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1 A Deterministic Algorithm for Summarizing Asynchronous Streams over a Sliding Window Costas Busch Rensselaer Polytechnic Institute Srikanta Tirthapura Iowa State University

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2 Introduction Algorithm Analysis Outline of Talk

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3 Time Data stream: For simplicity assume unit valued elements

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4 Current time Most recent time window of duration Compute the sum of elements with time stamps in time window Goal: Data stream:

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5 Example I: All packets on a network link, maintain the number of different ip sources in the last one hour Example II: Large database, continuously maintain averages and frequency moments

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6 Synchronous stream t i : In ascending order Asynchronous stream t i : No order guaranteed Data stream:

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7 Why Asynchronous Data Streams? Synchronous streamAsynchronous stream Synchronous Asynchronous Merge w/o control Network delay & multi-path routing

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8 Processing Requirements: One pass processing Small workspace: poly-logarithmic in the size of data Fast processing time per element Approximate answers are ok

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9 Our results: A deterministic data aggregation algorithm Time: Space: Relative Error:

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10 Previous Work: [Datar, Gionis, Indyk, Motwani. SIAM Journal on Computing, 2002] Deterministic, Synchronous [Tirthapura, Xu, Busch, PODC, 2006] Randomized, Asynchronous Merging buckets Random sampling

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11 Introduction Algorithm Analysis Outline of Talk

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12 Current time Time Data stream: For simplicity assume unit valued elements

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13 Current time Most recent time window of duration Data stream: Compute the sum of elements with time stamps in time window Goal:

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14 Divide time into periods of duration

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15 The sliding window may span at most two time periods sliding window

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16 sliding window Sum can be written as two sub-sums In two time periods

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17 sliding window Data structure that maintains an estimate of In left time period

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18 Without loss of Generality, Consider data structure in time period

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19 Data structure consists of various levels is an upper bound of the sum in a period

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20 Counts up to elements Time period Bucket at Level Consider level

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21 Increase counter value Stream:

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22 Increase counter value Stream:

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23 Stream: Increase counter value

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24 Increase counter value Stream:

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25 Stream: Counter threshold of reached Split bucket

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26 Stream: New buckets have threshold also

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27 Stream: Increase appropriate bucket

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28 Stream: Increase appropriate bucket

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29 Stream: Increase appropriate bucket

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30 Stream: Split bucket

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31 Stream:

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32 Stream: Increase appropriate bucket

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33 Stream: Split bucket

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34 Stream:

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35 Splitting Tree

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36 Leaf buckets of duration 1 are not split any further Max depth =

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37 The initial bucket may be split into many buckets Leaf buckets

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38 Due to space limitations we only keep the last buckets Leaf buckets

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39 Suppose we want to find the sum of elements in time period

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40 of splitting threshold Consider various levels

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41 First level with a leaf bucket that intersects timeline

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42 Estimate of S: Consider buckets on right of timeline

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43 First level with a leaf bucket On right timeline OR

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44 Introduction Algorithm Analysis Outline of Talk

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45 Suppose that we use level in order to compute the estimate

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46 Stream: A data element is counted in the appropriate bucket Consider splitting threshold level

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47 Stream: We can assume that the element is placed in the respective bucket

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48 Stream: We can assume that when bucket splits the element is placed in an arbitrary child bucket

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49 Stream: If: GOOD! Element counted in correct bucket

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50 Stream: If: BAD! Element counted in wrong bucket

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51 Consider Leaf Buckets If GOOD!

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52 Consider Leaf Buckets If BAD! Element counted in wrong bucket

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53 :elements of left part counted on right Consider Leaf Buckets :elements of right part counted on left

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54 elements of left part counted on right Must have been initially inserted in one of these buckets

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55 Since tree depth

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56 Since tree depth Similarly, we can prove Therefore:

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57 It can be proven Since

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58 It can be proven Since Combined with We obtain relative error :

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