Keyword Proximity Search on Graphs M.Sc. Systems Course The Hebrew University of Jerusalem, Winter...

Keyword Proximity Search on Keyword Proximity Search on Graphs Graphs

M.Sc. Systems CourseM.Sc. Systems CourseThe Hebrew University of Jerusalem, Winter 2006

Keyword Proximity Search on Graphs MSSYS 2006

A rapidly evolving paradigm for data extraction

Data have varying degrees of structure

Queries are sets of keywords− No structural constraints

Keyword Proximity Search

Relational Databases

Web SitesXML

Documents

The Goal:The Goal:

Extract meaningful parts of data w.r.t. the keywords


Recent Work on KPS (Keyword Proximity Search)

• DataSpotDataSpot (Sigmod 1998)

• Information Units Information Units (WWW 2001)

• BANKSBANKS (ICDE 2002, VLDB 2005)

• DISCOVERDISCOVER (VLDB 2002)

• DBXplorerDBXplorer (ICDE 2002)

• XKeyword XKeyword (ICDE 2003)

• ……


Systems for KPS on Relational Data

BANKS, DISCOVER and DBXplorer implemented KPS (Keyword Proximity Search) on relational databases Different algorithms are used Slight differences in semantics

G. Bhalotia, A. Hulgeri, C. Nakhe, S. Chakrabarti, and S. Sudarshan. Keyword searching and browsing in databases using BANKS. In ICDE, pages 431–440, 2002.

V. Hristidis and Y. Papakonstantinou. DISCOVER: Keyword search in relational databases. In VLDB, pages 670–681, 2002.

S. Agrawal, S. Chaudhuri, and G. Das. DBXplorer: enabling keyword search over relational databases. In SIGMOD Conference, page 627, 2002.


Example: KPS on RDB

IDNamePopulation

22Amsterdam1101407

73Brussels951580

IDNameHead Q.

135EU73

175ESA81

CountryOrg.

B135

NL135

search Belgium , Brussels

CodeNameAreaCapital

NLNetherlands3733022

BBelgium3051073

CitiesCities OrganizationsOrganizations

CountriesCountries MembershipsMemberships


IDNamePopulation

22Amsterdam1101407

73Brussels951580

IDNameHead Q.

135EU73

175ESA81

CountryOrg.

B135

NL135


CodeNameAreaCapital


BBelgium3051073



Brussels is the capital city of Belgium


IDNamePopulation

22Amsterdam1101407

73Brussels951580

IDNameHead Q.

135EU73

175ESA81

CountryOrg.

B135

NL135


CodeNameAreaCapital


BBelgium3051073



BBelgium3051073 73Brussels951580

Brussels is the capital city of Belgium


IDNamePopulation

22Amsterdam1101407

73Brussels951580

IDNameHead Q.

135EU73

175ESA81

CountryOrg.

B135

NL135

CodeNameAreaCapital


BBelgium3051073



Brussels hosts EU and Belgium is a member



IDNamePopulation

22Amsterdam1101407

73Brussels951580

IDNameHead Q.

135EU73

175ESA81

CountryOrg.

B135

NL135

CodeNameAreaCapital


BBelgium3051073



BBelgium3051073

73Brussels951580

Brussels hosts EU and Belgium is a member


B135 135EU73


XKeyword: KPS on XML

XKeyword implemented KPS on XML Architecture is based on that of DISCOVER

A demo over DBLP is available

• http://kebab.ucsd.edu:81/xkeyword

V. Hristidis, Y. Papakonstantinou, and A. Balmin. Keyword proximity search on XML graphs. In ICDE, pages 367–378, 2003.


Example: KPS on XML

dblp

title

author

article

MihalisYannakakis

On theApproximationof MaximumSatisfiability

title

author

article

ImprovedApproximationAlgorithms for

MAX SAT

TakaoAsano

David P.Williamson

authorreferences

cite

search Yannakakis , Approximation


Yannakakis wrote a paper about Approximation

dblp

title

author

article

MihalisYannakakis

On theApproximation

of MaximumSatisfiability

title

author

article


MAX SAT

TakaoAsano

David P.Williamson

authorreferences

cite



dblp

title

author

article

MihalisYannakakis


title

author

article


MAX SAT

TakaoAsano

David P.Williamson

authorreferences

cite

Yannakakis is cited by a paper about Approximation



KPS on Web Sites (Information Units)

• KPS can also be used for retrieving information from Web sites

• For a given query, results are collections of Web pages from the site

– Pages are relevant w.r.t. the keywords

– Pages are connected by hyperlinks

Wen-Syan Li, K. Selçuk Candan, Quoc Vu, and Divyakant Agrawal. Retrieving and organizing web pages by “information unit”. In WWW, pages 230-244, 2001.


Example: KPS in Web Sites

http://www.goisrael.com/http://www.goisrael.com/

search Hilton , Beach


Example: KPS in Web Sites

Eilat Beaches

Hilton Eilat Queen of Sheba

search Hilton , Beach

Eilat

A Formal Framework for KPSA Formal Framework for KPS


Data Graphs

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Data graphs have two types of nodes: Structural nodes

Keywords


Queries

K={ Summers , Cohen , coffee }company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Queries are sets of keywords from the data graph


Query Results

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq


Query Results

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Query results are subtrees of the data graph Contain all keywords in the query

Have no redundant edges

A subtree that isreduced w.r.t. thekeywords


Three Variants

Three variants of keyword proximity search are considered:

Rooted proximity

Undirected proximity

Strong proximity


Rooted Variant

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Used in BANKS BANKS

Results are rooted trees


Undirected Variant

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Used in Interconnection Interconnection Semantics for XMLSemantics for XML

Results are undirected trees


Strong Variant

company

supplies

supply

product

supplier

papersA4

company

supplies

supply

product

supplier

coffee

president

Cohen

department

Summers

manager

Parishqhq

Used in XKeywordXKeyword, Information Information UnitsUnits, DBXplorerDBXplorer and DISCOVERDISCOVER

Results are undirected treesand keywords are leaves


DataData

A data graph G

Problem Definition

QueryQuery

A set K of keywords in G

Query ResultsQuery Results

Subtrees of G that are reduced w.r.t. K

Input:Input:

Output:Output:

Rooted/Undirected/Strong


Creating Data Graphs from Relational Databases

Nodes are tuples

Edges are foreign-key references


Creating Data Graphs from Relational Databases

Edges from each tuple node to all the keywords in that tuple

Belgium 30510B 73

Belgium 30510B 73


Creating Data Graphs from XML

Nodes are XML elements

dblp

article article


titleMihalis

Yannakakis

authorTakao Asano

authorDavid P.

Williamson

authorImproved

ApproximationAlgorithms for

MAX SAT

titlecite



dblp

article article


titleMihalis

Yannakakis

authorTakao Asano

authorDavid P.

Williamson

authorImproved


MAX SAT

titlecite


Edges are nesting of elements …Edges represent

nesting of elements …



dblp

article article


titleMihalis

Yannakakis

authorTakao Asano

authorDavid P.

Williamson

authorImproved


MAX SAT

titlecite


Edges represent nesting of elements …

… and ID references


Creating Data Graphs from XMLKeywords appear in PCDATA

dblp

article article


titleMihalis

Yannakakis

authorTakao Asano

authorDavid P.

Williamson

authorImproved


MAX SAT

titlecite


… and ID references

Edges are nesting of elements …Edges represent

nesting of elements …


All Occurrences of a Keyword are Represented by One Node

dblp

article article


titleMihalis

Yannakakis

authorTakao Asano

authorDavid P.

Williamson

authorImproved


MAX SAT

titlecite

Approximation Approximation

A keywords is represented by a single node


Creating Data Graphs from Web Sites

Nodes are Web pages …

Keywords appear in these pages …

Edges are hyperlinks/XLinks

http://www.goisrael.com/http://www.goisrael.com/

A keywords is represented by a single

node

Ranking and Enumeration OrderRanking and Enumeration Order


Ranking Results

Yannakakis

Approximation

title

Yannakakis

Approximation

dblp

article

title

article

title

Yannakakis

Approximation

article

title

article

title

cite

references

Ranking of results is determined by size

2 13


Edges Have Weights

Yannakakis

Approximation

dblp2

article

2

title

1

1

article

title1

1

Yannakakis

Approximation

article

title

article

title

cite

references1

1.5

1

1

1

1 1

1

Yannakakis

Approximation

title

edges incident to dblp have a large weight

edges from cite to article have a medium weight

2 13


Order of Results

Arbitrary Order

Exact Order ji RRji ,


Order of Results (cont’d)

Heuristic Order

C-Approximate Order

ji RCRji ,

Measuring the Efficiency of Measuring the Efficiency of EnumerationsEnumerations


Polynomial Runtime is not Appropriate for KPS

• In the theory of CS, the usual notion of efficiency is polynomial running time That is, the algorithm terminates in time that is

polynomial in the size of the input

• However, in KPS the number of results can be exponential in the size of the input Algorithms cannot be expected to terminate in

polynomial time

Even for two keywords

• Therefore, other notions are required


Time Efficiency

Polynomial Total TimePolynomial Total Time

Polynomial runtime in the combined size of the input and the output

Polynomial DelayPolynomial Delay

The runtime between two successive results is polynomial in the size of the input


About Polynomial Delay

• With polynomial delay you can: Generate the first few results quickly

Efficiently return results in pages

• In most cases of keyword search, this is the suitable notion of efficiency

• Goal: develop algorithms that enumerate KPS results with polynomial delay


Space Efficiency

Polynomial Space

Linearly-Incremental Space i results require i times polynomial space in

the input


Data and Query-and-Data Complexity

• Under query-and-data complexity, we assume that both the query and the data are of unbounded size Many problems in database theory, e.g.,

computing joins of relational tables, are intractable under this measure

• In practice, however, queries are very small compared to the data

• Under data complexity, the size of the query is assumed to be fixed

Enumerating Results of KS with Enumerating Results of KS with Polynomial DelayPolynomial Delay


Keyword Search with Polynomial Delay

• The following algorithm enumerates reduced subtrees (i.e., results of keyword search) with polynomial delay Results are not ranked

• A different version of the algorithm for each of the three variants: rooted

undirected

strong


Importance of the Algorithm

• An upper bound for ranked keyword search: Results can be enumerated in ranked order in polynomial total time Generate all the results and then sort them

• In some cases, ranking is not required

• A basis for developing efficient heuristics that enumerate in an “almost” ranked order (discussed later)

The Algorithm for Enumerating The Algorithm for Enumerating Rooted Reduced SubtreesRooted Reduced Subtrees


Overview

• The algorithm uses two reductions

• Each reduction alone either does not solve the problem or runs in exponential total time

• However, the two reductions can be combined together to enumerate reduced subtrees with polynomial delay


Data Reduction

1. Choose an arbitrary node v in K

2. For each parent p of v do:

I. In K: replace v with p

II. In G: remove v

III. Generate all results for the new input

IV. Add p→v to each result of the new input

A

KKGG

A B

p

vA B v

pvv

A B

p

v

p

A B

p

v A B A Bv v

B


Example Showing Failure

B

AC

KK

A

BC

Four results!

Two with this

root

Two with this

root


Failure Example

B

AC

KK

A

BC


Failure Example

B

C

KK BC


Failure Example

C

KK

C


Failure Example

KK


Failure Example

C

B

A

Only one result!

Three others are missing!


Why Data Reduction Fails

• We assumed that v is a leaf in every result

• It does not hold for structural nodes in recursive steps!

• Therefore, some results are not found!

• Solution(?): Repeat data reduction for every v in K Exponential total time in the worst case!


Query Reduction

1. Remove one keyword from the query

2. Find all results for the smaller query

3. Extend each result to include the missing keyword, in every possible way

A K= {A,B,C}

A

B

BA

A

BC

C

A

B

C BA A

C

B CBA


Extending Partial Results

• In query reduction, we need to extend a result T of the query K\{k} to all results of the query K

• This is done as follows: For all nodes v of T:

• Remove from G all nodes of T, except for v

• Find all simple directed paths P from v to k and print the concatenation of T and P

• If v is the root of T, we also need to concatenate T with all subtrees that are reduced w.r.t. v and k

• More details are can be found in the paper


Extensions by Directed Paths


Extensions by Directed Subtrees


Query Reduction is not Efficient!

• Query reduction completely solves the problem, but it is inefficient

• Problem: A subset of the query may have much more results than the query itself

Exponential total time!

A B CnA B CnA B CnA B Cn

2n results

for {A,B}

1 result for

{A,B,C}


Combining the Reductions

• In order to enumerate in polynomial total time, combine query and data reductions: If some node v of K is reachable, in the data

graph, from another node u of K, use query reduction

• remove v from K

Otherwise, use data reduction

• By combining the two reductions, results can be enumerated in polynomial total time

v

u


Achieving Polynomial Delay

• To achieve polynomial delay, we cannot wait until a recursive subroutine terminates

• Use coroutines instead of subroutines!

• That is, each recursive execution of the algorithm

stops after generating each result

resumes when the next result is required


routine 3 routine 2 routine 1

Subroutines

Base

Polynomial Polynomial Total TimeTotal Time


routine 3 routine 2 routine 1

Coroutines

Base

Polynomial Polynomial DelayDelay

For papers and projects related to this topic, see the home page of Benny Kimelfeld

http://www.cs.huji.ac.il/~bennyk

Keyword Proximity Search on Graphs M.Sc. Systems Course The Hebrew University of Jerusalem, Winter...

Documents

Transcript of Keyword Proximity Search on Graphs M.Sc. Systems Course The Hebrew University of Jerusalem, Winter...