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Algorithms for Large Data Sets

Ziv Bar-YossefLecture 10

June 4, 2006

http://www.ee.technion.ac.il/courses/049011

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Random Samplingof

Web Pages

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Outline

Problem definition Random sampling of web pages according

to their PageRank Uniform sampling of web pages

(Henzinger et al) Uniform sampling of web pages (Bar-

Yossef et al)

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Random Sampling of Web Pages W = a snapshot of the “indexable

web” Consider only “static” HTML web

pages

= a probability distribution over W

Goal: Design an efficient algorithm for generating samples from W distributed according to .

Our focus: = PageRank = Uniform

Indexable web

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Random Sampling of Web Pages: Motivation Compute statistics about the web

Ex: What fraction of web pages belong to .il? Ex: What fraction of web pages are written in Chinese? Ex: What fraction of hyperlinks are advertisements?

Compare coverage of search engines Ex: Is Google larger than MSN? Ex: What is the overlap between Google and Yahoo?

Data mining of the web Ex: How frequently computer science pages cite

biology pages? Ex: How are pages distributed by topic?

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Random Sampling of Web Pages: Challenges Naïve solution: crawl, index, sample

Crawls cannot get complete coverage Web is constantly changing Crawling is slow and expensive

Our goals: Accuracy: generate samples from a snapshot of the

entire indexable web Speed: samples should be generated quickly Low cost: sampling procedure should run on a desktop

PC

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A Random Walk Approach

Design a random walk on W whose stationary distribution is P = Random walk’s probability transition matrix P =

Run random walk for sufficiently many steps Recall: For any initial distribution q, Mixing time: # of steps required to get close to the limit

Use reached node as a sample

Repeat for as many samples as needed

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A Random Walk Approach:Advantages & Issues Advantages:

Accuracy: random walk can potentially visit every page on the web

Speed: no need to scan the whole web Low cost: no need for large storage or multiple

processors

Issues: How to design the random walk so it converges to ? How to analyze the mixing time of the random walk?

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PageRank Sampling [Henzinger et al 1999]

Use the “random surfer” random walk:Start at some initial node v0

When visiting a page v Toss a coin with heads probability If coin is heads, go to a uniformly chosen page If coin is tails, go to a random out-neighbor of v

Limit distribution: PageRank Mixing time: fast (will see later)

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PageRank Sampling:Reality Problem: how to pick a page uniformly at

random? Solutions:

Jump to a random page from the history of the walk

Creates bias towards dense web-sitesPick a random host from the hosts in the walk’s

history and jump to a random page from the pages visited on that host

Not converging to PageRank anymore Experiments indicate it is still fine

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Uniform Sampling via PageRank Sampling [Henzinger et al 2000]

Sampling algorithm:1. Use previous random walk to generate a sample w

according to the PageRank distribution

2. Toss a coin with heads probability

3. If coin is heads, output w as a sample

4. If coin is tails, goto step 1

Analysis: Need C/|W| iterations until getting a single sample

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Uniform Sampling via PageRank Sampling: Reality How to estimate PR(w)?

Use the random walk itself: VR(w) = visit ratio of w (# of times w was visited by

the walk divided by length of the walk) Approximation is very crude

Use the subgraph spanned by nodes visited to compute PageRank

Bias towards the neighborhood of the initial pageUse Google

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Uniform Sampling by RW on Regular Graphs [Bar-Yossef et al 2000]

Fact: A random walk on an undirected, connected, non-bipartite, and regular graph converges to a uniform distribution.

Proof: P: random walk’s probability transition matrix

P is stochastic 1 is a right eigenvector with e.v. 1: P1 = 1

Graph is connected RW is irreducible Graph is non-bipartite RW is aperiodic Hence, RW is ergodic, and thus has a stationary

distribution : is a left eigenvector of P with e.v. 1: P =

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Random Walks on Regular Graphs

Proof (cont.):d: graph’s degreeA: graph’s adjacency matrix

Symmetric, because graph is undirected

P = (1/d) A Hence, also P is symmetric Its left eigenvectors and right eigenvectors are the

same = (1/n) 1

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Web as a Regular Graph

Problems Web is not connected Web is directed Web is non-regular

Solutions Focus on indexable web, which is connected Ignore directions of links Add a weighted self loop to each node

weight(w) = degmax – deg(w)

All pages then have degree degmax

Overestimate on degmax doesn’t hurt

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Mixing Time Analysis

Theorem

Mixing time of a random walk is log(|W|) / (1 - 2) 1 - 2: spectral gap of P

Experiment (over a large web crawl): 1 – 2 ~ 1/100,000 log(|W|) ~ 34

Hence: mixing time ~ 3.4 million steps Self loop steps are free About 1 in 30,000 steps is not a self loop step (degmax

~ 300,000, degavg ~ 10) Actual mixing time: ~ 115 steps!

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Random Walks on Regular Graphs: Reality How to get incoming links?

Search engines Potential bias towards search engine index Do not provide full list of in-links? Costly communication

Random walk’s history Important for avoiding dead ends Requires storage

How to estimate deg(w)? Solution: run random walk on the sub-graph of W

spanned by the available links Sub-graph may no longer have the good mixing time

properties

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Top 20 Internet Domains (Summer 2003)

10.36%

5.57%4.15%3.01%

0.61%

9.19%

51.15%

0%

10%

20%

30%

40%

50%

60%

.com

.org

.net

.edu .d

e .uk

.au .u

s.e

s .jp .ca .nl .it .ch .p

l .il .nz

.gov

.info .m

x

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68%

54%50% 50%

48%

38%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Google AltaVista Fast Lycos HotBot Go

Search Engine Coverage(Summer 2000)

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Random Samplingfrom

a Search Engine’s Index

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Search Engine Samplers

IndexPublicInterface

PublicInterface

Search Engine

Sampler

Web

D

Queries

Top k results

Random document x D

Indexed Documents

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Motivation Useful tool for search engine evaluation:

Freshness Fraction of up-to-date pages in the index

Topical bias Identification of overrepresented/underrepresented topics

Spam Fraction of spam pages in the index

Security Fraction of pages in index infected by viruses/worms/trojans

Relative Size Number of documents indexed compared with other search

engines

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Size Wars

August 2005

: We index 20 billion documents.

So, who’s right?

September 2005

: We index 8 billion documents, but our index is 3 times larger than our competition’s.

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The Bharat-Broder Sampler: Preprocessing Step

C

Large corpusL

t1, freq(t1,C)t2, freq(t2,C)……

Lexicon

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The Bharat-Broder Sampler

Search Engine

BB Sampler

t1 AND t2Top k results

Random document from top k results

LTwo random terms t1, t2

Only if:• all queries return the same number of results ≤ k • all documents are of the same lengthThen, samples are uniform.

Only if:• all queries return the same number of results ≤ k • all documents are of the same lengthThen, samples are uniform.

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The Bharat-Broder Sampler:Drawbacks Documents have varying lengths

Bias towards long documents

Some queries have more than k matchesBias towards documents with high static rank

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Search Engines as Hypergraphs

results(q) = { documents returned on query q } queries(x) = { queries that return x as a result } P = query pool = a set of queries Query pool hypergraph:

Vertices: Indexed documents Hyperedges: { result(q) | q P }

www.cnn.com

www.foxnews.com

news.google.com

news.bbc.co.ukwww.google.com

maps.google.com

www.bbc.co.uk

www.mapquest.com

maps.yahoot.com

“news”

“bbc”

“google”

“maps”

en.wikipedia.org/wiki/BBC

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Query Cardinalities and Document Degrees

Query cardinality: card(q) = |results(q)| Document degree: deg(x) = |queries(x)| Examples:

card(“news”) = 4, card(“bbc”) = 3 deg(www.cnn.com) = 1, deg(news.bbc.co.uk) = 2

www.cnn.com

www.foxnews.com

news.google.com

news.bbc.co.ukwww.google.com

maps.google.com

www.bbc.co.uk

www.mapquest.com

maps.yahoot.com

“news”

“bbc”

“google”

“maps”

en.wikipedia.org/wiki/BBC

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Pool-Based Sampler [Bar-Yossef,Gurevich] Preprocessing Step

C

Large corpusPq1

……

Query Pool

Example: P = all 3-word phrases that occur in C If “to be or not to be” occurs in C, P contains:

“to be or”, “be or not”, “or not to”, “not to be”

Choose P that “covers” most documents in D

q2

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Monte Carlo Simulation

Don’t know how to generate uniform samples from D directly

How to use biased samples to generate uniform samples?

Samples with weights that represent their bias can be used to simulate uniform samples

Monte Carlo Simulation Methods

Rejection Sampling

Rejection Sampling

Importance Sampling

Importance Sampling

Metropolis-Hastings

Metropolis-Hastings

Maximum-Degree

Maximum-Degree

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Document Degree Distribution

Can generate biased samples from the “document degree distribution”

Advantage: Can compute weights representing the bias of p:

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Unnormalized forms of distributions p: a distribution on a domain D Unnormalized form of p:

: (unknown) normalization constant

Examples: p = uniform: p = degree distribution:

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Monte Carlo Simulation : Target distribution

In our case: =uniform on D p: Trial distribution

In our case: p = document degree distribution

Bias weight of p(x) relative to (x): In our case:

Monte Carlo Simulator

Monte Carlo Simulator

Samples from p

Sample from

x

Sampler

(x1,w(x)), (x2,w(x)),… p-Samplerp-Sampler

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C: envelope constant C ≥ w(x) for all x

The algorithm: accept := false while (not accept)

generate a sample x from p toss a coin whose heads probability is if coin comes up heads,

accept := true

return x

In our case: C = 1 and acceptance prob = 1/deg(x)

Rejection Sampling [von Neumann]

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Pool-Based Sampler

Degree distribution sampler

Degree distribution sampler

Search EngineSearch Engine

Rejection Sampling

Rejection Sampling

q1,q2,…results(q1), results(q2),…

x

Pool-Based Sampler

(x(x11,1/deg(x,1/deg(x11)),)),

(x(x22,1/deg(x,1/deg(x22)),…)),…

Uniform sample

Documents sampled from degree distribution with corresponding weights

Degree distribution: p(x) = deg(x) / x’deg(x’)

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Sampling documents by degree

Select a random q P Select a random x results(q) Documents with high degree are more likely to be sampled If we sample q uniformly “oversample” documents that

belong to narrow queries We need to sample q proportionally to its cardinality

www.cnn.com

www.foxnews.com

news.google.com

news.bbc.co.ukwww.google.com

maps.google.com

www.bbc.co.uk

www.mapquest.com

maps.yahoot.com

“news”

“bbc”

“google”

“maps”

en.wikipedia.org/wiki/BBC

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Query Cardinality Distribution

Unrealistic assumptions: Can sample queries from the cardinality distribution

In practice, don’t know a priori card(q) for all q P q P, card(q) ≤ k

In practice, some queries overflow (vol(q) > k)

results(q) = results returned on query q card(q) = |results(q)| Cardinality distribution:

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Degree Distribution Sampler

Search EngineSearch Engine

results(q)

xCardinality Distribution Sampler

Cardinality Distribution Sampler

Sample x uniformly from results(q)

Sample x uniformly from results(q)

q

Analysis:

Degree Distribution Sampler

Query sampled from cardinality

distribution

Document sampled from

degree distribution

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Sampling queries by cardinality

Sampling queries from pool uniformly:Easy

Sampling queries from pool by cardinality: Hard Requires knowing cardinalities of all queries in the

search engine

Use Monte Carlo methods to simulate biased sampling via uniform sampling: Target distribution: the cardinality distribution Trial distribution: uniform distribution on P

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Sampling queries by cardinality

Bias weight of cardinality distribution relative to the uniform distribution:

Can be computed using a single search engine query

Use rejection sampling: Envelope constant for rejection sampling:

Queries are sampled uniformly from P Each query q is accepted with probability

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Degree Distribution Sampler

Degree Distribution Sampler

Complete Pool-Based Sampler

Search EngineSearch Engine

Rejection Sampling

Rejection Sampling

x(x,1/deg(x)),…

Uniform document

sample

Documents sampled from degree distribution with corresponding weights

Uniform Query Sampler

Uniform Query Sampler

Rejection Sampling

Rejection Sampling

(q,card(q)),…

Uniform query

sampleQuery

sampled from cardinality distribution

(q,results(q)),…

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Dealing with Overflowing Queries

Problem: Some queries may overflow (card(q) > k) Bias towards highly ranked documents

Solutions: Select a pool P in which overflowing queries are rare

(e.g., phrase queries) Skip overflowing queries Adapt rejection sampling to deal with approximate weights

Theorem:

Samples of PB sampler are at most -away from uniform. ( = overflow probability of P)

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Relative Sizes of Google, MSN and Yahoo!

Google = 1

Yahoo! = 1.28

MSN Search = 0.73

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End of Lecture 10