mehdi kargar aijun an york university, toronto, canada keyword search in graphs: finding r-cliques

Mehdi KargarAijun An

York University, Toronto, Canada

Keyword Search in Graphs:Finding r-cliques

Overview

• Keyword Search in Graphs/Relational Databases• r-clique Definition• Challenges in Finding r-clique• Approximation Algorithm for Finding r-cliques• Enumerating Top-k r-cliques in Polynomial Delay

• Empirical Results• Conclusion

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VLDB’11 Keyword Search in Graphs: Finding r-cliques

Keyword Search in Graphs/Relational Databases

• Keyword search is a well known mechanism for retrieving relevant information from a set of documents.• Google is a familiar example !

• What about structured data?• Such as XML documents or Relational Databases?

• Current enterprise search engines in structured data requires:• Knowledge of schema• Knowledge of a query language• Knowledge of the role of the keywords

• Do users have all of the above Knowledge ? • The answer is NO !

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Keyword Search in Graphs/Relational Databases

• Users need a simple system that receives some keywords as input and returns a set of nodes that together cover all or part of the input keywords as output.

• Relational databases can be modeled using graphs:• Tuples are nodes of the graph. • Foreign key relationships are edges that connect two nodes

(tuples) to each other.

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Example: Search in Relational Databases

ID Name Country

22 Toronto CA

16 New York US

ID Name Head Q.

135 UN 16

175 EU 81

Country Org.

CA 135

US 135

Code Name

CA Canada

US United States

CitiesCities OrganizationsOrganizations

CountriesCountries MembershipsMemberships

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New York is Located in United States

ID Name Country

22 Toronto CA

16 New York US

ID Name Head Q.

135 UN 16

175 EU 81

Country Org.

CA 135

US 135

Code Name

CA Canada

US United States



Keywords : “New York” “United States”

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New York hosts UN and Canada is a member

ID Name Country

22 Toronto CA

16 New York US

ID Name Head Q.

135 UN 16

175 EU 81

Country Org.

CA 135

US 135

Code Name

CA Canada

US United States



Keywords : “New York” “Canada”

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Previous Approaches

• Most of the works find minimal connected trees that contain all or part of the input keywords.• The tree is called Steiner Tree.

• Recently, methods that produce sub-graphs are proposed. They might provide more informative answers• One of the recent approaches is called multi-center community

(ICDE 2009).

• So, what is the problem with previous approaches?

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Problems with Previous Approaches

1. There might be some content nodes that are far away from each other.

• It means that weak relationships among content nodes might exist.• There is no guarantee on the closeness of the nodes.• Since all keywords are equally important, all of them should be close

to each other. They are also equally important in the ranking function.

2. While searching for the answers, current methods explore both content and non-content nodes.

• This might lead to poor performance.

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r-cliques• To solve the problem of previous approaches, we propose to

find r-cliques.• An r-clique is a set of content nodes that together contain all

of the input keywords and in which the shortest distance between each pair of nodes is no longer than r.

• Weight of r-clique: Suppose that the nodes of an r-clique are denoted as {v1, v2, … , vn}. The weight of the r-clique is defined as:

• dist(vi,vj) is the shortest distance between vi and vj .

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Benefits of Finding r-cliques• Finding r-cliques as the answers for keyword search in

graphs does not have the problems of previous approaches.

• All of the content nodes are reasonably close to each other.• The weight function evaluates all of the content nodes

equally.• The algorithm (to be discussed later) for finding r-cliques

concentrate on the content nodes rather than all of the nodes in the graph. So, it is faster and more efficient.

• For presenting the relationships, the final answer has less irrelevant nodes than a multi-center community.

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An Example

Input Keyword: James John Jack

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r-clique weight: 12tree weight: 8community weight: 8

r-clique weight: 14tree weight: 7community weight: 7


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Challenges in Finding r-cliques• Problem 1: Given a distance threshold r, a graph G and a

set of input keywords, find an r-clique in G whose weight is minimum.

• Theorem: Problem 1 is NP-hard.• Proved in the paper by reduction from 3-satisfiability (3-SAT).

• Solution : Approximation algorithm with guaranteed ratio.

• Total number of answers is exponential regarding the number of input keywords.• It is not efficient to generate all answers and then sort them.

• Solution : Enumerating answers in polynomial delay.


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What We Need …• Producing r-cliques in a ranking order

• r-cliques with lower weights should be presented before ones with higher weights.

• Producing top-k r-cliques efficiently with a bound on approximation ratio • Each r-clique must be generated efficiently in polynomial time.• There must be a bound on the quality of a generated r-clique

• The weight of a generated r-clique should be within some factor of the current optimal solution

• Generating all the r-cliques if needed• No r-clique should be missed


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Heuristic and Approximate Order

It is close to the optimal answer with a provable guarantee

It is expected to be close to the optimal answer.

But, we have no guarantee

Heuristic Order

Approximate Order

Desired Choice

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Enumerating in Approximate Order• The Lawler’s technique is used for finding the top-k

answers.• In each iteration, the next r-clique is generated by finding

the top answer under constraints.• Two problems should be solved

1- What are the constraints?

2- How top answer can be found efficiently under the constraints?


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Overview of the SystemInput

Keywords+

Value of k

Find best Answer with

no Constraint

Insert the best r-clique with the search space in priority queue

Fetch the best r-clique from

priority queue and print it

Divide the related search space of the top answer

into sub-spaces

Find best r-clique in each sub-space

with associated constrains

Insert each answer with the related

search space into priority queue

Top-k already printedOR

Empty priority queue

?YESTerminate

NO


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Constraints and Search Space• Let’s do it using an example !

• Suppose that the input keywords are {k1, k2, k3, k4}.

• Ci = {set of nodes that contains keyword ki }.

• The search space that contains the best r-clique can be represented as {C1 ᵡ C2 ᵡ C3 ᵡ C4}.

• Assume that the best r-clique is (v1, v2, v3, v4), where vi is a node containing keyword ki .


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The whole search space

Finding Best Approximate r-clique• Step 1: for all content nodes n in the search space, for all

keywords ki, find the closest node in the search space which contains ki.

• Step 2: for all content nodes n, for all keywords ki, calculate the sum of distances from n to the holder of ki.

• Step 3: Find the content node with the minimum sum of distances among other content nodes.

• Step 4: Return the set of content nodes with the minimum sum of distances.


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Properties of the Approximation Algorithm• Only content nodes are searched for finding the best

answer in the search space.

• The approximation ratio of the algorithm is equal to 2.• The weight of the answer is at most twice of the weight of the

optimal answer.

• Proof can be found in the paper.

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Presenting r-cliques to the User• To show the relationship between the nodes in an r-clique,

a Steiner tree is found and presented to the user.

Distributed Parallel Algorithm For Nonlinear Optimization

Without Derivatives

A Binding Number Computation of Graph

Xuping Zhang

w

w

Keywords : (in DBLP dataset)“Parallel” “Algorithm” “Optimization” “Graph”

Distributed Parallel Algorithm For Nonlinear Optimization

Without Derivatives

A Binding Number Computation of Graph

Congying Han

w w

A New Non-interior Continuation Method for Second-Order Cone Programming

Guoping He

Xuping Zhang

w w w w

w

r-clique

communityIrrelevant

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Experimental Results• The r-clique is compared with the multi-center community

method (it is called com-k).

• Our approximation algorithm is called poly-delay-k.

• Two datasets are used: DBLP and IMDb.

• The set of input keywords and parameters are the same as the community paper.

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Running Time

DBLP Dataset

IMDb Dataset24/28


Quality of the Answers DBLP Dataset

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Search Accuracy from a User Study

DBLP Dataset 26/28


• Top-k precision: the percentage of the answers in the top-k answers that are relevant to the query.

• The users are asked to evaluate the answers using two methods.• In the first approach the scores (0-1) are assigned to the nodes.

Then, the average is used as the precision.• In the second approach, the whole answer is evaluated and a score

is assigned to it.

• The results of both of the methods are similar.

Conclusion• A novel and efficient approach for keyword search in

graphs has been proposed.• All of the content nodes are reasonably close to each

other.• An approximation algorithm with bounded guarantee has

been proposed.• Only content nodes are explored during the search

process.• A Steiner tree which has as small as possible number of

middle nodes has been generated to reveal relations among content nodes.

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Thank you!

Any Questions?

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mehdi kargar aijun an york university, toronto, canada keyword search in graphs: finding r-cliques

Documents

rcliqueskeyword search

28vldb11 keyword search

canadakeyword search

set of nodes

set of content nodes

nodes tuples

noncontent nodes

input keywords