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

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Data Streams and Sliding 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)

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Sliding Window (SW) Model … … Time Increases Current Time Window Size N = 7

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Variance and k-Medians Variance: Σ(x i – μ) 2, μ = Σ x i /N k-median clustering: Given: N points (x 1… x N ) in a metric space Find k points C = {c 1, c 2, …, c k } that minimize Σ d(x i, C) (the assignment distance)

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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)(log 2 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)

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Results – Variance (1 ± ε) approximation Space: O((1/ε 2 ) log N) words Update Time: O(1) amortized, O((1/ε 2 ) log N) worst case

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Results – k-medians 2 O(1/ τ ) approximation of assignment distance (0 < τ < ½) Space: O((k/ τ 4 )N 2 τ ) Update time: O(k) amortized, O((k 2 / τ 3 )N 2 τ ) worst case Query time: O((k 2 / τ 3 )N 2 τ ) ~ ~ ~ ~

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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

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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 … … Bucket Sums = {3,2,1,2} Bucket Sums = {2,1,2}

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

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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}Bucket sums = {4, 2, 2, 1, 1}Bucket sums = {4, 2, 2, 1, 1,1}Bucket sums = {4, 2, 2, 2, 1}Bucket sums = {4, 4, 2, 1}

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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

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Notation V i = Variance of the ith bucket n i = number of elements in ith bucket μ i = mean of the ith bucket B1B1 B m B2B2 ……………… Current window, size = N B m-1

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Variance – composition B i,j = concatenation of buckets i and j

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

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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 Variance of Bucket i Internal Variance of Bucket j External Variance

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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

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

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Variance – error bound Theorem: Relative error ≤ ε, provided V m ≤ (ε 2 /9) V m* Aim: Maintain V m ≤ (ε 2 /9) V m* using as few buckets as possible B1B1 B m B2B2 ……………… Current window, size = N B m-1 B m*

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Variance – algorithm EH algorithm for variance: Initially buckets contain 1 item each Merge adjacent buckets i, i+1 whenever the following condition holds: (9/ε 2 ) V i,i-1 ≤ V i-1* (i.e. variance of merged bucket is small compared to combined variance of later buckets)

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

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Update and Query time Query Time: O(1) We maintain n, V & μ values for m and m* Update Time: O((1/ε 2 ) 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

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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

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k-medians summary (2/2) Merging buckets Combine medians from each level i If they exceed N τ in 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!

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Open Problems Variance: Close gap between upper and lower bounds (1/ε log N vs. 1/ε 2 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?

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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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