Maintaining Variance and k-Medians over Data Stream Windows

Brian Babcock, Mayur Datar, Rajeev Motwani, Liadan

O’CallaghanStanford University

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)

Sliding Window (SW) Model

….1 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1…

Time Increases

Current Time

Window Size N = 7

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)

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)

Results – Variance

(1 ± ε) approximation Space: O((1/ε2) log N) words Update Time: O(1) amortized,

O((1/ε2) log N) worst case

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)

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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

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

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}

Exponential Histograms

Exponential Histogram algorithm: Initially buckets contain 1 item each Merge adjacent buckets once the sum

of later buckets is large enough

….1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1…

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}

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

Notation

Vi = Variance of the ith bucketni = number of elements in ith bucketμi = mean of the ith bucket

B1 Bm B2………………

Current window, size = N

Bm-1

Variance – composition

Bi,j = concatenation of buckets i and j

ji

jjiiji, n + n

μn + μn = μ

jiji, n n n

2ji

ji

jijiji, )μ - (μ

n + n

nn + V + V = V

Failure of “Weak Additivity”

Time

ValueVariance of each bucket is small

Variance of combinedbucket is large

Cannot afford to neglect contribution of last bucket

Main Solution Idea More careful estimation of last bucket’s

contribution Decompose variance into two parts

“Internal” variance: within bucket “External” variance: between buckets

2ji

ji

jijiji, )μ - (μ

n + n

nn + V + V = V

Internal Varianceof Bucket i

Internal Varianceof Bucket j

External Variance

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

Main Idea – Illustration

Time

Value

Spread

Spread is small external variance is small Spread is large error from “bucket

averaging” insignificant

Variance – error bound

Theorem: Relative error ≤ ε, provided Vm ≤ (ε2/9) Vm*

Aim: Maintain Vm ≤ (ε2/9) Vm* using as few buckets as possible

B1 Bm B2………………

Current window, size = N

Bm-1

Bm*

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)

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

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

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

k-medians summary (2/2) Merging buckets

Combine medians from each level i If they exceed Nτ 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!

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?

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”