A Fast Algorithm for Permutation Pattern Matching
Based on Alternating Runs
Marie-Louise Bruner and Martin Lackner
Vienna University of Technology
July 6, 2012
SWAT 2012, Helsinki
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
1
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
X (342 is a matching)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
X (342 is a matching)
Does 53142 contain 123 as a pattern?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
X (342 is a matching)
Does 53142 contain 123 as a pattern?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
X (342 is a matching)
Does 53142 contain 123 as a pattern?
Does 53142 contain 4231 as a pattern?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Permutation patterns
A permutation T (the text) contains P as a pattern if we can find a
subsequence of T that is order-isomorphic to P.
Does 53142 contain 231 as a pattern?
X (342 is a matching)
Does 53142 contain 123 as a pattern?
Does 53142 contain 4231 as a pattern?
X (5342 is a matching)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
2
Enumerative combinatorics
Theorem
The n-permutations that do not contain the pattern 123 are counted by
the n-th Catalan number.
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
3
Enumerative combinatorics
Theorem
The n-permutations that do not contain the pattern 123 are counted by
the n-th Catalan number.
The same holds for every other pattern of length 3.
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
3
Enumerative combinatorics
Theorem
The n-permutations that do not contain the pattern 123 are counted by
the n-th Catalan number.
The same holds for every other pattern of length 3.
Stanley-Wilf conjecture, shown by Marcus and Tardos (2004)
For every permutation P there is a constant c such that the number of
n-permutations that do not contain P as a pattern is bounded by c n .
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
3
Permutation Pattern Matching
Permutation Pattern Matching (PPM)
Instance: A permutation T of length n (the
text) and a permutation P of length
k ≤ n (the pattern).
Question: Is there a matching of P into T ?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
4
Permutation Pattern Matching
Permutation Pattern Matching (PPM)
Instance: A permutation T of length n (the
text) and a permutation P of length
k ≤ n (the pattern).
Question: Is there a matching of P into T ?
1993 (Bose, Buss, Lubiw): PPM is in general NP-complete.
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
4
Tractable cases of PPM
I
Pattern avoids both 3142 and 2413
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
O(kn4 )
July 6, 2012
5
Tractable cases of PPM
I
Pattern avoids both 3142 and 2413
I
P = 12 . . . k or P = k . . . 21
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
O(kn4 )
O(n log log n)
July 6, 2012
5
Tractable cases of PPM
I
Pattern avoids both 3142 and 2413
I
P = 12 . . . k or P = k . . . 21
I
P has length at most 4
M.L. Bruner, M. Lackner
O(kn4 )
O(n log log n)
O(n log n)
A Fast Algorithm for PPM
July 6, 2012
5
Tractable cases of PPM
I
Pattern avoids both 3142 and 2413
I
P = 12 . . . k or P = k . . . 21
I
P has length at most 4
I
Pattern and Text avoid 321
M.L. Bruner, M. Lackner
O(kn4 )
O(n log log n)
O(n log n)
A Fast Algorithm for PPM
O(k 2 n6 )
July 6, 2012
5
The general case
Anything better than the
O∗ (2n )
runtime of brute-force search?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
6
Parameterized Complexity Theory
Idea: confine the combinatorial explosion to a parameter of the input
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
7
Parameterized Complexity Theory
Idea: confine the combinatorial explosion to a parameter of the input
Fixed-parameter tractability
A problem is fixed-parameter tractable with respect to a parameter k if
there is a computable function f and an integer c such that there is an
algorithm solving the problem in time O(f (k) · |I |c ).
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
7
Alternating runs
12
11
10
9
8
7
6
5
4
3
2
1
1 8 12 (up), 4 (down), 7 11 (up), 6 3 2 (down), 9 (up), 5 (down), 10 (up)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
8
Alternating runs
12
11
10
9
8
7
6
5
4
3
2
1
1 8 12 (up), 4 (down), 7 11 (up), 6 3 2 (down), 9 (up), 5 (down), 10 (up)
Notation
run(π)...the number of alternating runs in π,
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
8
The alternating run algorithm
I
Matching functions:
Reduce the search space
I
Dynamic programming algorithm:
Checks for every matching function whether there is a compatible
matching
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
9
Matching functions
Pattern P
↓ matching function ↓
Text T
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
10
Matching functions - an example
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
11
Matching functions - an example
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (2)
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
F (3)
July 6, 2012
11
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (2)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (2)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (1)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (3)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (3)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
The algorithm - finding a matching
Pattern P
Text T
12
11
10
9
4
8
7
3
6
5
2
4
3
1
2
1
F (3)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
12
Runtime
√
Matching functions:
Dynamic programming algorithm:
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
run(T )
2
O∗ (1.2611run(T ) )
July 6, 2012
13
Runtime
√
Matching functions:
run(T )
2
Dynamic programming algorithm:
O∗ (1.2611run(T ) )
In total:
O∗ (1.784run(T ) )
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
13
Runtime
√
Matching functions:
run(T )
2
Dynamic programming algorithm:
O∗ (1.2611run(T ) )
In total:
O∗ (1.784run(T ) )
→ This is a fixed-parameter tractable (FPT) algorithm,
i.e. a runtime of f (k) · nc .
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
13
Runtime
√
Matching functions:
run(T )
2
Dynamic programming algorithm:
O∗ (1.2611run(T ) )
In total:
O∗ (1.784run(T ) )
→ This is a fixed-parameter tractable (FPT) algorithm,
i.e. a runtime of f (k) · nc .
Since run(T ) ≤ n, we also obtain
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
O∗ (1.784n )
July 6, 2012
13
Alternating runs in the pattern run(P)
O∗ (1.784run(T ) )
O∗
run(P) n2
2run(P)
M.L. Bruner, M. Lackner
FPT, i.e. f (k) · nc
XP, i.e. nf (k)
A Fast Algorithm for PPM
July 6, 2012
14
Alternating runs in the pattern run(P)
O∗ (1.784run(T ) )
O∗
run(P) n2
2run(P)
FPT, i.e. f (k) · nc
XP, i.e. nf (k)
no FPT result possible (W[1]-hardness)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
14
Conclusion
Main results
I
O∗ (1.784run(T ) ) → FPT result
I
O∗ (1.784n ) → fastest general algorithm
I
W[1]-hardness for run(P)
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
15
Conclusion
Main results
I
O∗ (1.784run(T ) ) → FPT result
I
O∗ (1.784n ) → fastest general algorithm
I
W[1]-hardness for run(P)
Future work
I
PPM parameterized by some other parameter of P? By k = |P|?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
15
Conclusion
Main results
I
O∗ (1.784run(T ) ) → FPT result
I
O∗ (1.784n ) → fastest general algorithm
I
W[1]-hardness for run(P)
Future work
I
PPM parameterized by some other parameter of P? By k = |P|?
I
Kernelization results
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
15
Conclusion
Main results
I
O∗ (1.784run(T ) ) → FPT result
I
O∗ (1.784n ) → fastest general algorithm
I
W[1]-hardness for run(P)
Future work
I
PPM parameterized by some other parameter of P? By k = |P|?
I
Kernelization results
I
Other permutation statistics of the text?
M.L. Bruner, M. Lackner
A Fast Algorithm for PPM
July 6, 2012
15