Finding small patterns in permutations in linear time

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Finding small patterns in permutations in linear time

Sylvain Guillemot Dániel Marx

Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI)

Budapest, Hungary

SODA 2014 January 5, 2014

Portland, OR

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Different interpretations:

Bijective mapping σ: [n]→[n]:

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 Ordering of [n]:

3 2 7 8 4 6 1 5 n points in the plane “in general position”:

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Subpermutations

There is a natural way of defining the meaning ofσ being a subpermutation ofπ (or a “permutation pattern” in π).

Example:

σ π

1 4 2 3 3 2 7 8 4 6 1 5

is a subpermutation of

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There is a natural way of defining the meaning ofσ being a subpermutation ofπ (or a “permutation pattern” in π).

Example:

σ π

1 4 2 3 3 2 7 8 4 6 1 5

is a subpermutation of

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Excluding subpermutations

There aren!permutations of lengthn, but their number is much smaller if we exclude a fixed permutation:

Theorem[Marcus and Tardos 2004]

For every fixed permutationσ, the number of permutations of lengthn avoidingσ is at most cⁿ for some constant c depending onσ.

Example:

The number of permutations of lengthn avoiding231 is exactly the Catalan numberCn= _n+1¹ ²ⁿ_n

<4ⁿ.

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Finding patterns in permutations

Permutation Pattern

Input: Two permutationsσ andπ.

Decide: Isσ a subpattern ofπ?

NP-hard in general [Bose, Buss, and Lubiw 1998].

Can be solved in time n^`+O(1) by brute force, where`=|σ|

andn=|π|.

Can be solved in time n^0.47`+o^(`) [Ahal and Rabinovich 2008].

Theorem

Permutation Patterncan be solved in time 2^O^(`²^log^`)·n, where`=|σ|andn =|π|.

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Finding patterns in permutations

Permutation Pattern

Input: Two permutationsσ andπ.

Decide: Isσ a subpattern ofπ?

NP-hard in general [Bose, Buss, and Lubiw 1998].

Can be solved in time n^`+O(1) by brute force, where`=|σ|

andn=|π|.

Can be solved in time n^0.47`+o^(`) [Ahal and Rabinovich 2008]. Main result:

Theorem

can be solved in time 2^O^(`²^log^`)·n,

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We define the notion of d-wide decompositionand solve the problem using the following win/win strategy:

(`=|σ|,n =|π|)

1 There is an algorithm that either findsσinπor

finds a2^O(`^log^`)-wide decomposition ofπ.

2 There is an algorithm that, given σ and ad-wide

decomposition ofπ, decides ifσ is a subpattern ofπ in time (d`)^O(`)·n.

These two algorithms together give a2^O^(`²^log^`)·n time algorithm forPermutation Pattern.

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Decompositions

Starting with a set of points (degenerate rectangles), the

decomposition is a sequence of merges, where a merge consists of replacing two rectangles with their bounding box:

1 2

3 4

5 6

7 8

(3,4) (9,12)

(1,2)

→10

(13,11)

→14

(6,7)

→11

(10,14)

→15

(5,8)

→12

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Decompositions

1 2

3 4

5 6

7 8

(3,4)

→9 (9,12) (1,2)

→10

(13,11)

→14

(6,7)

→11

(10,14)

→15

(5,8)

→12

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Decompositions

1 2

5 6

7 9

8

(3,4)→9

(9,12) (1,2)

→10

(13,11)

→14

(6,7)

→11

(10,14)

→15

(5,8)

→12

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Decompositions

1 2

5 6

7 9

8

(3,4)→9

(9,12)

(1,2)

→10 (13,11)

→14

(6,7)

→11

(10,14)

→15

(5,8)

→12

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Decompositions

5 6

7 10

9

8

(3,4)→9

(9,12)

(1,2)→10

(13,11)

→14

(6,7)

→11

(10,14)

→15

(5,8)

→12

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Decompositions

5 6

7 10

9

8

(3,4)→9

(9,12)

(1,2)→10

(13,11)

→14

(6,7)

→11 (10,14)

→15

(5,8)

→12

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Decompositions

5 10

9

11 8

(3,4)→9

(9,12)

(1,2)→10

(13,11)

→14

(6,7)→11

(10,14)

→15

(5,8)

→12

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Decompositions

5 10

9

11 8

(3,4)→9

(9,12)

(1,2)→10

(13,11)

→14

(6,7)→11

(10,14)

→15

(5,8)

→12

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Decompositions

10 9

11 12

(3,4)→9

(9,12)

(1,2)→10

(13,11)

→14

(6,7)→11

(10,14)

→15

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Decompositions

10 9

11 12

(3,4)→9 (9,12)

→13

(1,2)→10

(13,11)

(6,7)→11

(10,14)

→15

(5,8)→12

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Decompositions

10

11 13

(3,4)→9 (9,12)→13 (1,2)→10

(13,11)

(6,7)→11

(10,14)

→15

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Decompositions

10

11 13

(3,4)→9 (9,12)→13 (1,2)→10 (13,11)

→14

(6,7)→11

(10,14)

(5,8)→12

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Decompositions

10

14

(3,4)→9 (9,12)→13 (1,2)→10 (13,11)→14 (6,7)→11

(10,14)

15

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Decompositions

10

14

(3,4)→9 (9,12)→13 (1,2)→10 (13,11)→14 (6,7)→11 (10,14) (5,8)→12

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Decompositions

15

(3,4)→9 (9,12)→13 (1,2)→10 (13,11)→14 (6,7)→11 (10,14)→15

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15

R₁ sees R₂ horizontally (resp., vertically) if there is a horizontal (resp., vertical) line intersecting both.

A rectangle family isd-wideif every rectangle sees less than d other rectangles horizontally and less thand other rectangles vertically.

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Grids

r×r-grid: partitioning the rows and the columns into r classes such that every cell contains a point.

Observation: If a point set has an r×r-grid, then it contains every permutation of lengthr.

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Finding decompositions

Large grids imply large width:

Fact

Ifπ contains anr×r-grid, then every decomposition of π is Ω(r)-wide.

Large width implies large grids:

Theorem

There is anO(n) time algorithm that finds either an r×r grid in π or a2^O^(r^log^r⁾-wide decomposition ofπ.

Theorem (essentially[Marcus and Tardos 2004]) IfM is a point set in [p]×[q]with |M|>r⁴ ^r_r²

(p+q), then we can find anr×r-grid in M.

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Finding decompositions

Large grids imply large width:

Fact

Ifπ contains anr×r-grid, then every decomposition of π is Ω(r)-wide.

Large width implies large grids:

Theorem

There is anO(n) time algorithm that finds either an r×r grid in π or a2^O^(r^log^r⁾-wide decomposition ofπ.

The algorithm relies on the following previous result:

Theorem (essentially[Marcus and Tardos 2004])

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We maintain a partition of rows and columns that is compatible with every current rectangle and satisfies that

every row/column contains<r⁰ rectangles and

every two adjacent rows/columns contain ≥r⁰ rectangles.

<r⁰

≥r⁰

p rows

q columns

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Finding decompositions

<r⁰

≥r⁰

p rows

q columns

Case 1: If a cell contains two rectangles, merge them. If two adja-

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<r⁰

!!! p rows

q columns

Case 1: If a cell contains two rectangles, merge them. If two adjacent rows/columns contain<r⁰ rectangles, then merge them.

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Finding decompositions

<r⁰

≥r⁰ p rows

q columns

Case 1: If a cell contains two rectangles, merge them. If two adja-

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<r⁰

≥r⁰

p rows

q columns

Case 2: Every cell contains at most one rectangle

⇒There are Ω((p+q)r⁰) nonempty cells

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Finding patterns

We would like to design a dynamic programming algorithm to solve Permutation Patternusing a given decomposition.

Problem:

The decomposition does not break the problem into independent subproblems.

R

1 2

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Finding patterns

Problem:

R1

R2

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Finding patterns

Problem:

R1

R2

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Problem:

R1

R2

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Connected components

Visibility graph: two rectangles are adjacent if they see each other horizontally or vertically.

Observation: The degree of the visibility graph is less than 2d if the rectangle family isd-wide. Therefore, there ared^O(`)·n setsK of size≤`that are connected in the visibility graph.

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Visibility graph: two rectangles are adjacent if they see each other horizontally or vertically.

Subproblems defined by

stepi of the decomposition,

a connected set K of size ≤`in the visibility graph, a subpermutationσ⁰ ifσ, and

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Connected components

How to solve a subproblem at stepi using the subproblems at step i−1?

1

R₂, then we have to distribute the points assigned toR in every possible way between R₁ andR₂.

In step i−1, set K falls apart into some number of connected sets, but the interaction between them is completely

understood.

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R

If a rectangle R of K was created at step i by mergingR₁ and R₂, then we have to distribute the points assigned toR in every possible way between R₁ andR₂.

In step i−1, set K falls apart into some number of connected sets, but the interaction between them is completely

understood.

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Connected components

R1 R2

If a rectangle R of K was created at step i by mergingR₁ and R₂, then we have to distribute the points assigned toR in every possible way between R₁ andR₂.

In step i−1, set K falls apart into some number of connected

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Theorem

Permutation Patterncan be solved in time 2^O^(`²^log^`)·n, where`=|σ|andn =|π|.

Win/win strategy:

1 There is an algorithm that either findsσinπor

finds a2^O(`^log^`)-wide decomposition ofπ.

2 There is an algorithm that, given σ and ad-wide

decomposition ofπ, decides ifσ is a subpattern ofπ in time (d`)^O(`)·n.

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Hardness

Partitioned Permutation Pattern: for every i ∈σ, a subsetS_i ⊆π is given where it can be mapped.

Theorem

Partitioned Permutation Patternis W[1]-hard parameterized by|σ|.

3-Dimensional Permutation Pattern: natural generalization to 3-dimensional points in general position.

Theorem

3-Dimensional Permutation Patternis W[1]-hard parameterized by|σ|.

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Finding patterns in permutations is fixed-parameter tractable.

Algorithm is based on a novel width measure for permutations.

Win/win situation similar to certain algorithms based on minors and bidimensionality.

But our decomposition is strictly speaking not a decomposition.