (2)

Limitations of Traditional Clustering Algorithms

Figure A. Data sets on which centroid and medoid approaches fail: Clusters (1) of widely differentsizes and (2) with concave shapes.

Figure 1. Algorithms based only on closeness will incorrectly merge (a) the dark-blueand green clusters because these two clusters are closer together than (b) the red andcyan clusters.

Figure 2. Algorithms based only on the interconnectivity of two clusters will incorrectlymerge the dark-blue and red clusters rather than dark-blue and green clusters.

(a) (b)

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Figure 4 illustrates the 1-, 2-, and 3-nearest-neigh-bor graphs of a simple data set. There are severaladvantages to representing the items to be clusteredusing a k-nearest-neighbor graph. Data items that arefar apart are completely disconnected, and the weightson the edges capture the underlying population den-sity of the space. Items in denser and sparser regionsare modeled uniformly, and the sparsity of the repre-sentation leads to computationally efficient algo-rithms. Because Chameleon operates on a sparsegraph, each cluster is nothing more than a subgraphof the data set’s original sparse-graph representation.

Modeling cluster similarityChameleon uses a dynamic modeling framework to

determine the similarity between pairs of clusters by look-ing at their relative interconnectivity (RI) and relativecloseness (RC). Chameleon selects pairs to merge forwhich both RI and RC are high. That is, it selects clus-ters that are well interconnected as well as close together.

Relative interconnectivity. Clustering algorithms typ-ically measure the absolute interconnectivity betweenclusters Ci and Cj in terms of edge cut—the sum of theweight of the edges that straddle the two clusters,which we denote EC(Ci, Cj). Relative interconnectivitybetween clusters is their absolute interconnectivity nor-malized with respect to their internal interconnectivi-ties. To get the cluster’s internal interconnectivity, wesum the edges crossing a min-cut bisection that splitsthe cluster into two roughly equal parts. Recentadvances in graph partitioning have made it possibleto efficiently find such quantities.10

Thus, the relative interconnectivity between a pairof clusters Ci and Cj is

By focusing on relative interconnectivity, Chameleon

RI C CEC C C

EC C EC C( , )

( , )

( ) ( )i j

i j

i j

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ter is represented by a centroid or medoid—a data point that is the closest to the centerof the cluster—and the similarity betweentwo clusters is measured by the similaritybetween the centroids/medoids. Both ofthese schemes fail for data in which pointsin a given cluster are closer to the center ofanother cluster than to the center of theirown cluster. This situation occurs in manynatural clusters,4,8 for example, if there is alarge variation in cluster sizes, as in FigureA1, or when cluster shapes are concave, as inFigure A2.

The single-link hierarchical method2 mea-sures the similarity between two clusters bythe similarity of the closest pair of datapoints belonging to different clusters. Unlikethe centroid/medoid-based methods, thismethod can find clusters of arbitrary shapeand different sizes. However, it is highly sus-ceptible to noise, outliers, and artifacts.

Researchers have proposed CURE9

(Clustering Using Representatives) to rem-edy the drawbacks of both of these meth-ods while combining their advantages. InCURE, a cluster is represented by selectinga constant number of well-scattered pointsand shrinking them toward the cluster’s

centroid, according to a shrinking factor.CURE measures the similarity between twoclusters by the similarity of the closest pairof points belonging to different clusters.

Unlike centroid/medoid-based methods,CURE can find clusters of arbitrary shapesand sizes, as it represents each cluster viamultiple representative points. Shrinking therepresentative points toward the centroidallows CURE to avoid some of the problemsassociated with noise and outliers. However,these techniques fail to account for specialcharacteristics of individual clusters. Theycan make incorrect merging decisions whenthe underlying data does not follow theassumed model or when noise is present.

In some algorithms, the similaritybetween two clusters is captured by theaggregate of the similarities (that is, theinterconnectivity) among pairs of itemsbelonging to different clusters. The ratio-nale for this approach is that subclustersbelonging to the same cluster will tend tohave high interconnectivity. But the aggre-gate interconnectivity between two clustersdepends on the size of the clusters; in gen-eral, pairs of larger clusters will have higherinterconnectivity.

Many such schemes normalize the aggre-gate similarity between a pair of clusterswith respect to the expected intercon-nectivity of the clusters involved. For ex-ample, the widely used group-averagemethod2 assumes fully connected clusters,and thus scales the aggregate similaritybetween two clusters by n × m, where n andm are the number of members in the twoclusters.

Rock8 (Robust Clustering Using Links),a recently developed algorithm that oper-ates on a derived similarity graph, scales theaggregate interconnectivity with respect toa user-specified interconnectivity model.However, the major limitation of all suchschemes is that they assume a static, user-supplied interconnectivity model. Suchmodels are inflexible and can easily lead toincorrect merging decisions when the modelunder- or overestimates the interconnectiv-ity of the data set or when different clustersexhibit different interconnectivity charac-teristics. Although some schemes allow theconnectivity to vary for different problemdomains (as does Rock8), it is still the samefor all clusters irrespective of their densitiesand shapes.

Figure 3. Chameleonuses a two-phasealgorithm, which firstpartitions the dataitems into subclus-ters and then repeat-edly combines thesesubclusters to obtainthe final clusters.

Data set

Construct asparse graph

Partitionthe graph

Mergepartitions

k-nearestneighbor graph Final clusters

can overcome the limitations of existing algorithmsthat use static interconnectivity models. Relative inter-connectivity can account for differences in clustershapes as well as differences in the degree of intercon-nectivity for different clusters.

Relative closeness. Relative closeness involves con-cepts that are analogous to those developed for rela-tive interconnectivity. The absolute closeness ofclusters is the average weight (as opposed to the sumof weights for interconnectivity) of the edges that con-nect vertices in Ci to those in Cj. Since these connec-tions come from the k-nearest-neighbor graph, theiraverage strength provides a good measure of the affin-ity between the data items along the interface layer ofthe two clusters. At the same time, this measure is tol-erant of outliers and noise.

To get a cluster’s internal closeness, we take theaverage of the edge weights across a min-cut bisectionthat splits the cluster into two roughly equal parts.

The relative closeness between a pair of clusters is theabsolute closeness normalized with respect to the inter-nal closeness of the two clusters:

where SEC(Ci) and SEC(Cj) are the average weights ofthe edges that belong in the min-cut bisector of clustersCi and Cj, and SEC(Ci, Cj) is the average weight of theedges that connect vertices in Ci and Cj. Terms |Ci| and|Cj| are the number of data points in each cluster. Thisequation also normalizes the absolute closeness of thetwo clusters by the weighted average of the internalcloseness of Ci and Cj. This discourages the mergingof small sparse clusters into large dense clusters.

In general, the relative closeness between two clus-ters is less than one because the edges that connectvertices in different clusters have a smaller weight.

By focusing on the relative closeness, Chameleon canovercome the limitations of existing algorithms that lookonly at the absolute closeness. By looking at the relativecloseness, Chameleon correctly merges clusters so thatthe resulting cluster has a uniform degree of closenessbetween its items.

ProcessThe dynamic framework for modeling cluster sim-

ilarity is applicable only when each cluster contains asufficiently large number of vertices (data items). Thereason is that to compute the relative interconnectiv-ity and closeness of clusters, Chameleon needs to com-pute each cluster’s internal interconnectivity andcloseness, neither of which can be accurately calcu-lated for clusters containing a few data points. For this

RC(C CSEC(C ,C )

C

C CSEC C

C

C CSEC C

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i

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j

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reason, Chameleon has a first phase that clusters thedata items into several subclusters that contain a suf-ficient number of items to allow dynamic modeling. Ina second phase, it discovers the genuine clusters in thedata set by using the dynamic modeling frameworkto hierarchically merge these subclusters.

Chameleon finds the initial subclusters usinghMetis,10 a fast, high-quality graph-partitioning algo-rithm. hMetis partitions the k-nearest-neighbor graphof the data set into several partitions such that it min-imizes the edge cut. Since each edge in the k-nearest-neighbor graph represents the similarity among datapoints, a partitioning that minimizes the edge cuteffectively minimizes the relationship (affinity) amongdata points across the partitions.

After finding subclusters, Chameleon switches toan algorithm that repeatedly combines these smallsubclusters, using the relative interconnectivity andrelative closeness framework. There are many waysto develop an algorithm that accounts for both ofthese measures. Chameleon uses two differentschemes.

User-specified thresholds. The first merges onlythose pairs of clusters exceeding user-specified thresh-olds for relative interconnectivity (TRI) and relativecloseness (TRC). In this approach, Chameleon visitseach cluster Ci and checks for adjacent clusters Cj

with RI and RC that exceed these thresholds.If more than one adjacent cluster satisfies these con-

ditions, then Chameleon will merge Ci with the clus-ter that it is most connected to—that is, the pair Ci

and Cj with the highest absolute interconnectivity.Once Chameleon chooses a partner for every clus-

ter, it performs these mergers and repeats the entireprocess. Users can control the characteristics of thedesired clusters with TRI and TRC.

Function-defined optimization. The second schemeimplemented in Chameleon uses a function to com-bine the relative interconnectivity and relative close-ness. Chameleon selects cluster pairs that maximizethis function. Since our goal is to merge pairs for whichboth the relative interconnectivity and relative close-ness are high, a natural way of defining such a functionis to take their product. That is, select clusters thatmaximize RI(Ci,Cj) × RC(Ci,Cj). This formula givesequal importance to both parameters. However, quiteoften we may prefer clusters that give a higher prefer-ence to one of these two measures. For this reason,Chameleon selects the pair of clusters that maximizes

RI(Ci,Cj) × RC(Ci,Cj)α

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Figure 4. K-nearest-neighbor graphs from original data in two dimensions: (a) originaldata, (b) 1-, (c) 2-, and (d) 3-nearest neighbor graphs.

(a) (b) (c) (d)

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where α is a user-specified parameter. If α > 1, thenChameleon gives a higher importance to the relativecloseness, and when α < 1, it gives a higher impor-tance to the relative interconnectivity.

RESULTS AND COMPARISONSWe compared Chameleon’s performance against

that of CURE9 and DBScan7 on four different datasets. These data sets contain from 6,000 to 10,000points in two dimensions; the points form geometricshapes as shown in Figure 5. These data sets representsome difficult clustering instances because they con-tain clusters of arbitrary shape, proximity, orienta-tion, and varying densities. They also containsignificant noise and special artifacts. Many of theseresults are available at http://www.cs.umn.edu/~han/chameleon.html.

Chameleon is applicable to any data set for whicha similarity matrix is available (or can be constructed).

However, we evaluated it with two-dimensional datasets because similar data sets have been used to eval-uate other state-of-the-art algorithms. Clusters in 2Ddata sets are also easy to visualize and compare.

Figure 6 shows the clusters that Chameleon foundfor each of the four data sets, using the same set ofparameter values. In particular, we used k = 10 in afunction-defined optimization with α = 2.0. Points indifferent clusters are represented using a combinationof colors and glyphs. As a result, points that belongto the same cluster use both the same color and glyph.

For example, in the DS3 clusters, there are two cyanclusters. One contains the points in the region betweenthe two circles inside the ellipse, and the other, the linebetween the two horizontal bars and the c-shaped clus-ter. Two clusters are dark blue—one corresponds to theupside-down c-shaped cluster and the other, the circleinside the candy cane. Their points use different glyphs(bells and squares for the first pair, and squares and

Figure 5. Four datasets used in ourexperiments: (a) DS1has 8,000 points; (b) DS2 has 6,000points; (c) DS3 has10,000 points; and(d) DS4 has 8,000points.

Figure 6. Clustersdiscovered byChameleon for thefour data sets.

(a) (b) (c) (d)

bells for the second), so they denote different clusters.The clustering shown in Figure 6 corresponds to the

earliest point at which Chameleon was able to find thegenuine clusters in the data set. That is, they corre-spond to the earliest iteration at which Chameleonidentifies the genuine clusters and places each in a sin-gle cluster. Thus for a given data set, Figure 6 showsmore than the number of genuine clusters; the addi-tional clusters contain outliers.

Chameleon correctly identifies the genuine clus-ters in all four data sets. For example, in DS1,Chameleon finds six clusters, five of which are gen-uine clusters, and the sixth (shown in brown) corre-sponds to outliers that connect the two ellipsoidclusters. In DS3, Chameleon finds the nine genuineclusters and two clusters that contain outliers. In DS2and DS4, Chameleon accurately finds only the gen-uine cluster.

CUREWe also evaluated CURE, which has been shown to

effectively find clusters in two-dimensional data sets.9

CURE was able to find the right clusters for DS1 andDS2, but it failed to find the right clusters on theremaining two data sets. Figure 7 shows the results

obtained by CURE for the DS3 and DS4 data sets.For each of the data sets, Figure 7 shows two dif-

ferent clustering solutions containing different num-bers of clusters. The clustering solution in Figure 7acorresponds to the earliest point in the process inwhich CURE merges subclusters that belong to twodifferent genuine clusters. As we can see from Figure7b, in the case of DS3, CURE makes a mistake whengoing down to 25 clusters, as it merges one of the cir-cles inside the ellipse with a portion of the ellipse.

Similarly, in the case of DS4 (Figure 7c), CUREalso makes a mistake when going down to 25 clus-ters by merging the small circular cluster with a por-tion of the upside-down Y-shaped cluster. The secondclustering solution (Figure 7d) corresponds to solu-tions that contain as many clusters as those discov-ered by Chameleon. These solutions are considerablyworse than the first set of solutions, indicating thatCURE’s merging scheme makes multiple mistakesfor these data sets.

DBSCANFigure 8 shows the clusters found by DBScan for

DS1 and DS2, using different values of the Eps para-meter. Following the recommendation of DBScan’s

August 1999 73

(a) (b)

(c) (d)

Figure 7. Clustersidentified by CUREwith shrinking factor0.3 and number ofrepresentative pointsequal to 10. For DS3,CURE first mergessubclusters thatbelong to two differ-ent subclusters at (a) 25 clusters. With (b) 11 clustersspecified—the samenumber thatChameleon found—CURE obtains theresult shown. CURE produced these results for DS4 with (c) 25 and (d) 8 clusters specified.

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developers,7 we fixed the MinPts at 4 and varied Eps inthese experiments.

The clusters produced for DS1 illustrate that DBScancannot effectively find clusters of different density. InFigure 8a, DBScan puts the two red ellipses (at the topof the image) into the same cluster because the outlierpoints satisfy the density requirements as dictated bythe Eps and MinPts parameters. Decreasing the valueof Eps can separate these clusters, as shown in Figure8b for Eps at 0.4. Although DBScan separates theellipses without fragmenting them, it now fragmentsthe largest—but lower density—cluster into severalsmall subclusters. Our experiments have shown thatDBScan exhibits similar characteristics on DS4.

The clusters produced for DS2 illustrate that DBScancannot effectively find clusters with variable internaldensities. Figure 9a through Figure 9c show a sequenceof three clustering solutions for decreasing values ofEps. As we decrease Eps in the hope of separating thetwo clusters, the natural clusters in the data set are frag-mented into several smaller clusters.

DBScan finds the correct clusters in DS3 given theright combination of Eps and MinPts. If we fix MinPtsat 4, then the algorithm works fine on this data set aslong as Eps is from 5.7 to 6.1.

Even though we chose to model the data using a k-nearest-neighbor graph, it is entirely possible touse other graph representations—such as those

based on mutual shared neighbors,2,8,11—which aresuitable for particular application domains. Fur-thermore, different domains may require differentmodels for capturing relative closeness and intercon-nectivity. In any of these situations, we believe that thetwo-phase framework of Chameleon would still behighly effective. Our future research includes the veri-

fication of Chameleon on different application do-mains. We also want to study the effectiveness of dif-ferent techniques for modeling data as well as clustersimilarity. ❖

AcknowledgmentsThis work was partially supported by Army

Research Office contract DA/DAAG55-98-1-0441and the Army High-Performance Computing Re-search Center under the auspices of the Department ofthe Army, Army Research Laboratory cooperativeagreement number DAAH04-95-2-0003/contractnumber DAAH04-95-C-0008, the content of whichdoes not necessarily reflect the government’s positionor policy, and should infer no official endorsement.This work was also supported by an IBM PartnershipAward. Access to computing facilities was providedby the Army High-Performance Computing ResearchCenter and the Minnesota Supercomputer Institute.

We also thank Martin Ester, Hans-Peter Kriegel,Jorg Sander, and Xiaowei Xu for providing theDBScan program.

References1. M.S. Chen, J. Han, and P.S. Yu, “Data Mining: An

Overview from a Database Perspective,” IEEE Trans.Knowledge and Data Eng., Dec. 1996, pp. 866-883.

2. A.K. Jain and R.C. Dubes, Algorithms for Clustering Data,Prentice Hall, Upper Saddle River, N.J., 1988.

3. D. Boley et al., “Document Categorization and Query Gen-eration on the World Wide Web Using WebACE,” to bepublished in AI Review, 1999.

4. E.H. Han et al., “Hypergraph-Based Clustering in High-Dimensional Data Sets: A Summary of Results,” Bull. Tech.

Figure 9. DBScanresults for DS2 withMinPts at 4 and Eps at(a) 5.0, (b) 3.5, and(c) 3.0.

Figure 8. DBScanresults for DS1 withMinPts at 4 and Eps at(a) 0.5 and (b) 0.4.

(a) (b)

(a) (b) (c)

Committee on Data Eng., Vol. 21, No. 1, 1998, pp. 15-22.Also a longer version available as Tech. Report TR-97-019, Dept. of Computer Science, Univ. of Minnesota,1997.

5. V. Ganti et al., “Clustering Large Datasets in Arbitrary Met-ric Spaces,” Proc. 15th Int’l Conf. Data Eng., IEEE CSPress, Los Alamitos, Calif., 1999, pp. 502-511.

6. R. Ng and J. Han, “Efficient and Effective ClusteringMethod for Spatial Data Mining,” Proc. 20th Int’l Conf.Very Large Data Bases, Morgan Kaufmann, San Francisco,1994, pp. 144-155.

7. M. Ester et al., “A Density-Based Algorithm for Discover-ing Clusters in Large Spatial Databases with Noise,” Proc.2nd Int’l Conf. Knowledge Discovery and Data Mining,AAAI Press, Menlo Park, Calif., 1996, pp. 226-231.

8. S. Guha, R. Rastogi, and K. Shim, “ROCK: A RobustClustering Algorithm for Categorical Attributes,” Proc.15th Int’l Conf. Data Eng., IEEE CS Press, Los Alamitos,Calif., 1999, pp. 512-521.

9. S. Guha, R. Rastogi, and K. Shim, “CURE: An EfficientClustering Algorithm for Large Databases,” Proc. ACMSIGMOD Int’l Conf. Management of Data, ACM Press,New York, 1998, pp. 73-84.

10. G. Karypis and V. Kumar, “hMETIS 1.5: A HypergraphPartitioning Package,” Tech. Report, Dept. of ComputerScience, Univ. of Minnesota, 1998; http://winter.cs.umn.edu/~karypis/metis.

11. K.C. Gowda and G. Krishna, “Agglomerative ClusteringUsing the Concept of Mutual Nearest Neighborhood,”Pattern Recognition, Feb. 1978, pp. 105-112.

George Karypis is an assistant professor of computerscience at the University of Minnesota. His researchinterests include data mining, bio-informatics, parallelcomputing, graph partitioning, and scientific comput-ing. Karypis has a PhD in computer science from theUniversity of Minnesota. He is a member of the IEEEComputer Society and the ACM.

Eui-Hong (Sam) Han is a PhD candidate at the Uni-versity of Minnesota. His research interests include datamining and information retrieval. Han has an MS incomputer science from the University of Texas, Austin.He is a member of the ACM.

Vipin Kumar is a professor of computer science at theUniversity of Minnesota. His research interests includedata mining, parallel computing, and scientific comput-ing. Kumar has a PhD in computer science from the Uni-versity of Maryland, College Park. He is a member ofthe IEEE, ACM, and SIAM.

Contact the authors at the Univ. of Minnesota, Dept. ofComputer Science and Eng., 4-192 EECS Bldg., 200Union St. SE, Minneapolis, MN 55455; {karypis,han,kumar}@cs.umn.edu.

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