Mining Complex Patterns

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1 Mining Complex Data COMP Seminar Spring 0 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Mining Complex Patterns Common Pattern Mining Tasks: Itemsets (transactional, unordered data) Sequences (temporal/positional: text, bioseqs) Tree patterns (semi-structured/xml data, web mining) Graph patterns (protein structure, web data, social network)

2 Example Pattern Types Itemsett Sequence A B C D A B C D Tree A Graph A Can add attributes To nodes To edges B C B C Attributes Labels Type (directed or undirected ) Set-valued D D Induced vs Embedded Sub-trees Id Induced dsbt Sub-trees: S = (V s, E s )i is a sub-tree bt of ft = (V,E) if and only if V s V e = (n x, n y ) E s iff (n x, n y ) E (n x directly connected to n y ) Embedded Sub-trees: S = (V s, E s ) is a sub-tree of T = (V,E) if and only if V s V e = (n x, n y ) E s iff n x l n y in T (n x connected to n y ) An induced sub-tree is a special case of embedded subtree. We say S occurs in T and T contains S if S is an embedded sub-tree of T If S has k nodes, we call it a k-sub-tree

3 Mining Frequent Trees Support: the support of a subtree in a database of trees, is the number of trees containing the subtree. A subtree is frequent if its support is at least the minimum support. TreeMiner: Given a database of trees (a forest) and a minimum support, find all frequent subtrees. 5 String Representation of Trees n n n n0 0 n n5 n With N nodes, M branches, F max fanout Adjacency Matrix requires: N(F+) space Adjacency List requires: N- space Tree requires (node, child, sibling): N space String representation requires: N- space 6

4 Tree: String Representation Like an itemset - as the backtrack item Assuming only labels on nodes For trees labels on edges can be treated as labels on nodes: edge-label+node-label = new label! 7 Match labels Tree [0,6] Subtree n0 [,5] [6,6] n n6 6 [,] [5,5] 5 n n5 8 [,] n n vector < id, match label, scope > [,] Match Label: 056 Support:

5 An example 9 Generic Mining Algorithms Horizontal pattern matching based Vertical intersection based BFS or DFS 0

6 Candidate Generation & Support Counting Candidate Generation Extend by a node or an edge Avoid duplicates as far as possible Trees: Systematic Candidate Generation Two subtrees are in the same class iff they share a common prefix string P up to the (k-)th node Not valid position: Prefix x A valid element x attached to only the nodes lying on the path from root to rightmost leaf in prefix P

7 Candidate generation Given an equivalence class of k-subtrees, how do we generate candidate (k+)- subtrees? Main idea: consider each ordered pair of elements in the class for extension, including self extension Sort elements by node label and position Class extension

8 Candidate Generation (Join operator) Self Join Equivalence Class Prefix:, Elements: (,) (,0) New Candidates Join 5 New Equivalence Class Prefix: Elements: (,) (,) (,0) Candidate Generation (Join operator) Join Equivalence Class Prefix:, Elements: (,) (,0) Self Join New Equivalence Class Prefix: Elements: (,0) (,) 6

9 Candidate Generation (Join operator) Self Join Equivalence Class Prefix:, Elements: (,) (,) New Candidates Join New Equivalence Class Prefix: Elements: ()()()() (,) (,) (,) (,) 7 Candidate Generation (Join operator) Equivalence Class Join Prefix:, Elements: (,) (,) New Candidates SelfJoin New Equivalence Class Prefix: Elements: ()()()() (,) (,) (,) (,) 8

10 Apriori Style TreeMiner 9

Efficiently Mining Frequent Trees in a Forest

Efficiently Mining Frequent Trees in a Forest Mohammed J. Zaki Computer Science Department, Rensselaer Polytechnic Institute, Troy NY 8 zaki@cs.rpi.edu, http://www.cs.rpi.edu/ zaki ABSTRACT Mining frequent