BIDE: Efficient Mining of Frequent Closed Sequences

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BIDE Efficient Mining of Frequent Closed
Sequences

• JianyongWang and Jiawei Han

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outline

• Frequent closed sequences
• BIDE
• Save space
• Speed up
• Conclusion

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frequent closed sequences
Sequence identifier Sequence
1 CAABC
2 ABCB
3 CABC
4 ABBCA

• frequent sequences
• A4, AA2, AB4, ABB2, ABC4, AC4, B4,
  BB2,BC4, C4, CA3, CAB2, CABC2, CAC2, CB3,
  CBC2,CC2
• frequent closed sequences
• AA2, ABB2, ABC4, CA3, CABC2,CB3

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space usage

• Most of the frequent closed pattern mining
  algorithms need to maintain the set of already
  mined frequent closed patterns (or just
  candidates) in memory
• subpattern checking
• super-pattern checking

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mining frequent closed sequences

• Closed frequent pattern
• Compact
• Efficient
• Costly in both runtime and space usage
• BI-Directional Extension
• mining frequent closed sequences without
  candidate maintenance

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sequence closure checking scheme

• if an n-sequence, Se1e2. . . en, is non-closed
• e is a forward-extension event
• S e1e2 . . . ene and supSDB(S) supSDB(S)
• e is a backward-extension event
• ?i (1 i lt n), S e1e2 . . . eieei1 . . .en
  and
• supSDB(S) supSDB(S)
• S ee1e2 . . . en and supSDB(S) supSDB(S)

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Theorem

• BI-Directional Extension closure checking
• If there exists no forward-extension event nor
  backward extension event w.r.t. a prefix sequence
  Sp, Sp must be a closed sequence otherwise, Sp
  must be non-closed.

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Lemma

• Forward-extension event checking
• For a pre-fix sequence Sp, its complete set of
  forward-extension events is equivalent to the set
  of its locally frequent items whose supports are
  equal to SUPSDB(Sp)

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Definition

• Projected sequence of a prefix sequence
• the projected sequence of prefix sequence AB in
  sequence ABBCA is BCA
• Projected database of a prefix sequence
• the projected database of prefix sequence AB in
  our running example is C, CB, C, BCA

Sequence identifier Sequence
1 CAABC
2 ABCB
3 CABC
4 ABBCA
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Lemma

• Backward-extension event checking
• First, we assume SpAC4, it is easy to find that
  item B appears in each of the 2nd maximum periods
  of Sp. As a result AC4 is not closed.
• let SpABC4, we cannot find any
  backward-extension item for it. Also there is no
  forward-extension item for it, therefore ABC4 is
  a frequent closed sequence.

Sequence identifier Sequence
1 CAABC
2 ABCB
3 CABC
4 ABBCA
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Definition

• First instance of a prefix sequence
• the first instance of the prefix sequence AB in
  sequence CAABC is CAAB
• Last instance of a prefix sequence
• the subsequence from the beginning of S to the
  last appearance of item ei in S
• the last instance of the prefix sequence AB in
  sequence ABBCA is ABB

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Definition

• Given an input sequence S which contains a prefix
  i-sequence e1e2 . . . ei
• The i-th last-in-last appearance w.r.t. a pre-fix
  sequence
• it is the last appearance of ei in the last
  instance of the prefix Sp in S
• SCAABC and SpAB, the 1st last-in-last
  appearance w.r.t. prefix Sp in S is the second A
  in S

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Definition

• Given an input sequence S which contains a prefix
  i-sequence e1e2 . . . ei
• The i-th maximum period of a prefix sequence
• if 1 lt i n, it is the piece of sequence between
  the end of the first instance of prefix e1e2 . .
  . ei-1 in S and the i-th last-in-last appearance
  w.r.t. prefix Sp
• if i 1, it is the piece of sequence in S
  locating before the 1st last-in-last appearance
  w.r.t. prefix Sp
• SABCB and the prefix sequence SpAB
• 2nd maximum period of prefix Sp in S is BC,
• 1st maximum period of prefix Sp in S is f

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EXAMPLE

• SCAABCBC and SpAB
• First instance of a prefix sequence
• CAAB
• Last instance of a prefix sequence
• CAABCB
• The i-th last-in-last appearance w.r.t. a pre-fix
  sequence
• 1st ---- 2nd A
• 2nd ---- 2nd B
• The i-th maximum period of a prefix sequence
• 1st ---- CA
• 2nd ---- ABC

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Definition

• Given an input sequence S which contains a prefix
  i-sequence e1e2 . . . ei
• The i-th last-in-first appearance w.r.t. a prefix
  sequence
• it is the last appearance of ei in the first
  instance of the prefix Sp in S
• SCAABC and SpCA
• 2nd last-in-first appearance w.r.t. prefix Sp in
  S is the first A in S

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Definition

• Given an input sequence S which contains a prefix
  i-sequence e1e2 . . . ei
• The i-th semi-maximum period of a prefix sequence
• if 1 lt i n, it is the piece of sequence between
  the end of the first instance of prefix e1e2 . .
  . ei-1 in S and the i-th last-in-first appearance
  w.r.t. prefix Sp
• if i 1, it is the piece of sequence in S
  locating before the 1st last-in-first appearance
  w.r.t. pre-fix Sp
• SABCB and the prefix sequence SpAC
• 2nd semi-maximum period of prefix AC in S is B
• 1st semi-maximum period of prefix AC in S is f

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Theorem

• BackScan search space pruning
• Let the pre-fix sequence be an n-sequence,
  Spe1e2 . . . en. If ?i (1 i n) and there
  exists an item e which appears in each of the
  i-th semi-maximum periods of the prefix Sp in
  SDB, we can safely stop growing prefix Sp

Sequence identifier Sequence
1 CAABC
2 ABCB
3 CABC
4 ABBCA
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ScanSkip

• The ScanSkip optimization technique
• closure checking scheme needs to scan backward a
  set of maximum-periods w.r.t. a certain prefix
• SpABC 4
• 3rd maximum periods is f, f, f, B gt skip last
  three
• 2nd maximum periods is A, f, f, B gt skip last
  two
• 1st maximum periods is CA, f, C, f gt skip last
  two

Sequence identifier Sequence
1 CAABC
2 ABCB
3 CABC
4 ABBCA
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BIDE

• scans the database once to find the frequent
  1-sequences
• projected database for each frequent 1- sequence
• BackScan pruning method
• backward-extension-item forward-extension-item
• output Sp as a frequent closed sequence
• grow Sp to get a new prefix
• projected database for the new pre-fix
• ?BackScan and backward extension use the ScanSkip
  to speed up the mining process

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Conclusion

• BIDE
• closed pattern mining
• Memory space usage
• It does not need to maintain the set of historic
  closed patterns
• Time cost
• BackScan
• ScanSkip