Distributed and Parallel High Utility Sequential Pattern Mining

吳易倫

Yi-Lun Wu

Problem Description

Definition:

  • Item: id[price]

    • 1[4], 2[3]

  • Itemset: a set of item

    • {1[4], 2[9], 3[2]}

  • ​Sequence: a sequence of itemset

    • ​({1[4], 2[9]}, {2[3], 3[3], 4[1]}, {4[4]})

Problem Description cont.

  • Itemset: a set of item

    • {1[4], 2[9], 3[2]}
    • sorted in item id w.l.o.g.

  • ​Sequence: a sequence of itemset

    • ​({1[4], 2[9]}, {2[3], 3[3], 4[1]}, {4[4]})
    • order of itemsets in sequence is meaningful
S_1=(\{1[4], 2[9]\}, \{2[3], 3[3], 4[1]\}, \{4[4]\})
S1=({1[4],2[9]},{2[3],3[3],4[1]},{4[4]})S_1=(\{1[4], 2[9]\}, \{2[3], 3[3], 4[1]\}, \{4[4]\})
P_1=(\{1, 2\}, \{4\})
P1=({1,2},{4})P_1=(\{1, 2\}, \{4\})

Two subsequence in     are similar to 

S_1
S1S_1
P_1
P1P_1
S_2=(\{1[4], 2[9]\}, \{2[3], 3[3], 4[1]\}) \sim P_1
S2=({1[4],2[9]},{2[3],3[3],4[1]})∼P1S_2=(\{1[4], 2[9]\}, \{2[3], 3[3], 4[1]\}) \sim P_1
S_3=(\{1[4], 2[9]\}, \{4[4]\}) \sim P_1
S3=({1[4],2[9]},{4[4]})∼P1S_3=(\{1[4], 2[9]\}, \{4[4]\}) \sim P_1
Utility(P_1,S_2)=4+9+1=14
Utility(P1,S2)=4+9+1=14Utility(P_1,S_2)=4+9+1=14
Utility(P_1,S_3)=4+9+4=16
Utility(P1,S3)=4+9+4=16Utility(P_1,S_3)=4+9+4=16
Utility(P_1,S_1)=Max(14,16)
Utility(P1,S1)=Max(14,16)Utility(P_1,S_1)=Max(14,16)

Problem Description cont.

S_1,S_1,S_3,\cdots,S_n
S1,S1,S3,⋯,SnS_1,S_1,S_3,\cdots,S_n
Utility(P, Database) = \sum_{i=0}^{i=n}Utility(P,S_i)
Utility(P,Database)=∑i=0i=nUtility(P,Si)Utility(P, Database) = \sum_{i=0}^{i=n}Utility(P,S_i)

Utility of pattern P in database:

Purpose: find all

P
PP

s.t.

Utility(P, Database)>threshold
Utility(P,Database)>thresholdUtility(P, Database)>threshold

Problem Description cont.

Find Utility of Patterns in a Sequence

  • USpan algorithm

    • Concept: Brute Force

    • Strategy: Depth First Search

      • Each node in space search tree represents a pattern state

      • Calculating utility of next pattern is based on previous utility

S_1=(\{1[4], 2[9]\}, \{1[3], 3[1], 4[2]\}, \{3[10]\})
S1=({1[4],2[9]},{1[3],3[1],4[2]},{3[10]})S_1=(\{1[4], 2[9]\}, \{1[3], 3[1], 4[2]\}, \{3[10]\})
\{4,3\}
{4,3}\{4,3\}
\{9\}
{9}\{9\}
\{1,10\}
{1,10}\{1,10\}
(\{3\})
({3})(\{3\})
(\{1\})
({1})(\{1\})
(\{2\})
({2})(\{2\})
\{13\}
{13}\{13\}
(\{1,2\})
({1,2})(\{1,2\})
\{5,14,13\}
{5,14,13}\{5,14,13\}
(\{1\},\{3\})
({1},{3})(\{1\},\{3\})
\{4\}
{4}\{4\}
\{2\}
{2}\{2\}
(\{4\})
({4})(\{4\})
\cdots\cdots
⋯⋯\cdots\cdots
\{7\}
{7}\{7\}
(\{1\},\{3,4\})
({1},{3,4})(\{1\},\{3,4\})
\{15\}
{15}\{15\}
(\{1\},\{3\},\{3\})
({1},{3},{3})(\{1\},\{3\},\{3\})
\{0\}
{0}\{0\}
(\{\})
({})(\{\})

Concat-2

Concat-3

Concat-3

Concat-4

Concat-3

No downward closure property

Concat-1

(\{1,3\})
({1,3})(\{1,3\})

USpan Algorithm

Algorithm in Spark

RDD1

USpan

USpan

USpan

USpan

Global Filter

Local Filter

Maintain all Local High Utility Pattern in RDD2

Calculate Global Utility of each pattern

Output

Similar to USpan

Alpha-Beta pruning in DFS

Experimental Results

Candidates

Candidates

Time(s)

Time(s)

Threshold(%)

Threshold(%)

Threshold(%)

Threshold(%)

Dataset1 1000 sequence

Dataset2 100 sequence

Extra Results

  • Algorithm(named PG-HUSP mining) which check whether a pattern is Global High Utility Pattern is surprisingly fast

    • PG-HUSP
      •   
      •  
      •  
    • My algorithm(Dynamic programming)
      • ​
      • ​
      •  
worst\ case:O(n^m)
worst case:O(nm)worst\ case:O(n^m)
n = length\ of\ sequence, m = length\ of\ pattern
n=length of sequence,m=length of patternn = length\ of\ sequence, m = length\ of\ pattern
worst\ case:O(n\times m)
worst case:O(n×m)worst\ case:O(n\times m)
n = length\ of\ sequence, m = length\ of\ pattern
n=length of sequence,m=length of patternn = length\ of\ sequence, m = length\ of\ pattern
depend\ on\ the\ number\ of\ subsequence
depend on the number of subsequencedepend\ on\ the\ number\ of\ subsequence
average:\Theta (n\times m)
average:Θ(n×m)average:\Theta (n\times m)

  • One of Global filter does not work

Extra Results cont.

Remove filter

Apply filter

=
==

Q&A

Project at

https://github.com/w86763777/ParallelizedUSpan
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