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Maintaining Variance and k-Medians over Data Stream Windows

Maintaining Variance and k-Medians over Data Stream Windows. Brian Babcock, Mayur Datar, Rajeev Motwani, Liadan O’Callaghan Stanford University. Data Streams and Sliding Windows. Streaming data model Useful for applications with high data volumes, timeliness requirements

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Maintaining Variance and k-Medians over Data Stream Windows

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  1. Maintaining Variance and k-Medians over Data Stream Windows Brian Babcock, Mayur Datar, Rajeev Motwani, Liadan O’Callaghan Stanford University

  2. Data Streams andSliding Windows • Streaming data model • Useful for applications with high data volumes, timeliness requirements • Data processed in single pass • Limited memory (sublinear in stream size) • Sliding window model • Variation of streaming data model • Only recent data matters • Parameterized by window size N • Limited memory (sublinear in window size)

  3. Sliding Window (SW) Model Time Increases ….1 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1… Window Size N = 7 Current Time

  4. Variance and k-Medians • Variance: Σ(xi – μ)2, μ = Σ xi/N • k-median clustering: • Given: N points (x1… xN) in a metric space • Find k points C = {c1, c2, …, ck} that minimize Σ d(xi, C) (the assignment distance)

  5. Previous Results in SW Model • Count of non-zero elements /Sum of positive integers [DGIM’02] • (1 ± ε) approximation • Space: θ((1/ε)(log N)) words θ((1/ε)(log2 N)) bits • Update time: θ(log N) worst case, θ(1) amortized • Improved to θ(1) worst case by [GT’02] • Exponential Histogram (EH) data structure • Generalized SW model [CS’03] (previous talk)

  6. Results – Variance • (1 ± ε) approximation • Space: O((1/ε2) log N) words • Update Time: O(1) amortized, O((1/ε2) log N) worst case

  7. Results – k-medians • 2O(1/τ)approximation of assignment distance (0 < τ < ½) • Space: O((k/τ4)N2τ) • Update time: O(k) amortized, O((k2/τ3)N2τ) worst case • Query time: O((k2/τ3)N2τ) ~ ~ ~ ~

  8. Remainder of the Talk • Overview of Exponential Histogram • Where EH fails and how to fix it • Algorithm for Variance • Main ideas in k-medians algorithm • Open problems

  9. Sliding Window Computation • Main difficulty: discount expiring data • As each element arrives, one element expires • Value of expiring element can’t be known exactly • How do we update our data structure? • One solution: Use histograms ….1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 … Bucket Sums = {2,1,2} Bucket Sums = {3,2,1,2}

  10. Containing the Error • Error comes from last bucket • Need to ensure that contribution of last bucket is not too big • Bad example: … 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0… Bucket Sums = {4,4,4} Bucket Sums = {4}

  11. Exponential Histograms • Exponential Histogram algorithm: • Initially buckets contain 1 item each • Merge adjacent buckets once the sum of later buckets is large enough Bucket sums = {4, 2, 2, 1, 1} Bucket sums = {4, 2, 2, 1, 1 ,1} Bucket sums = {4, 2, 2, 1} Bucket sums = {4, 2, 2, 2, 1} Bucket sums = {4, 4, 2, 1} ….1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1…

  12. Where EH Goes Wrong • [DGIM’02] Can estimate any function f defined over windows that satisfies: • Positive:f(X) ≥ 0 • Polynomially bounded:f(X) ≤ poly(|X|) • Composable: Can compute f(X +Y) from f(X), f(Y) and little additional information • Weakly Additive:(f(X) + f(Y)) ≤ f(X +Y) ≤ c(f(X) + f(Y)) • “Weakly Additive” condition not valid for variance, k-medians

  13. Notation Current window, size = N ……………… Bm-1 Bm B2 B1 Vi = Variance of the ith bucket ni = number of elements in ith bucket μi = mean of the ith bucket

  14. Variance – composition • Bi,j = concatenation of buckets i and j

  15. Variance of each bucket is small Cannot afford to neglect contribution of last bucket Failure of “Weak Additivity” Value Variance of combinedbucket is large Time

  16. Main Solution Idea • More careful estimation of last bucket’s contribution • Decompose variance into two parts • “Internal” variance: within bucket • “External” variance: between buckets Internal Varianceof Bucket i External Variance Internal Varianceof Bucket j

  17. Main Solution Idea • When estimating contribution of last bucket: • Internal variance charged evenly to each point • External variance • Pretend each point is at the average for its bucket • Variance for bucket is small  points aren’t too far from the average • Points aren’t far from the average  average is a good approx. for each point

  18. Main Idea – Illustration • Spread is small  external variance is small • Spread is large  error from “bucket averaging” insignificant Value Spread Time

  19. Variance – error bound Current window, size = N • Theorem: Relative error ≤ ε, provided Vm ≤ (ε2/9) Vm* • Aim: Maintain Vm ≤ (ε2/9) Vm*using as few buckets as possible ……………… Bm-1 Bm B2 B1 Bm*

  20. Variance – algorithm • EH algorithm for variance: • Initially buckets contain 1 item each • Merge adjacent buckets i, i+1 whenever the following condition holds: (9/ε2)Vi,i-1 ≤ Vi-1*(i.e. variance of merged bucket is small compared to combined variance of later buckets)

  21. Invariants • Invariant 1: (9/ε2)Vi ≤ Vi* • Ensures that relative error is ≤ ε • Invariant 2: (9/ε2)Vi,i-1 > Vi-1* • Ensures that number of buckets = O((1/ε2)log N) • Each bucket requires O(1) space

  22. Update and Query time • Query Time: O(1) • We maintain n, V & μ values for m and m* • Update Time: O((1/ε2) log N) worst case • Time to check and combine buckets • Can be made amortized O(1) • Merge buckets periodically instead of after each new data element

  23. k-medians summary (1/2) • Assignment distance substitutes for variance • Assignment distance obtained from an approximate clustering of points in the bucket • Use hierarchical clustering algorithm [GMMO’00] • Original points cluster to give level-1 medians • Level-i medians cluster to give level-(i+1) medians • Medians weighted by count of assigned points • Each bucket maintains a collection of medians at various levels

  24. k-medians summary (2/2) • Merging buckets • Combine medians from each level i • If they exceed Nτin number, cluster to get level i+1 medians. • Estimation procedure • Weighted clustering of all medians from all buckets to produce k overall medians • Estimating contribution of last bucket • Pretend each point is at the closest median • Relies on approximate counts of active points assigned to each median • See paper for details!

  25. Open Problems • Variance: • Close gap between upper and lower bounds (1/ε log N vs. 1/ε2 log N) • Improve update time from O(1) amortized to O(1) worst-case • k-median clustering: • [COP’03] give polylog N space approx. algorithm in streaming data model • Can a similar result be obtained in the sliding window model?

  26. Conclusion • Algorithms to approximately maintain variance and k-median clustering in sliding window model • Previous results using Exponential Histograms required “weak additivity” • Not satisfied by variance or k-median clustering • Adapted EHs for variance and k-median • Techniques may be useful for other statistics that violate “weak additivity”

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