11

Apriori AlgorithmApriori Algorithm

Rakesh AgrawalRakesh Agrawal

Ramakrishnan SrikantRamakrishnan Srikant

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Association RulesAssociation Rules(Agrawal, Imielinski & Swami: SIGMOD ’93)(Agrawal, Imielinski & Swami: SIGMOD ’93)

I = { i1, i2 …, im }: a set of literals, called items.

Transaction T: a set of items such that T I. Database D: a set of transactions.

A transaction T contains X, a set of some items in I, if X T.

An association rule is an implication of the form X Y, where X, Y I.– Support: % of transactions in D that contain X Y.– Confidence: Among transactions that contain X, what % also

contain Y.

Find all rules that have support and confidence greater than user-specified minimum support and minimum confidence.

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1. Find all sets of items that have minimum support (frequent itemsets).

2. Use the frequent itemsets to generate the desired rules. confidence ( X Y ) = support ( X Y ) / support ( X )

Computing Association Rules: Problem Decomposition

What itemsets should you count?

How do you count them efficiently?

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What itemsets do you count?What itemsets do you count?

Search space is exponential.– With n items, nCk potential candidates of size k.

Anti-monotonicity: Any superset of an infrequent itemset is also infrequent (SIGMOD ’93).– If an itemset is infrequent, don’t count any of its extensions.

Flip the property: All subsets of a frequent itemset are frequent.

Need not count any candidate that has an infrequent subset (VLDB ’94)– Simultaneously observed by Mannila et al., KDD ’94

Broadly applicable to extensions and restrictions.

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Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c d

66

Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c d

77

Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c

a b a d

d

b d

88

Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c

a b a d

d

b d

99

Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c

a b a d

a b d

d

b d

1010

Apriori Algorithm: Apriori Algorithm: Breadth First SearchBreadth First Search

{ }

a b c

a b a d

a b d

d

b d

1111

APRIORI Candidate GenerationAPRIORI Candidate Generation(VLDB 94)(VLDB 94)

Lk : Frequent itemsets of size k, Ck :

Candidate itemsets of size k

Given Lk, generate Ck+1 in two steps:

1. Join Step : Join Lk with Lk, with the

join condition that the first k-1 items should be the same and l1[k] < l2[k].

L3

{ a b c }

{ a b d }

{ a c d }

{ a c e }

{ b c d }

C4

{ a b c d }

{ a c d e }

1212

APRIORI Candidate GenerationAPRIORI Candidate Generation(VLDB 94)(VLDB 94)

Lk : Frequent itemsets of size k, Ck :

Candidate itemsets of size k

Given Lk, generate Ck+1 in two steps:

1. Join Step : Join Lk with Lk, with the

join condition that the first k-1 items should be the same and l1[k] < l2[k].

2. Prune Step : Delete all candidates which have a non-frequent subset.

C4

{ a b c d }

{ a c d e }

L3

{ a b c }

{ a b d }

{ a c d }

{ a c e }

{ b c d }

1313

How do you count?How do you count?

Given a set of candidates Ck, for each transaction T:

– Find all members of Ck which are contained in T.

Hash-tree data structure [VLDB ’94]– C2 : { }– T : {c, e, f}

a b c d e f g

{a, b} {e, f} {e, g}

1414

How do you count?How do you count?

Given a set of candidates Ck, for each transaction T:

– Find all members of Ck which are contained in T.

Hash-tree data structure [VLDB ’94]– C2 : { }– T : {c, e, f}

a b c d e f g

{a, b} {e, f} {e, g}

1515

How do you count?How do you count?

Given a set of candidates Ck, for each transaction T:

– Find all members of Ck which are contained in T.

Hash-tree data structure [VLDB ’94]– C2 : { }– T : {c, e, f}

a b c d e f g

{a, b} {e, f} {e, g}

Recursively construct hash tables if number of itemsets is above a

threshold.

f g

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ImpactImpact

Concepts in Apriori also applied to many generalizations, e.g., Concepts in Apriori also applied to many generalizations, e.g., taxonomies, quantitative Associations, sequential Patterns, taxonomies, quantitative Associations, sequential Patterns, graphs, …graphs, …

Over 3000 citations in Google Scholar.Over 3000 citations in Google Scholar.

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Subsequent Algorithmic InnovationsSubsequent Algorithmic Innovations

Reducing the cost of checking whether a candidate itemset is Reducing the cost of checking whether a candidate itemset is contained in a transaction:contained in a transaction:– TID intersection.TID intersection.– Database projection, FP GrowthDatabase projection, FP Growth

Reducing the number of passes over the data:Reducing the number of passes over the data:– Sampling & Dynamic CountingSampling & Dynamic Counting

Reducing the number of candidates counted:Reducing the number of candidates counted:– For maximal patterns & constraints.For maximal patterns & constraints.

Many other innovative ideas …Many other innovative ideas …