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Massive Data Sets:Theory & Practice

Ziv Bar-Yossef

IBM Almaden Research Center

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What are Massive Data Sets?

Technology

The World-Wide WebIP packet flowsPhone call logs

Science

Genomic dataAstronomical sky surveys

Weather data

Business

Credit card transactionsBilling records

Supermarket salesPetabytes

Terabytes

Gigabytes

• Huge • Distributed• Dynamic• Heterogeneous• Noisy• Unstructured / semi-structured

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Nontraditional Challenges

Traditionally

Cope with the complexity of the problem

New challenges• How to efficiently compute on massive data sets?

– Restricted access to the data– Not enough time to read the whole data– Tiny fraction of the data can be held in main memory

• How to find desired information in the data?• How to summarize the data?• How to clean the data?

Massive Data Sets

Cope with the complexity of the data

4Algorithm

• Sampling Query a small number of data elements

• Data streams Stream through the data;limited main memory storage

• Sketching Compress data chunks into small “sketches”; compute over the sketches

Computational Models for Massive Data Sets

Algorithm

Data Set

Algorithm

Data Set

Data Set

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Outline of the Talk

• Web statistics

• Sampling lower bounds

• Hamming distance sketching

• Template detection

“Theory”“Practice”

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Web Statistics(with A. Berg, S. Chien, J. Fakcharoenphol, D. Weitz, VLDB 2000)

The “BowTie” Structure of the Web

[Broder et al, 2000]

crawlable web

• What fraction of the web is covered by Google?

• Which is the largest country domain on the web?

• What is the percentage of French language pages?

• How large is the web?

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Our Approach• Straightforward solution:

– Crawl the crawlable web– Generate statistics based on the crawl

• Drawbacks:– Expensive– Complicated implementation– Slow– Inaccurate

• Our approach: uniform sampling by random walks– Random walk on an undirected & regular version of the crawlable web

• Advantages:– Provably uniform samples from the crawlable web– Runs on a desktop PC in a couple of days

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Undirected Regular Random Walk

Fact:

A random walk on a connected (non-bipartite) undirected regular graph converges to a uniform limit distribution.

w(v) = degmax - deg(v)

1

2

31

4

02 3

03

2

2

4

4

3

3

3

1

2

5

Follow a random out-link or a random in-link at each step

Use weighted self loops to even out page degrees

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Convergence Rate (“Mixing Time”)

Theorem Mixing time log(N)/

(N = graph size, = transition matrix’s spectral gap)

Experiment (based on a crawl)

For the web, 10-5

Mixing time: 3.3 million steps

• Self loop steps are free• 29,999 out of 30,000 steps are self loop steps

Actual mixing time is only 110 steps

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Realization of the Random Walk

Problems• The in-links of pages are not readily available• The degree of pages is not available

Available sources of in-links:• Previously visited nodes • Reverse link services of search engines

Experiments indicate samples are still nearly uniform.

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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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Search Engine Coverage (summer 2000)

68%

54%50% 50%

48%

38%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Google AltaVista Fast Lycos HotBot Go

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Subsequent Extensions • Focused Sampling

(with T. Kanungo and R. Krauthgamer, 2003)

– “Focused statistics” about web communities:• Statistics about the .uk domain• Statistics about pages on bicycling• Statistics about Arabic language pages

– Based on a sophisticated extension of the above random walk.

• Study of the web’s decay (with A. Broder, R. Kumar, and A. Tomkins, 2003)

– A measure for how well-maintained web pages are.– Based on a random walk idea.

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Sampling Lower Bounds (STOC 2003)

Q1. How many samples are needed to estimate:– The fraction of pages covered by Google?– The number of distinct web-sites?– The distribution of languages on the web?

Q2. Can we save samples by sampling non-uniformly?

A2. For “symmetric” functions, uniform sampling is the best possible.(“symmetric” – invariant under permutations of data elements)

A1. A “recipe” for obtaining sampling lower bounds for symmetric functions.

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Algorithm

Optimality of Uniform Sampling(with R. Kumar and D. Sivakumar, STOC 2001)

Theorem

When estimating symmetric functions, uniform sampling is the best possible.

Proof idea

X1 X2 X3 X4 X5 X6 X7 X8X1 X2 X3 X4 X5 X6 X7 X8X1 X2 X3 X4 X5 X6 X7 X8

X2 X7 X5

original algorithmsimulation

x

x) X2 X7 X5

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Preliminaries

Bf(a) f(b)

pairwise “disjoint inputs“f(c)

f: An B : symmetric function

approximation parameter

1 1 1 2 2 3x1) = 1/2 (2) = 1/3 (3) = 1/6

input “sample distribution”

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The Lower Bound Recipe

x1,…,xm: “pairwise disjoint” inputs

1,…,m: “sample distributions” on x1,…,xm

Theorem:Any algorithm approximating f requires q samples, where

Proof steps:• Reduction from statistical classification• Lower bound for statistical classification

( 0 · JS(1,…,m) · log m )

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Reduction from Statistical Classification

Bf(a) f(b)pairwise

f(c)

“disjoint inputs”

Statistical classification:

Given uniform samples from x { a, b, c }, decide whether x = a or x = b or x = c.

f: An B: symmetric function

Can be solved by any sampling algorithm approximating f

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The “Election Problem”• input: a sequence x of n votes to k parties

7/18 4/18 3/18 2/18 1/18 1/18

(n = 18, k = 6)

• Want to get s.t. || - x|| < .Vote Distribution x

Theorem

A poll of size (k/2) is required for estimating the election problem.

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Combinatorial Designs

1. Each of them constitutes half of U.2. The intersection of each two of them is

relatively small.

B1

B2

B3U

A family of subsets B1,…,Bm of a universe U s.t.

Fact: There exist designs of size exponential in |U|.

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Proof of the Lower Bound for the Election Problem

Step 1: Identification of a set S of pairwise disjoint inputs:

B1,…,Bm µ {1,…,k}: a design of size m = 2(k).

S = { x1,…,xm }, where in xi:

Bi Bic

Step 2: JS(1,…,m) = O(2).

By our theorem, # of queries is at least (k/2).

• ½ + of the votes are split among parties in Bi.

• ½ - of the votes are split among parties in Bi

c.

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Hamming Distance Sketching(with T.S. Jayram and R. Kumar, 2003)

Alice Bob

Referee

Ham(x,y) > k

x y

x)

y)

Ham(x,y) · k

$$

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Hamming Distance Sketching

Applications• Maintenance of large crawls• Comparison of large files over the network

Previous schemes:• Sketches of size O(k2)

[Kushilevitz, Ostrovsky, Rabani, 98], [Yao 03]

• Lower bound: (k)

Our scheme:• Sketches of size O(k log k)

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Preliminaries

Balls and Bins:

• When throwing n balls into n/log n bins, then with high probability the fullest bin has O(log n) balls.

• When throwing n balls into n2 bins, then with high probability no two balls fall into the same bin.

• Using KOR scheme, can assume Ham(x,y) · 2k.

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First Level Hashing

1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0x

1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1y

1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0

1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1

k/log k bins

k/log k bins

y1 y2 y3

x2x1 x3

Ham(x,y) =

i Ham(xi,yi)

8i, Ham(xi,yi) · O(log k)

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Second Level Hashing

y3

x3

1 1

0 1

00

0 1

1 1

10

1 1

0 1

00

0 1

1 1

10

log2 k bins

log2 k bins

3,1 3,2 3,3

3,4 3,5 3,6

3,1 3,2 3,3

3,4 3,5 3,6

3,j = 3,j iff # of “pink positions” in the j-th bin is even.

• If no collisions, Ham(3,3) = Ham(x3,y3)

• If collisions, Ham(3,3) · Ham(x3,y3)

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The Sketch

• (x) = { ij | i = 1,…,k/log k, j = 1,…,t }

• (y) = { ij | i = 1,… k/log k, j = 1,…,t }

• Referee decides Ham(x,y) · k if and only if

i maxj Ham(ij, i

j) · k

• Probability of collision: a small constant

• For each i = 1,…,k/log k, repeat second level hashing t = O(log k) times, obtaining (i

1,i1),…,(i

t,it).

• With probability at least 1 – 1/k,

Ham(xi,yi) = maxj Ham(ij,i

j)

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Other Sketching Results

• A sketching scheme for the edit distance– Leads to the first almost-linear time

approximation algorithm for the edit distance.

• Sketch lower bounds for (compressed) pattern matching.

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Template Detection (with S. Rajagopalan, WWW 2002)

Template – Master HTML shell page used for composing new pages.

Our contributions:

• Efficient algorithm for template detection

• Application to improvement of search engine precision

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Templates are Bad for Web IR

• Pose a significant source of “noise” in web pages– Their content is not related to the topics of pages

in which they reside– Create spurious linkage to unimportant pages

• Extremely common– Became standard in website design

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Pagelets [Chakrabarti 01]

• has a single theme

• not nested within a bigger region with the same theme

Navigational bar pagelet

Search pagelet

Directory pagelet

News headlines pagelet

Pagelet – a region in a page that:

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Template Definition

Template = a collection of pagelets that:

1.Belong to the same website.

2.Are nearly-identical.

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Template Detection

Template Detection Algorithm• Group the pages in S according to website.• For each website w:

– For each page p 2 w: • Partition p into pagelets p1,…,pk

• Compute a “shingle” sketch for each pagelet [Broder et al. 1997]

– Group the resulting pagelets by their sketches.– Output all the pagelet groups of size > 1.

Template Detection Problem:

Given a set of pages S, find all the templates in S.

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HITS & Clever[Kleinberg 1997, Chakrabarti et al. 1998]

Hubs Authorities

h(p) = q 2 out(p) a(q)

a(p) = q 2 in(p) h(q)

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“Template” Clever

Hubs Authorities

• Hubs – all the non-templatized constituent pagelets of pages in the base set.

• Authorities – all pages in the base set.

Page

Pagelet

Templatized pagelet

Legend

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Classical Clever vs. Template Clever

Average Precision @ 50 for broad queries

0

20

40

60

80

100

120

10 20 30 40 50

Pre

csio

n

Classical Clever

Template Clever

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Template Proliferation

Template Frequency for ARC Set Queries

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

recycling_cans

gardening

mutual_funds

java

Zener

San_Francisco

field_hockey

Penelope_Fitzgerald

HIV

bicycling

affirmative_action

amusement_parks

Thailand_tourism

cruises

volcano

stamp_collecting

architecture

Shakespeare

Gulf_war

zen_buddhism

lyme_disease

Death_Valley

citrus_groves

cheese

table_tennis

blues

classical_guitar

telecommuting

parallel_architecture

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Summary

• Web data mining via random walks on the web graph:– Web statistics– Focused statistics– Web decay

• Sampling lower bounds– Optimality of uniform sampling for symmetric functions– A “recipe” for lower bounds

• Sketching of string distance measures– Hamming distance– Edit distance

• Template detection

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Some of My Other Work

• Database– Semi-structured data and XML

• Computational Complexity – Communication complexity– Pseudo-randomness and de-randomization– Space-bounded computations– Parallel computation complexity

• Algorithm Design– Data stream algorithms– Internet auctions

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Web Statistics(with A. Berg, S. Chien, J. Fakcharoenphol, D. Weitz, VLDB 2000)

The “BowTie” Structure of the Web

[Broder et al, 2000]

crawlable web

SCCOUTIN

• What fraction of the web is covered by Google?

• Which is the largest country domain on the web?

• What is the percentage of porn pages?

• How large is the web?

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Straightforward Random Walk

• Gets stuck in sinks and in dense web communities

• Biased towards popular pages

• Converges slowly, if at all

yahoo.com

amazon.com

www.almaden.ibm.com/cs/people/ziv

Follow a random out-link at each step

1

2

3

4

56

7

8

9

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Undirected Regular Random Walk

Fact:

A random walk on a connected (non-bipartite) undirected regular graph converges to a uniform limit distribution.

w(v) = degmax - deg(v)

yahoo.com1

2

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amazon.com

4

02 3

0

3

2

2

4

4

3

3

3

1

2

5

Follow a random out-link or a random in-link at each step

Use weighted self loops to even out page degrees

www.almaden.ibm.com/cs/people/ziv

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Evaluation: Bias towards High Degree Nodes

Deciles of nodes ordered by degree

High Degree

Low Degree

Percent of nodes from walk

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Evaluation: Bias towards the Search Engines

Search engine size30% 50%

Estimate of search engine size

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Link-Based Web IR Applications

• Search and ranking – HITS and Clever [Kleinberg 1997,Chakrabarti et al. 1998]– PageRank [Brin and Page 1998]– SALSA [Lempel and Moran 2000]

• Similarity search– Co-Citation [Dean and Henzinger 1999]

• Categorization– Hyperclass [Chakrabarti, Dom, Indyk 1998]

• Focused crawling– FOCUS [Chakrabarti, van der Berg, Dom 1999]

• …

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Hypertext IR Principles

• Relevant Linkage Principle [Kleinberg 1997]

– p links to q q is relevant to p

• Topical Unity Principle [Kessler 1963, Small 1973]

– q1 and q2 are co-cited in p q1 and q2 are related to each other

• Lexical Affinity Principle [Maarek et al. 1991]

– The closer the links to q1 and q2 are the stronger the relation between them.

Underlying principles of link analysis:

p q

pq1

q2

p

q1

q2

q3

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Example: HITS & Clever[Kleinberg 1997, Chakrabarti et al. 1998]

• Relevant Linkage Principle– All links propagate score from hubs

to authorities and vice versa.

• Topical Unity Principle– Co-cited authorities propagate

score to each other.

• Lexical Affinity Principle (Clever)– Text around links is used to weight

relevance of the links.

Hubs Authorities

h(p) = q 2 out(p) a(q)

a(p) = q 2 in(p) h(q)