Summarizing Spatial Data Streams Using ClusterHulls ∗

John Hershberger† Nisheeth Shrivastava‡ Subhash Suri‡

Abstract

We consider the following problem: given an on-line,possibly unbounded stream of two-dimensional points,how can we summarize its spatial distribution or shapeusing a small, bounded amount of memory? We pro-pose a novel scheme, called ClusterHull, which repre-sents the shape of the stream as a dynamic collection ofconvex hulls, with a total of at most m vertices, wherem is the size of the memory. The algorithm dynami-cally adjusts both the number of hulls and the numberof vertices in each hull to best represent the streamusing its fixed memory budget. This algorithm ad-dresses a problem whose importance is increasingly rec-ognized, namely the problem of summarizing real-timedata streams to enable on-line analytical processing.As a motivating example, consider habitat monitoringusing wireless sensor networks. The sensors produce asteady stream of geographic data, namely, the locationsof objects being tracked. In order to conserve their lim-ited resources (power, bandwidth, storage), the sensorscan compute, store, and exchange ClusterHull sum-maries of their data, without losing important geomet-ric information. We are not aware of other schemesspecifically designed for capturing shape informationin geometric data streams, and so we compare Cluster-Hull with some of the best general-purpose clusteringschemes such as CURE, k-median, and LSEARCH. Weshow through experiments that ClusterHull is able torepresent the shape of two-dimensional data streamsmore faithfully and flexibly than the stream versionsof these clustering algorithms.

∗A partial summary of this work will be presented as a posterat ICDE ’06, and represented in the proceedings by a three-pageabstract.

†Mentor Graphics Corp., 8005 SW Boeckman Road,Wilsonville, OR 97070, USA, and (by courtesy) Computer Sci-ence Department, University of California at Santa Barbara.john hershberger@mentor.com.

‡Computer Science Department, University of California,Santa Barbara, CA 93106, USA. {nisheeth,suri}@cs.ucsb.edu. The research of Nisheeth Shrivastava and Subhash Suriwas supported in part by National Science Foundation grantsIIS-0121562 and CCF-0514738.

1 Introduction

The extraction of meaning from data is perhaps themost important problem in all of science. Algorithmsthat can aid in this process by identifying useful struc-ture are valuable in many areas of science, engineer-ing, and information management. The problem takesmany forms in different disciplines, but in many set-tings a geometric abstraction can be convenient: forinstance, it helps formalize many informal but visuallymeaningful concepts such as similarity, groups, shape,etc. In many applications, geometric coordinates are anatural and integral part of data: e.g., locations of sen-sors in environmental monitoring, objects in location-aware computing, digital battlefield simulation, or me-teorological data. Even when data have no intrinsic ge-ometric association, many natural data analysis taskssuch as clustering are best performed in an appropri-ate artificial coordinate space: e.g., data objects aremapped to points in some Euclidean space using cer-tain attribute values, where similar objects (points) aregrouped into spatial clusters for efficient indexing andretrieval. Thus we see that the problem of finding asimple characterization of a distribution known onlythrough a collection of sample points is a fundamentalone in many settings.

Recently there has been a growing interest in de-tecting patterns and analyzing trends in data that aregenerated continuously, often delivered in some fixedorder and at a rapid rate. Some notable applica-tions of such data processing include monitoring andsurveillance using sensor networks, transactions in fi-nancial markets and stock exchanges, web logs andclick streams, monitoring and traffic engineering of IPnetworks, telecommunication call records, retail andcredit card transactions, and so on. Imagine, for in-stance, a surveillance application, where a remote en-vironment instrumented by a wireless sensor networkis being monitored through sensors that record themovement of objects (e.g., animals). The data gath-ered by each sensor can be thought of as a stream oftwo-dimensional points (geographic locations). Giventhe severe resource constraints of a wireless sensor net-work, it would be rather inefficient for each sensor tosend its entire stream of raw data to a remote base sta-

tion. Indeed, it would be far more efficient to computeand send a compact geometric summary of the trajec-tory. One can imagine many other remote monitoringapplications like forest fire hazards, marine life, etc.,where the shape of the observation point cloud is a nat-ural and useful data summary. Thus, there are manysources of “transient” geometric data, where the keygoal is to spot important trends and patterns, whereonly a small summary of the data can be stored, andwhere a “visual” summary such as shape or distribu-tion of the data points is quite valuable to an analyst.

A common theme underlying these data processingapplications is the continuous, real-time, large-volume,transient, single-pass nature of data. As a result, datastreams have emerged as an important paradigm fordesigning algorithms and answering database queriesfor these applications. In the data stream model,one assumes that data arrive as a continuous stream,in some arbitrary order possibly determined by anadversary; the total size of the data stream is quitelarge; the algorithm may have memory to store onlya tiny fraction of the stream; and any data notexplicitly stored are essentially lost. Thus, data streamprocessing necessarily entails data reduction, wheremost of the data elements are discarded and onlya small representative sample is kept. At the sametime, the patterns or queries that the applications seekmay require knowledge of the entire history of thestream, or a large portion of it, not just the mostrecent fraction of the data. The lack of access tofull data significantly complicates the task of dataanalysis, because patterns are often hidden, and easilylost unless care is taken during the data reductionprocess. For simple database aggregates, sub-samplingcan be appropriate, but for many advanced queries orpatterns, sophisticated synopses or summaries must beconstructed. Many such schemes have recently beendeveloped for computing quantile summaries [21], mostfrequent or top-k items [23], distinct item counts [3, 24],etc.

When dealing with geometric data, an analyst’sgoal is often not as precisely stated as many of thesenumerically-oriented database queries. The analystmay wish to understand the general structure of thedata stream, look for unusual patterns, or search forcertain “qualitative” anomalies before diving into amore precisely focused and quantitative analysis. The“shape” of a point cloud, for instance, can convey im-portant qualitative aspects of a data set more effec-tively than many numerical statistics. In a stream set-ting, where the data must be constantly discarded andcompressed, special care must be taken to ensure thatthe sampling faithfully captures the overall shape of

the point distribution.

Shape is an elusive concept, which is quite chal-lenging even to define precisely. Many areas of com-puter science, including computer vision, computergraphics, and computational geometry deal with rep-resentation, matching and extraction of shape. How-ever, techniques in those areas tend to be compu-tationally expensive and unsuited for data streams.One of the more successful techniques in processing ofdata streams is clustering. The clustering algorithmsare mainly concerned with identifying dense groups ofpoints, and are not specifically designed to extract theboundary features of the cluster groups. Neverthe-less, by maintaining some sample points in each clus-ter, one can extract some information about the geo-metric shape of the clusters. We will show, perhapsunsurprisingly, that ClusterHull, which explicitly aimsto summarize the geometric shape of the input pointstream using a limited memory budget, is more effec-tive than general-purpose stream clustering schemes,such as CURE, k-median and LSEARCH.

1.1 ClusterHull

Given an on-line, possibly unbounded stream of two-dimensional points, we propose a scheme for summa-rizing its spatial distribution or shape using a small,bounded amount of memory m. Our scheme, calledClusterHull, represents the shape of the stream as adynamic collection of convex hulls, with a total of atmost m vertices. The algorithm dynamically adjustsboth the number of hulls and the number of verticesin each hull to represent the stream using its fixedmemory budget. Thus, the algorithm attempts to cap-ture the shape by decomposing the stream of pointsinto groups or clusters and maintaining an approxi-mate convex hull of each group. Depending on theinput, the algorithm adaptively spends more pointson clusters with complex (potentially more interesting)boundaries and fewer on simple clusters. Because eachcluster is represented by its convex hull, the Cluster-Hull summary is particularly useful for preserving suchgeometric characteristics of each cluster as its bound-ary shape, orientation, and volume. Because hulls areobjects with spatial extent, we can also maintain addi-tional information such as the number of input pointscontained within each hull, or their approximate datadensity (e.g., population divided by the hull volume).By shading the hulls in proportion to their density, wecan then compactly convey a simple visual representa-tion of the data distribution. By contrast, such infor-mation seems difficult to maintain in stream clusteringschemes, because the cluster centers in those schemes

constantly move during the algorithm.

For illustration, in Figure 1 we compare the outputof our ClusterHull algorithm with those produced bytwo popular stream-clustering schemes, k-median [19]and CURE [20]. The top row shows the input data(left), and output of ClusterHull (right) with memorybudget set to m = 45 points. The middle row showsoutputs of k-median, while the bottom row shows theoutputs of CURE. One can see that both the boundaryshapes and the densities of the point clusters are quiteaccurately summarized by the cluster hulls.

Figure 1: The top row shows the input data (left) andthe output of ClusterHull (right) with memory budgetof m = 45. The hulls are shaded in proportion to theirestimated point density. The middle row shows twodifferent outputs of the stream k-medians algorithm,with m = 45: in one case (left), the algorithm simplycomputes k = 45 cluster centers; in the other (right),the algorithm computes k = 5 centers, but maintains9 (random) sample points from the cluster to get arough approximation of the cluster geometry. (This isa simple enhancement implemented by us to give moreexpressive power to the k-median algorithm.) Finally,the bottom row shows the outputs of CURE: in theleft figure, the algorithm computes k = 45 clustercenters; in the right figure, the algorithm computesk = 5 clusters, with c = 9 samples per cluster. CUREhas a tunable shrinkage parameter, α, which we setto 0.4, in the middle of the range suggested by itsauthors [20].

We implemented ClusterHull and experimentedwith both synthetic and real data to evaluate its per-formance. In all cases, the representation by Cluster-Hull appears to be more information-rich than thoseby clustering schemes such as CURE, k-medians, orLSEARCH, even when the latter are enhanced withsome simple mechanisms to capture cluster shape.Thus, our general conclusion is that ClusterHull can bea useful tool for summarizing geometric data streams.

ClusterHull is computationally efficient, and thuswell-suited for streaming data. At the arrival of eachnew point, the algorithm must decide whether thepoint lies in one of the existing hulls (actually, within acertain ring around each hull), and possibly merge twoexisting hulls. With appropriate data structures, thisprocessing can be done in amortized time O(log m) perpoint.

ClusterHull is a general paradigm, which can beextended in several orthogonal directions and adaptedto different applications. For instance, if the input dataare noisy, then covering all points by cluster hulls canlead to poor shape results. We propose an incrementalcleanup mechanism, in which we periodically discardlight-weight hulls, that deals with noise in the datavery effectively. Similarly, the performance of a shapesummary scheme can depend on the order in whichinput is presented. If points are presented in a badorder, the ClusterHull algorithm may create long,skinny, inter-penetrating hulls early in the streamprocessing. We show that a period-doubling cleanupis effective in correcting the effects of these earlymistakes. When there is spatial coherence withinthe data stream, our scheme is able to exploit thatcoherence. For instance, imagine a point streamgenerated by a sensor field monitoring the movementof an unknown number of vehicles in a two-dimensionalplane. The data naturally cluster into a set of spatiallycoherent trajectories, which our algorithm is able toisolate and represent more effectively than general-purpose clustering algorithms.

1.2 Related Work

Inferring shape from an unordered point cloud is awell-studied problem that has been considered in manyfields, including computer vision, machine learning,pattern analysis, and computational geometry [4, 10,11, 26]. However, the classical algorithms from theseareas tend to be computationally expensive and requirefull access to data, making them unsuited for use in adata stream setting.

An area where significant progress has occurredon stream algorithms is clustering. Our focus is some-

what different from classical clustering—we are mainlyinterested in low-dimensional data and capturing the“surface” or boundary of the point cloud, while clus-tering tends to focus on the “volume” or density andmoderate and large dimensions. While classical clus-tering schemes of the past have focused on cluster cen-ters, which work well for spherical clusters, some recentwork has addressed the problem of non-spherical clus-ters, and tried to pay more attention to the geometryof the clusters. Still this attention to geometry doesnot extend to the shape of the boundary.

Our aim is not to exhaustively survey the clus-tering literature, which is immense and growing, butonly to comment briefly on those clustering schemesthat could potentially be relevant to the problemof summarizing shape of two- or three-dimensionalpoint streams. Many well-known clustering schemes(e.g., [5, 7, 16, 25]) require excessive computation andrequire multiple passes over the data, making them un-suited for our problem setting. There are machine-learning based clustering schemes [12, 13, 27], that useclassification to group items into clusters. These meth-ods are based on statistical functions, and not gearedtowards shape representation. Clustering algorithmsbased on spectral methods [8, 14, 18, 28] use the sin-gular value decomposition on the similarity graph ofthe data, and are good at clustering statistical data,especially in high dimensions. We are unaware of anyresults showing that these methods are particularly ef-fective at capturing boundary shapes, and, more im-portantly, streaming versions of these algorithms arenot available. So, we now focus on clustering schemesthat work on streams and are designed to capture someof the geometric information about clusters.

One of the popular clustering schemes for largedata sets is BIRCH [30], which also works on datastreams. An extension of BIRCH by Aggarwal et al. [2]also computes multi-resolution clusters in evolvingstreams. While BIRCH appears to work well forspherical-shaped clusters of uniform size, Guha etal. [20] experimentally show that it performs poorlywhen the data are clustered into groups of unequalsizes and different shapes. The CURE clusteringscheme proposed by Guha et al. [20] addresses thisproblem, and is better at identifying non-sphericalclusters. CURE also maintains a number of samplepoints for each cluster, which can be used to deduce thegeometry of the cluster. It can also be extended easilyfor streaming data (as noted in[19]). Thus, CUREis one of the clustering schemes we compare againstClusterHull.

In [19], Guha et al. propose two stream variants ofk-center clustering, with provable theoretical guaran-

tees as well as experimental support for their perfor-mance. The stream k-median algorithm attempts tominimize the sum of the distances between the inputpoints and their cluster centers. Guha et al. [19] alsopropose a variant where the number of clusters k can berelaxed during the intermediate steps of the algorithm.They call this algorithm LSEARCH (local search).Through experimentation, they argue that the streamversions of their k-median and LSEARCH algorithmsproduce better quality clusters than BIRCH, althoughthe latter is computationally more efficient. Since weare chiefly concerned with the quality of the shape, wecompare the output of ClusterHull against the resultsof k-median and LSEARCH (but not BIRCH).

1.3 Organization

The paper is organized in seven sections. Section 2describes the basic algorithm for computing clusterhulls. In Section 3 we discuss the cost function usedin refining and unrefining our cluster hulls. Section 4provides extensions to the basic ClusterHull algorithm.In Sections 5 and 6 we present some experimentalresults. We conclude in Section 7.

2 Representing Shape as a

Cluster of Hulls

We are interested in simple, highly efficient algorithmsthat can identify and maintain bounded-memory ap-proximations of a stream of points. Some techniquesfrom computational geometry appear especially well-suited for this. For instance, the convex hull is a usefulshape representation of the outer boundary of the wholedata stream. Although the convex hull accurately rep-resents a convex shape with an arbitrary aspect ratioand orientation, it loses all the internal details. There-fore, when the points are distributed non-uniformlywithin the convex hull, the outer hull is a poor rep-resentation of the data.

Clustering schemes, such as k-medians, partitionthe points into groups that may represent the distribu-tion better. However, because the goal of many clus-tering schemes is typically to minimize the maximumor the sum of distance functions, there is no explicit at-tention given to the shape of clusters—each cluster isconceptually treated as a ball, centered at the clustercenter. Our goal is to mediate between the two ex-tremes offered by the convex hull and k-medians. Wewould like to combine the best features of the convexhull—its ability to represent convex shapes with any

aspect ratio accurately—with those of ball-covering ap-proximations such as k-medians—their ability to repre-sent nonconvex and disconnected point sets. With thismotivation, we propose the following measure for rep-resenting the shape of a point set under the boundedmemory constraint.

Given a two-dimensional set of N points,and a memory budget of m, where m � N ,compute a set of convex hulls such that (1)the collection of hulls uses at most m vertices,(2) the hulls together cover all the points of S,and (3) the total area covered by the hulls isminimized.

Intuitively, this definition interpolates between a singleconvex hull, which potentially covers a large area,and k-medians clustering, which fails to represent theshape of individual clusters accurately. Later wewill relax the condition of “covering all the points”to deal with noisy data—in the relaxed problem, aconstant fraction of the points may be dropped fromconsideration. But the general goal will remain thesame: to compute a set of convex hulls that attemptsto cover the important geometric features of the datastream using least possible area, under the constraintthat the algorithm is allowed to use at most m vertices.

2.1 Geometric approximation in data

streams

Even the classical convex hull (outer boundary) com-putation involves some subtle and nontrivial issues inthe data stream setting. What should one do whenthe number of extreme vertices in the convex hull ex-ceeds the memory available? Clearly, some of the ex-treme vertices must be dropped. But which ones, andhow shall we measure the error introduced in this ap-proximation? This problem of summarizing the con-vex hull of a point stream using a fixed memory m hasbeen studied recently in computational geometry anddata streams [1, 6, 9, 17, 22]. An adaptive samplingscheme proposed in [22] achieves an optimal memory-error tradeoff in the following sense: given memory m,the algorithm maintains a hull that (1) lies within thetrue convex hull, (2) uses at most m vertices, and (3)approximates the true hull well—any input point notin the computed hull lies within distance O(D/m2) ofthe hull, where D is the diameter of the point stream.Moreover, the error bound of O(D/m2) is the best pos-sible in the worst case.

In our problem setting, we will maintain not onebut many convex hulls, depending on the geometry ofthe stream, with each hull roughly corresponding to a

cluster. Moreover, the locations of these hulls are notdetermined a priori—rather, as in k-medians, they aredynamically determined by the algorithm. Unlike k-medians clusters, however, each hull can use a differentfraction of the available memory to represent its clusterboundary. One of the key challenges in designing theClusterHull algorithm is to formulate a good policy forthis memory allocation. For this we will introduce acost function that the various hulls use to decide howmany hull vertices each gets. Let us first begin with anoutline of our scheme.

2.2 The basic algorithm

The available memory m is divided into two pools: afixed pool of k groups, each with a constant number ofvertices; and a shared pool of O(k) points, from whichdifferent cluster hulls draw additional vertices. Thenumber k has the same role as the parameter fed to k-medians clustering—it is set to some number at least aslarge as the number of native clusters expected in theinput. (Thus our representation will maintain a morerefined view of the cluster structure than necessary, butsimple post-processing can clean up the unnecessarysub-clustering.) The exact constants in this divisionare tunable, and we show their effect on the perfor-mance of the algorithm through experimentation. Forthe sake of concreteness, we can assume that each ofthe k groups is initially allocated 8 vertices, and thecommon pool has a total of 8k vertices. Thus, if theavailable memory is m, then we must have m ≥ 16k.

a

b

cd

Triangle ofuncertainty

Figure 2: An approximate hull, with 6 samplingdirections. The sample hull’s vertices are a, b, c, d.

Our algorithm approximates the convex hull ofeach group by its extreme vertices in selected (sam-ple) directions: among all the points assigned to thiscluster group, for each sample direction, the algorithmretains the extreme vertex in that direction. See Fig-ure 2 for an example. Each edge of this sampled hullsupports what we call an uncertainty triangle—the tri-angle formed by the edge and the tangents at the two

endpoints of the edge in the sample directions for whichthose endpoints are extreme. A simple but importantproperty of the construction is that the boundary ofthe true convex hull is sandwiched in the ring of un-certainty triangles defined by the edges of the com-puted hull. See Figure 3 for an illustration. The ex-tremal directions are divided into two sets, one contain-ing uniformly-spaced fixed directions, corresponding tothe initial endowment of memory, and another contain-ing adaptively chosen directions, corresponding to ad-ditional memory drawn from the common pool. Theadaptive directions are added incrementally, bisectingpreviously chosen directional intervals, to minimize theerror of the approximation.

pq

Figure 3: The true hull is sandwiched in a ring ofuncertainty triangles.

Each hull has an individual cost associated withit, and the whole collection of k hulls has a total costthat is the sum of the individual costs. Our goalis to choose the cost function such that minimizingthe total cost leads to a set of approximate convexhulls that represent the shape of the point set well.Furthermore, because our minimization is performedon-line, assigning each new point in the stream to aconvex hull when the point arrives, we want our costfunction to be robust : as much as possible, we want itto reduce the chance of assigning early-arriving pointsto hulls in a way that forces late-arriving points to incurhigh cost. We leave the technical details of our choiceof the cost function to the following section.

Let us now describe the high-level organization ofour algorithm. Suppose that the current point set Sis partitioned among k convex hulls H1, . . . , Hk. Thecost of hull Hi is w(Hi), and the total cost of thepartition H = {H1, . . . , Hk} is w(H) =

∑H∈H

w(H).We process each incoming point p with the followingalgorithm:

Algorithm ClusterHull

if p is contained in any H ∈ H, or in the ring ofuncertainty triangles for any such H , then

Assign p to H without modifying H .else

Create a new hull containing only p and add it to H.

if |H| > k then

Choose two hulls H, H ′ ∈ H such that mergingH and H ′ into a single convex hull will resultin the minimum increase to w(H).

Remove H and H ′ from H, merge them to forma new hull H∗, and put that into H.

If H∗ has an uncertainty triangle over either edgejoining points of the former H and H ′ whoseheight exceeds the previous maximumuncertainty triangle height, refine (repeatedlybisect) the angular interval associated with thatuncertainty triangle by choosing new adaptivedirections until the triangle height is less thanthe previous maximum.

while the total number of adaptive directionsin use exceeds ck

Unrefine (discard one of the adaptive directionsfor some H ∈ H) so that the uncertainty trianglecreated by unrefinement has minimum height.

The last two steps (refinement and unrefinement)are technical steps for preserving the approximationquality of the convex hulls that were introduced in [22].The key observation is that an uncertainty trianglewith “large height” leads to a poor approximation of aconvex hull. Ideally, we would like uncertainty trianglesto be flat. The height of an uncertainty triangle isdetermined by two key variables: the length of theconvex hull edge, and the angle-difference between thetwo sampling directions that form that triangle. Moreprecisely, consider an edge pq. We can assume that theextreme directions for p and q, namely, θp and θq , pointtoward the same side of pq, and hence the intersectionof the supporting lines projects perpendicularly ontopq. Therefore the height of the uncertainty triangle isat most the edge length `(pq) times the tangent of thesmaller of the angles between pq and the supportinglines. Observe that the sum of these two angles equalsthe angle between the directions θp and θq . If we defineθ(pq) to be |θp−θq|, then the height of the uncertaintytriangle at pq is at most `(pq) · tan(θ(pq)/2), which isclosely approximated by

(2.1)`(pq) · θ(pq)

2.

This formula forms the basis for adaptively choos-ing new sampling directions: we devote more samplingdirections to cluster hull edges whose uncertainty tri-angles have large height. Refinement is the process ofintroducing a new sampling direction that bisects twoconsecutive sampling directions; unrefinement is theconverse of this process. The analysis in [22] showed

that if a single convex hull is maintained using m/2uniformly spaced sampling directions, and m/2 adap-tively chosen directions (using the policy of minimizingthe maximum height of an uncertainty triangle), thenthe maximum distance error between true and approx-imate hulls is O(D/m2). Because in ClusterHull weshare the refinement directions among k different hulls,we choose them to minimize the global maximum un-certainty triangle height explicitly. We point out thatthe allocation of adaptive directions is independent ofthe cost function w(H). The cost function guides thepartition into convex hulls; once that choice is made,we allocate adaptive directions to minimize the errorfor that partition. One could imagine making the as-signment of adaptive directions dependent on the costfunction, but for simplicity we have chosen not to doso.

3 Choosing a Cost Function

In this section we describe the cost function we applyto the convex hulls that ClusterHull maintains. Wediscuss the intuition behind the cost function, experi-mental support for that intuition, and variants on thecost function that we considered.

The α-hull is a well-known structure for represent-ing the shape of a set of points [15]. It can be viewed asan extension of the convex hull in which half-planes arereplaced by the complements of fixed-radius disks (i.e.,the regions outside the disks). In particular, the con-vex hull is the intersection of all half-planes containingthe point set, and the α-hull is the intersection of alldisk-complements with radius ρ that contain the pointset.1 See Figure 4 for examples of the convex hull andα-hull on an L-shaped point set. The α-hull minimizesthe area of the shape that covers the points, subject tothe radius constraint on the disks.

Figure 4: Shape representations for a set of points:(left) convex hull, (right) α-hull.

1In the definition of α-hulls, the disk radius ρ = 1/|α|, andα ≤ 0, but we are not concerned with these technical details.

The α-hull is not well suited to represent the shapeof a stream of points, because an unbounded number ofinput points may appear on the boundary of the shape.Our goal of covering the input points with bounded-complexity convex hulls of minimum total area is anattempt to mimic the modeling power of the α-hull ina data stream setting.

Although our goal is to minimize the total area ofour convex hull representation, we use a slightly morecomplex function as the cost of a convex hull H :

(3.2) w(H) = area(H) + µ · (perimeter(H))2.

Here µ is a constant, chosen empirically as describedbelow. Note that the perimeter is squared in thisexpression to match units: if the perimeter termentered linearly, then simply changing the units ofmeasurement would change the relative importanceof the area and perimeter terms, which would beundesirable.

Figure 5: Input distributions: L-shaped and ellipses.

We want to minimize total area, and so definingw(H) = area(H) seems natural; however, this provesto be infeasible in a stream setting. If a point set hasonly two points, the area of its convex hull is zero;thus all such hulls have the same cost. The first 2kpoints that arrive in a data stream are paired up intok two-point convex hulls, each with cost zero, and thepairing will be arbitrary. In particular, some convexhulls are likely to cross natural cluster boundaries.When these clusters grow as more points arrive, theywill have higher cost than the optimal hulls thatwould have been chosen by an off-line algorithm. Thiseffect is clearly visible in the clusters produced byour algorithm in Figure 6 (right) for the ellipses dataset of Figure 5 (right). By contrast, the L-shapeddistribution of Figure 5 (left) is recovered well usingthe area cost function, as shown in Figure 6 (left).

We can avoid the tendency of the area cost tocreate long thin needles in the early stages of thestream by minimizing the perimeter. If we choosew(H) = perimeter(H), then the well-separated clus-ters of the ellipses data set are recovered perfectly, evenwhen the points arrive on-line—see Figure 7 (right).

Figure 6: With the area cost function, ClusterHullfaithfully recovers the L-shaped distribution of points.But it performs poorly on a set of n = 10, 000points distributed among ten elliptical clusters; itmerges pairs of points from different groups and createsintersecting hulls.

Figure 7: With the perimeter cost function, Cluster-Hull faithfully recovers the disjoint elliptical clusters,but performs poorly on the L-shaped distribution.

However, as the poor recovery of the L distributionshows (Figure 7 (left)), the perimeter cost has its ownliabilities. The total perimeter of two hulls that arerelatively near each other can often be reduced bymerging the two into one. Furthermore, merging twolarge hulls reduces the perimeter more than mergingtwo similar small ones, and so the perimeter costapplied to a stream often results in many small hullsand a few large ones that contain multiple “natural”clusters.

We need to incorporate both area and perimeterinto our cost function to avoid the problems shownin Figures 6 and 7. Because our overall goal is tominimize area, we choose to keep the area term primaryin our cost function (Equation 3.2). In that function(perimeter(H))2 is multiplied by a constant µ, whichis chosen to adjust the relative importance of area andperimeter in the cost. Experimentation shows thatchoosing µ = 0.05 gives good shape reconstruction ona variety of inputs. With µ substantially smaller than0.05, the perimeter effect is not strong enough, andwith µ greater than 0.1, it is too strong. (Intuitively,we want to add just enough perimeter dependence toavoid creating needle convex hulls in the early stagesof the stream.)

Figure 8: With the combined area and perimeter costfunction, the algorithm ClusterHull recovers both theellipse and L distributions. The choice of µ = 0.05gives good shape reconstruction.

We can understand the combined area-perimetercost by modeling it as the area of a fattened convexhull. If we let ρ = µ · perimeter(H), we see thatthe area-perimeter cost (3.2) is very close to the areaobtained by fattening H by ρ. The true area isarea(H)+ρ ·perimeter(H)+πρ2 = area(H)+ρ2( 1

µ+

π); if µ is small, then 1/µ is relatively large comparedto π, and the extra πρ2 term is not very significant.

Because the cost (3.2) may fatten long thinclusters more than is desirable, we also experi-mented with replacing the constant µ in (3.2) bya value inversely related to the aspect ratio of H .The aspect ratio of H is diam(H)/width(H) =Θ((perimeter(H))2/area(H)). Thus if we simply re-placed µ by 1/aspectRatio(H) in (3.2), we would es-sentially obtain the area cost. We compromised byusing the cost

w(H) = area(H) +

µ · (perimeter(H))2/(aspectRatio(H))x

for various values of x (x = 0.5, x = 0.1).The aspect ratio is conveniently approximated as(perimeter(H))2/area(H), since the quantities inthat expression are already maintained by our convexhull approximation. Except in extreme cases, the re-sults with this cost function were not enough differentfrom the basic area-perimeter cost to merit a separatefigure.

The cost (3.2) fattens each hull by a radius pro-portional to its own perimeter. This is appropriate ifthe clusters have different natural scales and we wantto fatten each according to its own dimensions. How-ever, in our motivating structure the α-hull, a uniformradius is used to define all the clusters. To fatten hullsuniformly, we could use the weight function

w(H) = area(H) + ρ · perimeter(H) + πρ2.

However, the choice of the fattening radius ρ is prob-lematic. We might like to choose ρ such that α-hulls

defined using radius-ρ disks form exactly k clusters,but then the optimum value of ρ would decrease andincrease as the stream points arrived. We can avoidthese difficulties by sticking to the simpler cost of def-inition (3.2).

4 Extensions and Enhancements

In this section we discuss how to enhance the basicClusterHull algorithm to improve the quality of shaperepresentation.

4.1 Spatial incoherence and

period-doubling cleanup

In many data streams the arriving points are orderedarbitrarily, possibly even adversarily. The ClusterHullscheme (and indeed any on-line clustering algorithm)is vulnerable to early errors, in which an early-arrivingpoint is assigned to a hull that later proves to be thewrong one.

Figure 9 (left) shows a particularly bad input con-sisting of five thin parallel stripes. We used ClusterHullwith µ = 0.05 to maintain five hulls, with the inputpoints ordered randomly. A low density sample fromthe stripe distribution (such as a prefix of the stream)looks to the algorithm very much like uniformly dis-tributed points. Early hull merges combine hulls fromdifferent stripes, and the ClusterHull algorithm cannotrecover from this mistake. See Figure 9 (right).

Figure 9: Processing the stripes input (left) in randomorder leads to errors for our algorithm (right).

If the input data arrive in random order, theidea of period-doubling cleanup may help identify andamplify the true clusters. The idea is to process theinput stream in rounds in which the number of pointsprocessed doubles in each round. At the end of eachround we identify low density hulls and discard them—these likely group points from several true clusters.The dense hulls are retained from round to round, andare allowed to grow.

Formally, the period-doubling cleanup operates asfollows: For each H ∈ H we maintain the number of

points it represents, denoted by count(H). The densityof any hull H is density(H) = count(H)/area(H).The algorithm also maintains an approximate convexhull G of all the input points. After each round, itdiscards from H every hull H for which any of thefollowing holds:

• count(H) < δ · N/k

• density(H) < density(G)

• density(H) < 1

2· median{density(A) : A ∈ H}

Here N is the number of points seen so far. In ourexperiments, we set the tunable parameter δ to 0.1.

The first test takes care of hulls with a very smallnumber of tightly clustered points (these may have highdensities because of their smaller area, and will not becaught by density pruning). The second test discardshulls that have less than average density. The intuitionis that each cluster should be at least as dense as theentire input space (otherwise it is not an interestingcluster). In case the points are distributed over avery large area, but the individual clusters are verycompact, the average density may not be very helpfulfor discarding hulls. Instead, we should discard hullsthat have low densities relative to other hulls in thedata structure; the third test takes care of this case—it discards any hull with density significantly less thanthe median density.

Figure 10 (left) shows the result of period-doublingcleanup on the stripes distribution; the sparse hullsthat were initially found have been discarded and fivedense hulls have been correctly computed. We notethat, with the same amount of memory, neither CUREnor the k-median clustering is able to represent thestripes distribution well (cf. Figure 10). Our exper-iments show that applying period-doubling cleanuphelps improve clustering on almost all data sets.

Figure 10: Period-doubling cleanup (left) on Cluster-Hull corrects the errors in the stripes distribution; themiddle figure shows the output of k-medians, and theright figure shows the output of CURE.

4.2 Noisy data and incremental

cleanup

Sampling error and outliers cause difficulty for nearlyall clustering algorithms. Likewise, a few outliers canadversely affect the ClusterHull shape summary. Analgorithm needs some way to distinguish between denseregions of the input distribution (the true clusters) andsparse ones (noise). In this section, we propose anincremental cleanup mechanism that can improve theperformance of our algorithm in the presence of noise.Both the period-doubling and the incremental cleanupare inspired by sampling techniques used in frequencyestimation in streams. In particular, period-doublingis inspired by sticky sampling and incremental cleanupis inspired by lossy counting [23]. The incrementalcleanup also processes the input in rounds, but therounds do not increase in length. This is becausenoise is not limited to the beginning of the input; ifwe increased the round length, all the hulls would becorrupted by noisy points. Instead, we fix the size ofeach round depending on the (estimated) noise in theinput.

Specifically, the incremental cleanup assumes thatthe input stream consists of points drawn randomlyfrom a fixed distribution, with roughly (1 − ε)N ofthem belonging to high density clusters and εN of thembeing low density noise. The expected noise frequencyε affects the quality of the output. We can estimateit conservatively if it is unknown. The idea is to setthe value of δ to be roughly equal to ε, and process theinput in rounds of k/(2ε) points. The logic is that inevery round, only about k/2 hulls will be corrupted bynoisy points, still leaving half of the hulls untouchedand free to track the true distribution of the input. Ifwe set k to be more than twice the expected numberof natural clusters in the input, we obtain a goodrepresentation of the clusters.

Figure 11: Incremental cleanup, with estimated noisefrequency ε = 0.1, applied to distributions with 1%,10%, and 20% actual background noise.

This scheme propagates the good hulls (those withhigh density and high cardinality) from one round ofthe algorithm to the next, while discarding hulls thatare sparse or belong to outliers. See Figure 11 foran example of how this scheme identifies true clusters

and discards noisy regions. Of course, if noise isunderestimated significantly (Figure 11 (right)), thequality of the cluster hulls suffers.

4.3 Spatial coherence and trajectory

tracking

Section 4.1 considered spatially incoherent inputstreams. If the input is spatially coherent, as occursin some applications, ClusterHull performs particularlywell. If the input stream consists of locations reportedby sensors detecting some moving entity (a light penon a tablet, a tank in a battlefield, animals in a remotehabitat), our algorithm effectively finds a covering ofthe trajectory by convex “lozenges.” The algorithmalso works well when there are multiple simultaneoustrajectories to represent, as might occur when sensorstrack multiple independent entities. If the stripes ofFigure 9 are fed to the algorithm in left-to-right orderthey are recovered perfectly; likewise in Figure 12 asynthetic trajectory is represented accurately.

(a) (b) (c)

Figure 12: Input along a trajectory in (a); the shape isrecovered well using m = 100 in (b), and using m = 150in (c).

4.4 Density estimation and display

Stream versions of traditional clustering schemes (in-cluding k-median and CURE) do not include an es-timate of the density of points associated with eachcluster center, whereas each cluster hull H can easilymaintain count(H). As in Section 4.1, this gives an es-timate of the density of the points in each hull. If thisinformation is displayed graphically (cf. Figure 13) itconveys more insight about the distribution of the in-put data than does the simple cluster-center output oreven cluster-sample output.

5 Implementation and

Experiments

We implemented the convex hull algorithm of [22]and the ClusterHull algorithm on top of it. The

convex hull algorithm takes logarithmic time per in-serted point, on the average, but our ClusterHull imple-mentation is more simple-minded, optimized more forease of implementation than for runtime performance.The bottleneck in our implementation is neighborhoodqueries/point location, taking time proportional to thenumber of hulls. By storing the hull edges in a quad-tree, we could speed up these operations to O(log m)time.

When a new point arrives, we must check whichhull it belongs to, if any. Using a quad-tree, thisreduces to a logarithmic-time search, followed bylogarithmic-time point-in-polygon tests with an ex-pected constant number of hulls. Each new hull Hmust compute its optimum merge cost—the minimumincrement to w(H) caused by merging H with anotherhull. On average, this increment is greater for moredistant hulls. Using the quad-tree we can compute theincrement for O(1) nearby hulls first. Simple lowerbounds on the incremental cost of distant merges thenlet us avoid computing the costs for distant hulls. Com-puting the incremental cost for a single pair of hulls re-duces to computing tangents between the hulls, whichtakes logarithmic time [22].

Merging hulls eliminates some vertices forever, andso we can charge the time spent performing the mergeto the deleted vertices. Thus a careful implementationof the ClusterHull algorithm would process streampoints in O(log m) amortized time per point, wherem is the total number of hull vertices.

In the remainder of this section we evaluate theperformance of our algorithm on different data sets.When comparing our scheme with k-median cluster-ing [19], we used an enhanced version of the latter.The algorithm is allowed to keep a constant number ofsample points per cluster, which can be used to deducethe approximate shape of that cluster. We ran the k-medians clustering using k clusters and total memory(number of samples) equal to m. CURE already hasa parameter for maintaining samples in each cluster,so we used that feature. In this section, we analyzethe output of these three algorithms (ClusterHull, k-median, CURE) on a variety of different data sets, andas a function of m, the memory.

Throughout this section, we use the period-doubling cleanup along with the area-perimeter cost(Equation 3.2) to compute the hulls. We use µ = .05and r, the number of initial sample directions per hull,equal to 8. The values of these parameters are criticalfor our algorithm; however, in this section we use thesame set of parameters for all data sets. This showsthat when tuned properly, our algorithm can generate

good quality clusters for a variety of input distributionsusing a single set of parameters. In the next section,we will analyze in detail the effects of these param-eters on the results of our scheme. To visualize theoutput, we also shade the hulls generated by our algo-rithm according to their densities (darker regions aremore dense).

5.1 West Nile virus spread

(a)

(b) (c)

(d) (e)

(f) (g)

Figure 13: The westnile data set is shown in the topfigure (a). Figures (b) and (c) show the outputs ofClusterHull for m = 256 and m = 512. Figures (d)and (e) show the corresponding outputs for k-medians.Figures (f) and (g) show the corresponding outputs forCURE.

Our first data set, westnile (Figure 13 (a)), con-tains about 68, 000 points corresponding to the loca-tions of the West Nile virus cases reported in the US, ascollected by the CDC and the USGS [29]. We random-ized the input order to eliminate any spatial coherencethat might give an advantage to our algorithm. We ranClusterHull to generate output of total size m = 256

and 512 (Figures (b) and (c)). The clustering algo-rithms k-medians and CURE were used to generateclusters with the same amount of memory. The resultsare shown in Figure 13.

All three algorithms are able to track high-densityregions in coherent clusters, but there was little infor-mation about the shapes of the clusters in the outputof k-medians or CURE. Visually the output of Cluster-Hull looks strikingly similar to the input set, offeringthe analyst a faithful yet compact representation of thegeometric shapes of important regions.

5.2 The circles and the ellipse data

sets

(a) (d)

(b) (e)

(c) (f)

Figure 14: The left column shows output of ClusterHull(top), k-medians (middle) and CURE (bottom) forcircles dataset with m = 64. The right columnshows corresponding outputs for ellipse dataset withm = 128.

In this experiment, we compared ClusterHull withk-median and CURE on the circles and the ellipsedata sets described earlier. The circles set containsn = 10, 000 points generated inside 3 circles of differentsizes. We ran the three algorithms with a total memorym = 64. The output of ClusterHull is shown inFigure 14 (a); the output of k-median is shown in (b);and the output of CURE is shown in (c).

Similarly, Figures 14 (d), (e), and (f), respectively,show the outputs of ClusterHull, k-median, and CURE

on the ellipse data set with memory m = 128. Theellipse data set contains n = 10, 000 points distributedamong ten ellipse-shaped clusters.

In all cases, ClusterHull output is more accurate,visually informative, and able to compute the bound-aries of clusters with remarkable precision. The out-puts of other schemes are ambiguous, inaccurate, andlacking in details of the cluster shape boundary. Forthe circles data, the k-median does a poor job in de-termining the true cluster structure. For the ellipsedata, CURE does a poor job in separating the clus-ters. (CURE needed a much larger memory—a “win-dow size” of at least 500—to separate the clusters cor-rectly.)

6 Tuning ClusterHull

Parameters

In this section, we study the effects of various param-eters on the quality of clusters.

6.1 Variation with r

r = 4 r = 6

r = 8 r = 16

Figure 15: The result of varying r on the ellipses dataset.

We first consider the effect of changing r, thenumber of initial directions assigned to each hull. Toisolate the effects of r, we fixed the values of µ = .05and k = 10. We ran the experiments on two data sets,ellipses and circles. The results are shown in Figures 15and 16, respectively.

The results show that the shape representationwith 4 initial directions is very crude: ellipses areturned into pointy polygons. As we increase r, therepresentation of clusters becomes more refined. Thiscontrast can be seen if we compare the boundary of

r = 4 r = 6

r = 8 r = 16

Figure 16: The result of varying r on the circles dataset.

the big circle in Figure 16 for r = 4 and 8. However,increasing the number of directions means that we needmore memory for the hulls (memory grows linearly withr). For r = 8, we get a good balance between memoryusage and the quality of the shapes.

6.2 Variation with µ

µ = 0.01 µ = 0.05

µ = 0.1 µ = 0.5

Figure 17: For the circles data set, ClusterHull recoversclusters correctly for µ ∈ [.05, .5], but fails for µ ≤ .01.

We considered values of µ in the range [.01, .5].We fixed r = 8, and ran our algorithm for two datasets, circles and stripes. We fixed the number of hulls,k = 10 (m = 128) for circles and k = 5 (m = 64) forstripes.

If the value of µ is too small, the area dominates thecost function. This causes distant hulls to merge intolong skinny hulls spanning multiple clusters. Althoughthe period-doubling cleanup gets rid of most of them

µ = 0.01 µ = 0.05

µ = 0.1 µ = 0.5

Figure 18: For the stripes data set, ClusterHull re-covers clusters correctly for µ ∈ [.01, .1], but fails forµ ≥ .5.

by discarding hulls with small densities, the output stillcontains some hulls spanning multiple natural clusters.Figure 17 shows this effect when µ = .01.

On the other hand, if µ is increased too much, thecost function prefers decreasing the total perimeter,and it is hard to prevent large neighboring clusters frommerging together. In Figure 18, neighboring stripes aremerged into a single hull for µ = .5. The results showthat choosing µ in the range [.05, .1] gives good clustersfor most input sets.

7 Conclusion

We developed a novel framework for summarizing thegeometric shape and distribution of a two-dimensionalpoint stream. We also proposed an area-based qualitymeasure for such a summary. Unlike existing stream-clustering methods, our scheme adaptively allocatesits fixed memory budget to represent different clusterswith different degrees of detail. Such an adaptivescheme can be particularly useful when the input haswidely varying cluster structures, and the boundaryshape, orientation, or volume of those clusters can beimportant clues in the analysis.

Our scheme uses a simple and natural cost func-tion to control the cluster structure. Experiments showthat this cost function performs well across widely dif-ferent input distributions. The overall framework ofClusterHull is flexible and easily adapted to differentapplications. For instance, we show that the schemecan be enhanced with period-doubling and incremental

cleanup to deal effectively with noise and extreme datadistributions. In those settings, especially when the in-put has spatial coherence, our scheme performs notice-ably better than general-purpose clustering methodslike CURE and k-medians.

Because our hulls tend to be more stable than, forinstance, the centroids of k-medians, we can maintainother useful data statistics such as population count ordensity of individual hulls. (Our hulls grow by mergingwith other hulls, whereas the centroids in k-medianspotentially shift after each new point arrival. The useof incremental cleanup may cause some of our hullsto be discarded, but that happens only for very low-weight, and hence less-interesting hulls.) Thus, thecluster hulls can capture some important frequencystatistics, such as which five hulls have the most points,or which hulls have the highest densities, etc.

Although ClusterHull is inspired by the α-hulland built on top of an optimal convex hull structure,the theoretical guarantees of those structures do notextend to give approximation bounds for ClusterHull.Providing a theoretical justification for ClusterHull’spractical performance is a challenge for future work.

References

[1] P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan.Approximating extent measures of points. J. ACM,51(4):606–635, 2004.

[2] C. C. Aggarwal, J. Han, J. Wang, and P. S. Yu. Aframework for clustering evolving data streams. InProc. of the 29th VLDB conference, 2003.

[3] N. Alon, Y. Matias, M. Szegedy. The space complexityof approximating the frequency moments. J. Comput.

Syst. Sci. 58 (1999), 137–147.

[4] N. Amenta, M. Bern, and D. Eppstein. The crust andthe β-skeleton: Combinatorial curve reconstruction.Graphical Models and Image Processing, 60:125–135,1998.

[5] P. S. Bradley, U. Fayyad, and C. Reina. ScalingClustering Algorithms to Large Databases. Proc. 4thInternational Conf. on Knowledge Discovery and DataMining (KDD-98), 1998.

[6] T. Chan. Faster core-set constructions and datastream algorithms in fixed dimensions. In Proc. 20th

Annu. ACM Sympos. Comput. Geom., pages 152–159,2004.

[7] M. Charikar, C. Chekuri, T. Feder, and R. Motwani.Incremental clustering and dynamic information re-trieval. In Proc. 29th Annu. ACM Symp. Theory Com-

puting, pages 626–635, 1997.

[8] D. Cheng, R. Kannan, S. Vempala, G. Wang ADivide-and-Merge Methodology for Clustering. InProc. of the ACM Symposium on Principles ofDatabase Systems, 2005.

[9] G. Cormode and S. Muthukrishnan. Radial histogramfor spatial streams. Technical Report 2003-11, DI-MACS, 2003.

[10] B. Curless and M. Levoy. A volumetric method forbuilding complex models from range images. In Proc.

SIGGRAPH 96, pages 303–312, 1996.

[11] T. K. Dey, K. Mehlhorn, and E. Ramos. Curvereconstruction: Connecting dots with good reason.Comp. Geom.: Theory and Appl., 15:229–244, 2000.

[12] P. Domingos and G. Hulten. A general method forscaling up machine learning algorithms and its appli-cation to clustering. In Proc. of the 18th InternationalConference on Machine Learning, ICML, 2001

[13] P. Domingos, G. Hulten. Learning from InfiniteData in Finite Time. In Proc. Advances in NeuralInformation Processing Systems (NIPS), 2002

[14] P. Drineas, A. Frieze, R. Kannan, S. Vempala, andV. Vinay. Clustering in large graphs and matrices.In Proc. of 10th Symposium on Discrete Algorithms(SODA), 1999.

[15] H. Edelsbrunner. Algorithms in Combinatorial Geom-

etry, volume 10 of EATCS Monographs on Theoretical

Computer Science. Springer-Verlag, Heidelberg, WestGermany, 1987.

[16] M. Ester, H. Kriegel and X. Xu. A database interfacefor clustering in large spatial databases. In Int.Conference on Knowledge Discovery in Databases andData Mining (KDD-95), Montreal, Cananda, August1995.

[17] J. Feigenbaum, S. Kannan, and J. Zhang. Computingdiameter in the streaming and sliding-window models,2002. Manuscript.

[18] A. Frieze, R. Kannan and S. Vempala. Fast Monte-Carlo algorithms for finding low-rank approximations.In Proc. of 39th Symposium on Foundations of Com-puter Science (FOCS), 1998.

[19] S. Guha, A. Meyerson, N. Mishra, R. Motwani, andL. O’Callaghan. Clustering data streams: Theory and

practice. IEEE Trans. Knowl. Data Engineering, Vol.15, pages 515–528, 2003.

[20] S. Guha, R. Rastogi, and K. Shim. CURE: An efficientclustering algorithm for large databases. In Proc.of International Conference on Management of Data(SIGMOD), 1998.

[21] M. Greenwald and S. Khanna. Space-efficient onlinecomputation of quantile summaries. Proc. of SIG-MOD, pages 58–66, 2001.

[22] J. Hershberger and S. Suri. Adaptive sampling forgeometric problems over data streams. In Proc. 23rd

ACM Sympos. Principles Database Syst., pages 252–262, 2004.

[23] G. S. Manku and R. Motwani. Approximate frequencycounts over data streams. In VLDB, pages 346–357,2002.

[24] S. Muthukrishnan. Data streams: Algorithms andapplications. Preprint, 2003.

[25] R. T. Ng and J. Han. Efficient and Effective ClusteringMethods for Spatial Data Mining. In Proceedings ofthe 20th VLDB Conference, 1994.

[26] J. O’Rourke and G. T. Toussaint. Pattern recognition.In Jacob E. Goodman and Joseph O’Rourke, editors,Handbook of Discrete and Computational Geometry,chapter 43, pages 797–813. CRC Press LLC, BocaRaton, FL, 1997.

[27] D. Pelleg, and A. W. Moore. X-means: ExtendingK-means with Efficient Estimation of the Number ofClusters. In Proc. of the 17th International Confer-ence on Machine Learning (ICML), 2000.

[28] S. Vempala R. Kannan and A. Vetta On clusterings -good, bad and spectral. In Proc. 41st Symposium onthe Foundation of Computer Science, FOCS, 2000.

[29] USGS West Nile Virus Maps - 2002. http://cindi.usgs.gov/hazard/event/west nile/.

[30] T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH:An Efficient Data Clustering Method for Very LargeDatabases. In Proc. of the International Conferenceon Management of Data (SIGMOD), 1996.