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Sublinear Algorihms for Big DataPowerPoint Presentation

Sublinear Algorihms for Big Data

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SOFSEM 2015

- URL: http://www.sofsem.cz
- 41st International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM’15)
- When and where?
- January 24-29, 2015. Czech Republic, Pec pod Snezkou

- Deadlines:
- August 1st (tomorrow!): Abstract submission
- August 15th: Full papers (proceedings in LNCS)

- I am on the Program Committee ;)

Today

- Count-Min (continued), Count Sketch
- Sampling methods
- -sampling
- -sampling

- Sparse recovery
- -sparse recovery

- Graph sketching

Recap

- Stream: elements from universe , e.g.
- = frequency of in the stream = # of occurrences of value

Count-Min

- are -wise independent hash functions
- Maintain counters with values:
elements in the stream with

- For every the value and so:
- If and then:
.

More about Count-Min

- Authors: Graham Cormode, S. Muthukrishnan [LATIN’04]
- Count-Min is linear:
Count-Min(S1 + S2) = Count-Min(S1) + Count-Min(S2)

- Deterministic version: CR-Precis
- Count-Min vs. Bloom filters
- Allows to approximate values, not just 0/1 (set membership)
- Doesn’t require mutual independence (only 2-wise)

- FAQ and Applications:
- https://sites.google.com/site/countminsketch/home/
- https://sites.google.com/site/countminsketch/home/faq

Fully Dynamic Streams

- Stream: updates that define vector where .
- Example: For
- Count Sketch: Count-Min with random signs and median instead of min:

Count Sketch

- In addition to use random signs
- Estimate:
- Parameters:

-Sampling

- Stream: updates that define vector where .
- -Sampling: Return random and :

Application: Social Networks

- Each of people in a social network is friends with some arbitrary set of other people
- Each person knows only about their friends
- With no communication in the network, each person sends a postcard to Mark Z.
- If Mark wants to know if the graph is connected, how long should the postcards be?

Optimal estimation

- Yesterday: -approximate
- space for
- space for

- New algorithm: Let be an -sample.
Return , where is an estimate of

- Expectation:

Optimal estimation

- New algorithm: Let be an -sample.
Return , where is an estimate of

- Variance:
- Exercise: Show that
- Overall:
- Apply average + median to copies

-Sampling: Basic Overview

- Assume Weight by where
where

- For some value return if there is a unique such that
- Probability is returned if is large enough:
- Probability some value is returned repeat times.

-Sampling: Part 1

- Use Count-Sketch with parameters to sketch
- To estimate
and

- Lemma: With high probability if
- Corollary: With high probability if and
- Exercise: for large

Proof of Lemma

- Let
- By the analysis of Count Sketch and by Markov:
- If then
- If , then
where the last inequality holds with probability

- Take median over repetitions high probability

-Sampling: Part 2

- Let if and otherwise
- If there is a unique with then return
- Note that if then and so
,

therefore

- Lemma: With probability there is a unique such that . If so then
- Thm: Repeat times. Space:

Proof of Lemma

- Let We can upper-bound
Similarly,

- Using independence of , probability of unique with :

Proof of Lemma

- Let We can upper-bound
Similarly,

- We just showed:
- If there is a unique i, probability is:

-sampling

- Maintain and -approximation to
- Hash items using for
- For each , maintain:
- Lemma: At level there is a unique element in the streams that maps to (with constant probability)
- Uniqueness is verified if . If so, then output as the index and as the count.

Proof of Lemma

- Let and note that
- For any
- Probability there exists a unique such that ,
- Holds even if are only 2-wise independent

Sparse Recovery

- Goal: Find such that is minimized among s with at most non-zero entries.
- Definition:
- Exercise: where are indices of largest
- Using space we can find such that and

Count-Min Revisited

- Use Count-Min with
- For let for some row
- Let be the indices with max. frequencies. Let be the event there doesn’t exist with
- Then for
- Because w.h.p. all ’s approx . up to

Sparse Recovery Algorithm

- Use Count-Min with
- Let be frequency estimates:
- Let be with all but the k-th largest entries replaced by 0.
- Lemma:

- Let be indices corresponding to largest value of and .
- For a vector and denote as the vector formed by zeroing out all entries of except for those in .
+ 2

+ 2

Thank you!

- Questions?

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