Outline. 1 Introduction. 2 Algorithm. 3 Sum Problem. 4 Trends. 5 References. Stream Statistics Over Sliding Window. Anil Maheshwari.

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Stream Statistics Over Sliding Window Anil Maheshwari Introduction Algorithm Sum Problem Trends References

Stream Statistics Over Sliding Window

Anil Maheshwari

School of Computer Science Carleton University

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Outline

1 Introduction 2 _Algorithm 3 _{Sum Problem} 4 Trends 5 _References

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Problem Setting

Main Problem

The input is an endless stream of binary bits. At any time,

among the lastN bits received, we are interested in

queries that seek an approximate count of the number of

1’s in the stream among the lastkbits, wherek≤N.

Result: A data structure of sizeO(1 log2N)that can

approximate the count of the number of1s within a factor

of1±

Reference: Maintaining stream statistics over sliding windows by Datar, Gionis, Indyk, and Motwani, SIAM Jl. Computing 2002

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Variants

1 A stream of positive numbers. The query consists of

a valuek∈ {1, . . . , N}, and we want to know the

(approximate) sum of the lastknumbers in the

stream. (Uses sublinear space.)

2 _{A stream consisting of numbers from the set}

{−1,0,+1}. We want to maintain the sum of lastN

numbers of the stream. (RequiresΩ(N)bits of

storage to approximate the sum that is within a constant factor of the exact sum.)

3 _{What are the most popular movies in the last week?}

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Main Problem

Report an approximate count of the number of1’s in the

stream of binary bits among the lastkbits, wherek≤N.

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Algorithm for Approximate Count

Algorithm uses two structures:

Time Stamps: To track the most recentN bits.

Buckets: With the following features:

O(logN)buckets maintain the1’s among the latest

N bits

Number of1’s in a bucket is a power of2

Each1-bit is assigned to exactly one bucket (0-bit may or may not be assigned to any bucket)

At most two buckets of a givensize(size= #1s)

Each bucket stores time stamp of its most recent bit Most recent bit of any bucket is1-bit

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Algorithm contd.

On receiving a new bit in the data stream:

0-bit: Increment the time stamp of each of the buckets by

1, and if any of the buckets time stamp exceedsN, we

discard that bucket.

1-bit: Following updates are done:

1 _{Create a bucket}_B

0 consisting of the newest1-bit

with a time stamp of1.

2 _{Scan the list of buckets in order of increasing size.}

Case 1: Two buckets of size1.

Increment time stamp of each bucket (and possibly

discard buckets whose time stamps exceedN)

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Illustration

1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 N A B 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 C 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 B0 B0 B1 B1 B2 B0 B0 B1 B1 B2 B0 B1 B2 B2 B0 B0 B1 B2 B2 B0 B0 B1 B2 D E Time Stamp 1 Time Stamp N

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Space Complexity

We have:

O(logN)buckets as the size of window isN

BucketBi stores2i 1-bits

For each bucket we store its time stamp and its size

Time stamps requiresO(logN)bits

Storingiwith bucketBiis sufficient for its size

As0≤i≤logN,ican represented using

O(log logN)bits

Total space required

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Time Complexity

On receiving a0-bit:

- We update time stamps of each of theO(logN)buckets

- RequiresO(logN)time

On receiving a1-bit:

- We update the time stamps of each bucket - Potentially merge & cascade buckets

- Time (merge & cascade)≈# of buckets

- Can be performed inO(logN)time

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Answering Query

Query Problem

For any query valuek∈ {1, . . . , N}, report an

approximate count of the number of1’s among the latest

kbits of the stream.

1 _Initialize_count_{:= 0}

2 _{Traverse buckets from right to left. For each bucket of}

typeBithat is encountered in the traversal:

1 B_iis completely contained in the window:

Increment count by2i

2 _B_iis completely outside the window:

count remains unchanged

3 Partially overlaps the window:

Increment count by2i

2

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Analysis of Approximation Factor

Observation: Except of one bucket, sayBj, that is

partially in the window of sizek, we know that all buckets

of typeB0, B1, . . . , Bj−1 are completely within the window.

- For those buckets, the count of the number of1-bits is

j−1

P

i=0

2i ≥2j −1

- The true count (and the approximate count) value is at least2j (as the last bit ofBj is in the window of interest)

- For the bucketBj that overlaps partially with the

window, the number of bits that can be in the true count

can be anywhere from0upto2j−1. But we only took a

contribution of2j−1 _{in the reported count value}

- Ratio of the true count to the reported count is within a factor of(1₂,2).

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Refining the Analysis

- Letr≥2be an integer parameter.

- Maintainr−1orr copies ofBi for eachi≥1(buckets

of typeB0 may be less thanr−1)

- At any time we exceedrcopies of any type of buckets,

we take the oldest two buckets and merge them to form a new bucket of the next size.

- For the query, assume that the bucket labelledBj is only

partially overlapping the query window. - At least1 +

j−1

P

i=1

(r−1)2i1-bits are in the query window. - True count and the reported value are within a factor of

1± 1

r−1

=⇒ By settingr= 1 +1, we obtain a data structure of

sizeO(1 log2N)that approximates the count of the number of1s within a factor of1±.

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Computation of Sum

The Sum Problem

A stream of positive numbers. The query consists of a valuek∈ {1, . . . , N}, and we want to know the

(approximate) sum of the lastknumbers in the stream.

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Approach I: Computation of Sum

Assumingd-bit numbers. For each bit position, maintain a

stream. Approximate number of10sin each stream.

Report approximate sum value as

d−1

P

i=0

count(i)2i

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Approach II: Computation of Sum

If the next number in the stream isx, insertx10s in the stream

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What is Trending?

Among the last1012movie tickets sold, list allpopular

movies?

Letc:= 10−3. Maintain (decaying) scores for movies

whose threshold is at leastτ ∈(0,1). For each new sale

of ticket (say for MovieM):

1 _{For each movie whose score is being maintained, its}

new score is reduced by a factor of(1−c)

2 If we have the score of_M, add₁to that score.

Otherwise, create a new score forM and initialize it

to1

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Questions

1 _{How many scores are maintained at any given time?}

2 _{What is sum of all scores at any point of time?}

3 _{Answer above questions for}_τ ₌ 1

2 and 1 3.

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Conclusions

Main References:

1 _{Maintaining stream statistics over sliding windows, by}

Datar, Gionis, Indyk, and Motwani, SIAM Jl. Computing 2002.

2 _{Chapter in MMDS book (mmds.org)}

3 _{Chapter on Data Streams in My Notes on Topics in}