Finding hierarchical heavy hitters in streaming data

ABSTRACT

Data items that arrive online as streams typically have attributes which take values from one or more hierarchies (time and geographic location, source and destination IP addresses, etc.). Providing an aggregate view of such data is important for summarization, visualization, and analysis. We develop an aggregate view based on certain organized sets of large-valued regions (“heavy hitters”) corresponding to hierarchically discounted frequency counts. We formally define the notion of hierarchical heavy hitters (HHHs). We first consider computing (approximate) HHHs over a data stream drawn from a single hierarchical attribute. We formalize the problem and give deterministic algorithms to find them in a single pass over the input.

In order to analyze a wider range of realistic data streams (e.g., from IP traffic-monitoring applications), we generalize this problem to multiple dimensions. Here, the semantics of HHHs are more complex, since a “child” node can have multiple “parent” nodes. We present online algorithms that find approximate HHHs in one pass, with provable accuracy guarantees. The product of hierarchical dimensions forms a mathematical lattice structure. Our algorithms exploit this structure, and so are able to track approximate HHHs using only a small, fixed number of statistics per stored item, regardless of the number of dimensions.

We show experimentally, using real data, that our proposed algorithms yields outputs which are very similar (virtually identical, in many cases) to offline computations of the exact solutions, whereas straightforward heavy-hitters-based approaches give significantly inferior answer quality. Furthermore, the proposed algorithms result in an order of magnitude savings in data structure size while performing competitively.

REFERENCES

• 1: Sameet Agarwal , Rakesh Agrawal , Prasad Deshpande , Ashish Gupta , Jeffrey F. Naughton , Raghu Ramakrishnan , Sunita Sarawagi, On the Computation of Multidimensional Aggregates, Proceedings of the 22th International Conference on Very Large Data Bases, p.506-521, September 03-06, 1996
• 2: Kevin Beyer , Raghu Ramakrishnan, Bottom-up computation of sparse and Iceberg CUBE, Proceedings of the 1999 ACM SIGMOD international conference on Management of data, p.359-370, May 31-June 03, 1999, Philadelphia, Pennsylvania, United States
• 3: Moses Charikar , Kevin Chen , Martin Farach-Colton, Finding Frequent Items in Data Streams, Proceedings of the 29th International Colloquium on Automata, Languages and Programming, p.693-703, July 08-13, 2002
• 4: Graham Cormode , Theodore Johnson , Flip Korn , S. Muthukrishnan , Oliver Spatscheck , Divesh Srivastava, Holistic UDAFs at streaming speeds, Proceedings of the 2004 ACM SIGMOD international conference on Management of data, June 13-18, 2004, Paris, France [doi>10.1145/1007568.1007575]
• 5: Graham Cormode , Flip Korn , S. Muthukrishnan , Divesh Srivastava, Finding hierarchical heavy hitters in data streams, Proceedings of the 29th international conference on Very large data bases, p.464-475, September 09-12, 2003, Berlin, Germany
• 6: Graham Cormode , Flip Korn , S. Muthukrishnan , Divesh Srivastava, Diamond in the rough: finding Hierarchical Heavy Hitters in multi-dimensional data, Proceedings of the 2004 ACM SIGMOD international conference on Management of data, June 13-18, 2004, Paris, France [doi>10.1145/1007568.1007588]
• 7: Graham Cormode , S. Muthukrishnan, What's hot and what's not: tracking most frequent items dynamically, Proceedings of the twenty-second ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, p.296-306, June 09-11, 2003, San Diego, California [doi>10.1145/773153.773182]
• 8: Graham Cormode , S. Muthukrishnan, An improved data stream summary: the count-min sketch and its applications, Journal of Algorithms, v.55 n.1, p.58-75, April 2005 [doi>10.1016/j.jalgor.2003.12.001]
• 9: Chuck Cranor , Theodore Johnson , Oliver Spataschek , Vladislav Shkapenyuk, Gigascope: a stream database for network applications, Proceedings of the 2003 ACM SIGMOD international conference on Management of data, June 09-12, 2003, San Diego, California [doi>10.1145/872757.872838]
• 10: Erik D. Demaine , Alejandro López-Ortiz , J. Ian Munro, Frequency Estimation of Internet Packet Streams with Limited Space, Proceedings of the 10th Annual European Symposium on Algorithms, p.348-360, September 17-21, 2002
• 11: Cristian Estan , Stefan Savage , George Varghese, Automatically inferring patterns of resource consumption in network traffic, Proceedings of the 2003 conference on Applications, technologies, architectures, and protocols for computer communications, August 25-29, 2003, Karlsruhe, Germany [doi>10.1145/863955.863972]
• 12: Prasanna Ganesan , Hector Garcia-Molina , Jennifer Widom, Exploiting hierarchical domain structure to compute similarity, ACM Transactions on Information Systems (TOIS), v.21 n.1, p.64-93, January 2003 [doi>10.1145/635484.635487]
• 13: Anna C. Gilbert , Yannis Kotidis , S. Muthukrishnan , Martin J. Strauss, How to summarize the universe: dynamic maintenance of quantiles, Proceedings of the 28th international conference on Very Large Data Bases, p.454-465, August 20-23, 2002, Hong Kong, China
• 14: Sudipto Guha , Nick Koudas , Kyuseok Shim, Data-streams and histograms, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.471-475, July 2001, Hersonissos, Greece [doi>10.1145/380752.380841]
• 15: John Hershberger , Nisheeth Shrivastava , Subhash Suri , Csaba D. Tóth, Space complexity of hierarchical heavy hitters in multi-dimensional data streams, Proceedings of the twenty-fourth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, June 13-15, 2005, Baltimore, Maryland [doi>10.1145/1065167.1065211]
• 16: Richard M. Karp , Scott Shenker , Christos H. Papadimitriou, A simple algorithm for finding frequent elements in streams and bags, ACM Transactions on Database Systems (TODS), v.28 n.1, p.51-55, March 2003 [doi>10.1145/762471.762473]
• 17: Laks V. S. Lakshmanan , Raymond T. Ng , Christine Xing Wang , Xiaodong Zhou , Theodore J. Johnson, The generalized MDL approach for summarization, Proceedings of the 28th international conference on Very Large Data Bases, p.766-777, August 20-23, 2002, Hong Kong, China
• 18: Ju-Hong Lee , Deok-Hwan Kim , Chin-Wan Chung, Multi-dimensional selectivity estimation using compressed histogram information, Proceedings of the 1999 ACM SIGMOD international conference on Management of data, p.205-214, May 31-June 03, 1999, Philadelphia, Pennsylvania, United States
• 19: Amit Manjhi , Vladislav Shkapenyuk , Kedar Dhamdhere , Christopher Olston, Finding (Recently) Frequent Items in Distributed Data Streams, Proceedings of the 21st International Conference on Data Engineering, p.767-778, April 05-08, 2005 [doi>10.1109/ICDE.2005.68]
• 20: Gurmeet Singh Manku , Rajeev Motwani, Approximate frequency counts over data streams, Proceedings of the 28th international conference on Very Large Data Bases, p.346-357, August 20-23, 2002, Hong Kong, China
• 21: Metwally, A., Agrawal, D., and Abbadi, A. E. 2005. Efficient computation of frequent and top-k elements in data streams. In Proceedings of the International Conference on Database Theory.
• 22: Misra, J. and Gries, D. 1982. Finding repeated elements. Sci. Comput. Program. 2, 143--152.
• 23: Raymond T. Ng , Alan Wagner , Yu Yin, Iceberg-cube computation with PC clusters, Proceedings of the 2001 ACM SIGMOD international conference on Management of data, p.25-36, May 21-24, 2001, Santa Barbara, California, United States
• 24: Nitin Thaper , Sudipto Guha , Piotr Indyk , Nick Koudas, Dynamic multidimensional histograms, Proceedings of the 2002 ACM SIGMOD international conference on Management of data, June 03-06, 2002, Madison, Wisconsin [doi>10.1145/564691.564741]
• 25: Jeffrey Scott Vitter , Min Wang , Bala Iyer, Data cube approximation and histograms via wavelets, Proceedings of the seventh international conference on Information and knowledge management, p.96-104, November 02-07, 1998, Bethesda, Maryland, United States [doi>10.1145/288627.288645]
• 26: Yin Zhang , Sumeet Singh , Subhabrata Sen , Nick Duffield , Carsten Lund, Online identification of hierarchical heavy hitters: algorithms, evaluation, and applications, Proceedings of the 4th ACM SIGCOMM conference on Internet measurement, October 25-27, 2004, Taormina, Sicily, Italy [doi>10.1145/1028788.1028802]

INDEX TERMS

Primary Classification:
  H. Information Systems
  H.2 DATABASE MANAGEMENT
      H.2.8 Database applications
          Subjects: Data mining

General Terms:
Algorithms, Experimentation, Performance, Theory

Keywords:
Data mining, approximation algorithms, network data analysis