research-article

Approximation algorithms for clustering uncertain data

Authors:

Graham Cormode,

Andrew McGregorAuthors Info & Claims

PODS '08: Proceedings of the twenty-seventh ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems

Pages 191 - 200

https://doi.org/10.1145/1376916.1376944

Published: 09 June 2008 Publication History

Abstract

There is an increasing quantity of data with uncertainty arising from applications such as sensor network measurements, record linkage, and as output of mining algorithms. This uncertainty is typically formalized as probability density functions over tuple values. Beyond storing and processing such data in a DBMS, it is necessary to perform other data analysis tasks such as data mining. We study the core mining problem of clustering on uncertain data, and define appropriate natural generalizations of standard clustering optimization criteria. Two variations arise, depending on whether a point is automatically associated with its optimal center, or whether it must be assigned to a fixed cluster no matter where it is actually located.

For uncertain versions of k-means and k-median, we show reductions to their corresponding weighted versions on data with no uncertainties. These are simple in the unassigned case, but require some care for the assigned version. Our most interesting results are for uncertain k-center, which generalizes both traditional k-center and k-median objectives. We show a variety of bicriteria approximation algorithms. One picks O(kε--¹log²n) centers and achieves a (1 + ε) approximation to the best uncertain k-centers. Another picks 2k centers and achieves a constant factor approximation. Collectively, these results are the first known guaranteed approximation algorithms for the problems of clustering uncertain data.

References

[1]

C. Aggarwal and P. S. Yu. Framework for clustering uncertain data streams. In IEEE International Conference on Data Engineering, 2008.

Digital Library

[2]

D. Arthur and S. Vassilvitskii. kmeans++: The advantages of careful seeding. In ACM-SIAM Symposium on Discrete Algorithms, pages 1027--1035, 2007.

Digital Library

[3]

V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544--562, 2004.

Digital Library

[4]

M. Badoiu, S. Har-Peled, and P. Indyk. Approximate clustering via core-sets. In ACM Symposium on Theory of Computing, pages 250--257, 2002.

Digital Library

[5]

O. Benjelloun, A. D. Sarma, A. Y. Halevy, and J. Widom. Uldbs: Databases with uncertainty and lineage. In International Conference on Very Large Data Bases, 2006.

Digital Library

[6]

M. Charikar, S. Khuller, D. M. Mount, and G. Narasimhan. Algorithms for facility location problems with outliers. In ACM-SIAM Symposium on Discrete Algorithms, pages 642--651, 2001.

Digital Library

[7]

M. Chau, R. Cheng, B. Kao, and J. Ngai. Uncertain data mining: An example in clustering location data. In Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), 2006.

Digital Library

[8]

G. Cormode and M. N. Garofalakis. Sketching probabilistic data streams. In Proceedings of ACM SIGMOD International Conference on Management of Data, pages 281--292, 2007.

Digital Library

[9]

N. N. Dalvi and D. Suciu. Efficient query evaluation on probabilistic databases. VLDB J., 16(4):523--544, 2007.

Digital Library

[10]

A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1):1--38, 1977.

[11]

J. C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. Journal of Cybernetics, 3:32--57, 1973.

[12]

M. Dyer and A. Frieze. A simple heuristic for the p-center problem. Operations Research Letters, 3:285--288, 1985.

Digital Library

[13]

M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases with noise. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, page 226, 1996.

Digital Library

[14]

D. Feldman, M. Monemizadeh, and C. Sohler. A PTAS for k-means clustering based on weak coresets. In Symposium on Computational Geometry, 2007.

Digital Library

[15]

T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38(2-3):293--306, 1985.

[16]

S. Guha, R. Rastogi, and K. Shim. CURE: An efficient clustering algorithm for large databases. In Proceedings of ACM SIGMOD International Conference on Management of Data, pages 73--84, 1998.

Digital Library

[17]

S. Har-Peled. Geometric approximation algorithms. http://valis.cs.uiuc.edu/~sariel/teach/notes/aprx/book.pdf, 2007.

Digital Library

[18]

S. Har-Peled and S. Mazumdar. On coresets for k-means and k-median clustering. In ACM Symposium on Theory of Computing, pages 291--300, 2004.

Digital Library

[19]

D. Hochbaum and D. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10(2):180--184, May 1985.

Digital Library

[20]

M. Inaba, N. Katoh, and H. Imai. Applications of weighted voronoi diagrams and randomization to variance-based k-clustering (extended abstract). In Symposium on Computational Geometry, pages 332--339, 1994.

Digital Library

[21]

P. Indyk. Algorithms for dynamic geometric problems over data streams. In ACM Symposium on Theory of Computing, 2004.

Digital Library

[22]

T. S. Jayram, A. McGregor, S. Muthukrishnan, and E. Vee. Estimating statistical aggregates on probabilistic data streams. In ACM Symposium on Principles of Database Systems, pages 243--252, 2007.

Digital Library

[23]

S. G. Kolliopoulos and S. Rao. A nearly linear-time approximation scheme for the euclidean k-median problem. In Proceedings of European Symposium on Algorithms, 1999.

Digital Library

[24]

A. Kumar, Y. Sabharwal, and S. Sen. A simple linear time (1+ε)-approximation algorithm for k-means clustering in any dimensions. In IEEE Symposium on Foundations of Computer Science, 2004.

Digital Library

[25]

J. B. MacQueen. Some method for the classification and analysis of multivariate observations. In Proceedings of the 5th Berkeley Symposium on Mathematical Structures, pages 281--297, 1967.

[26]

W. K. Ngai, B. Kao, C. K. Chui, R. Cheng, M. Chau, and K. Y. Yip. Efficient clustering of uncertain data. In IEEE International Conference on Data Mining, 2006.

Digital Library

[27]

R. Panigrahy and S. Vishwanathan. An O(log* n) approximation algorithm for the asymmetric p-center problem. J. Algorithms, 27(2):259--268, 1998.

Digital Library

[28]

T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH: an efficient data clustering method for very large databases. In Proceedings of ACM SIGMOD International Conference on Management of Data, pages 103--114, 1996.

Digital Library

Cited By

Chi GGuo LJia C(2025)A Local Search Algorithm for the Radius-Constrained k-Median ProblemTheory of Computing Systems10.1007/s00224-024-10211-w69:1Online publication date: 30-Jan-2025
https://doi.org/10.1007/s00224-024-10211-w
Alipour SShahbazi Gholiabad EAmin Raeisi M(2024)Uncertain k-Center Clustering, Revisited: Point AssignmentKnowledge Science, Engineering and Management10.1007/978-981-97-5495-3_21(280-291)Online publication date: 16-Aug-2024
https://dl.acm.org/doi/10.1007/978-981-97-5495-3_21
Pileggi S(2024)A Cross-Domain Perspective to Clustering with UncertaintyComputational Science – ICCS 202410.1007/978-3-031-63783-4_22(295-308)Online publication date: 29-Jun-2024
https://doi.org/10.1007/978-3-031-63783-4_22
Show More Cited By

Index Terms

Approximation algorithms for clustering uncertain data
1. Information systems
  1. Information retrieval
    1. Retrieval tasks and goals
  2. Information systems applications
    1. Data mining
      1. Clustering

Recommendations

Representative clustering of uncertain data
KDD '14: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining

This paper targets the problem of computing meaningful clusterings from uncertain data sets. Existing methods for clustering uncertain data compute a single clustering without any indication of its quality and reliability; thus, decisions based on their ...
Efficient clustering of uncertain data streams

Clustering uncertain data streams has recently become one of the most challenging tasks in data management because of the strict space and time requirements of processing tuples arriving at high speed and the difficulty that arises from handling ...
Mechanisms to improve clustering uncertain data with UKmeans
Abstract
Uncertain data in Kmeans clustering, namely UKmeans, have been discussed in decade years. UKmeans clustering, however, has some difficulties of time performance and effectiveness because of the uncertainty of objects. In this study, we ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

PODS '08: Proceedings of the twenty-seventh ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems

June 2008

330 pages

ISBN:9781605581521

DOI:10.1145/1376916

General Chair:
Phokion Kolaitis
IBM Almaden Research Center, USA
,
Program Chair:
Maurizio Lenzerini
SAPIENZA University of Rome, Italy

Copyright © 2008 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 June 2008

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

SIGMOD/PODS '08

Sponsor:

SIGMOD/PODS '08: SIGMOD/PODS '08 - International Conference on Management of Data

June 9 - 12, 2008

Vancouver, Canada

Acceptance Rates

PODS '08 Paper Acceptance Rate 28 of 159 submissions, 18%;

Overall Acceptance Rate 642 of 2,707 submissions, 24%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

85
Total Citations
View Citations
1,229
Total Downloads

Downloads (Last 12 months)34
Downloads (Last 6 weeks)0

Reflects downloads up to 07 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Chi GGuo LJia C(2025)A Local Search Algorithm for the Radius-Constrained k-Median ProblemTheory of Computing Systems10.1007/s00224-024-10211-w69:1Online publication date: 30-Jan-2025
https://doi.org/10.1007/s00224-024-10211-w
Alipour SShahbazi Gholiabad EAmin Raeisi M(2024)Uncertain k-Center Clustering, Revisited: Point AssignmentKnowledge Science, Engineering and Management10.1007/978-981-97-5495-3_21(280-291)Online publication date: 16-Aug-2024
https://dl.acm.org/doi/10.1007/978-981-97-5495-3_21
Pileggi S(2024)A Cross-Domain Perspective to Clustering with UncertaintyComputational Science – ICCS 202410.1007/978-3-031-63783-4_22(295-308)Online publication date: 29-Jun-2024
https://doi.org/10.1007/978-3-031-63783-4_22
Saldanha-da-Gama FWang SSaldanha-da-Gama FWang S(2024)Territory DesignFacility Location Under Uncertainty10.1007/978-3-031-55927-3_13(415-436)Online publication date: 26-Feb-2024
https://doi.org/10.1007/978-3-031-55927-3_13
Lee ENoh SSeo J(2023)SageProceedings of the VLDB Endowment10.14778/3565838.356584415:13(3897-3910)Online publication date: 20-Jan-2023
https://dl.acm.org/doi/10.14778/3565838.3565844
Goyal DJaiswal R(2023)Tight FPT approximation for constrained k-center and k-supplierTheoretical Computer Science10.1016/j.tcs.2022.11.001940:PA(190-208)Online publication date: 9-Jan-2023
https://dl.acm.org/doi/10.1016/j.tcs.2022.11.001
Chen DWang XXu XZhong CXu J(2022)Sparse non-negative matrix factorization for uncertain data clusteringIntelligent Data Analysis10.3233/IDA-20562226:3(615-636)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.3233/IDA-205622
Wilkinson LLuo H(2022)A Distance-Preserving Matrix SketchJournal of Computational and Graphical Statistics10.1080/10618600.2022.205024631:4(945-959)Online publication date: 13-May-2022
https://doi.org/10.1080/10618600.2022.2050246
Acharyya AJallu RKeikha VLöffler MSaumell M(2022)Minimum color spanning circle of imprecise pointsTheoretical Computer Science10.1016/j.tcs.2022.07.016930:C(116-127)Online publication date: 21-Sep-2022
https://dl.acm.org/doi/10.1016/j.tcs.2022.07.016
Hussain SButt IHanif MAnwar S(2022)Clustering uncertain graphs using ant colony optimization (ACO)Neural Computing and Applications10.1007/s00521-022-07063-134:14(11721-11738)Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.1007/s00521-022-07063-1
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten