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On recognizing words that are squares for the shuffle product

Laboratoire d’Informatique Gaspard-Monge Universit´ e Paris-Est Marne-la-Vall´ ee UMR CNRS 8049 Romeo Rizzi & St´ ephane Vialette

Technische Universit¨ at Berlin November 27, 2013

Shuffle

Shuffle

The shuffle u ✁ v of words u and v over A is the finite set of all words obtainable from merging the words u and v from left to right, but choosing the next symbol arbitrarily from u or v. ab ✁ cd = {abcd, acbd, acdb, cabd, cadb, cdab}

Shuffle: Concurrent Execution - Single Processor

Shuffle

A large literature is devoted to this matter:

◮ counting shuffles of two words, ◮ shuffle algebras, ◮ automata theory, ◮ . . .

Iterated shuffle

The iterated shuffle of u is the language ǫ ∪ u ∪ (u ✁ u) ∪ (u ✁ u ✁ u) ∪ . . .

Shuffle

There are basically two kinds of questions that can be addressed depending on whether or not the shuffled element is given as a part of the input:

◮ ”Given u, v ∈ A∗, is u in the iterated shuffle of v?”, and ◮ ”Given u ∈ A∗, is u in the iterated shuffle of some v ∈ A∗?”

For one application of the shuffle product, we obtain:

◮ ”Given u, v ∈ A∗, is u ∈ v ✁ v?”, and ◮ ”Given u ∈ A∗, does there exist v ∈ A∗ such that

u ∈ v ✁ v?”

Dynamic programming for “u ∈ x ✁ y”?

class String # @return [String] the last character of this string. def last self[-1,1] end # @return [String] all but the last character of this string. def init chop end # @return [Boolean] true if this string is in the shuffle # of x and y, and false otherwise. def shuffle_of?(x, y) # base cases return y == self if x.length.zero? return x == self if y.length.zero? (x.last == last and init.shuffle_of?(x.init, y)) or \ (y.last == last and init.shuffle_of?(x, y.init)) end end

Dynamic programming

$ irb 2.0.0-p247 :001 > require ’./shuffle’ => true 2.0.0-p247 :002 > "".shuffle_of?("", "") => true 2.0.0-p247 :003 > "abc".shuffle_of?("abc", "") => true 2.0.0-p247 :004 > "abc".shuffle_of?("ab", "c") => true 2.0.0-p247 :005 > "abc".shuffle_of?("a", "bc") => true 2.0.0-p247 :006 > "abc".shuffle_of?("b", "ac") => true 2.0.0-p247 :007 > "abc".shuffle_of?("", "abc") => true 2.0.0-p247 :008 > "abc".shuffle_of?("ba", "c") => false 2.0.0-p247 :009 > "abc".shuffle_of?("a", "cb") => false 2.0.0-p247 :010 > "abc".shuffle_of?("ca", "b") => false 2.0.0-p247 :011 > $

Some good news

◮ Given words u, v1 and v2, it can be tested in

O

|u|2/ log(|u|) time whether or not u ∈ v1 ✁ v2 1.

◮ The shuffle u ✁ v of words u and v can be computed in

O

• (|u| + |v|)

|u|+|v|

|u|

• time 2.

◮ Given words u1, u2, . . . , uk, the shuffle ✁k i=1ui can be

computed in O

|u1|+|u2|+...+|uk|

|u1|,|u2|,...,|uk|

• time 3.
• 1J. van Leeuwen and M. Nivat (1982). In: Information Processing

Letters 14.1

2J.-C. Spehner (1986). In: Theoretical Computer Science

• 3C. Allauzen (2000). Tech. rep. Institut Gaspard Monge, Universit´

e Marne-la-Vall´ ee

Some bad news

◮ Given words u, v1, v2, . . . , vn ∈ A∗, it is NP-complete to

decide whether or not u ∈ ✁k

i=1vi 4.

This remains true even if the alphabet has size 3.

◮ For two words u and v, it is NP-complete to decide

whether or not u is in the iterated shuffle of v 5. This remains true even if the alphabet has size 3.

• 4A. Mansfield (1983). In: Discrete Applied Mathematics 5

5M.K. Warmuth and D. Haussler (1984). In: Journal of Computer and

System Sciences 28.3

Our main result

Theorem

Given u ∈ A∗, it is NP-complete to decide whether or not u is the shuffle of some word v ∈ A∗ with itself (i.e., does there exist some v ∈ A∗ such that u ∈ v ✁ v?).

◮ This result was first claimed by K. Iwama 6but it turns out

that the proof has a serious flaw.

◮ This result was recently proved independently by Buss and

Soltys 7.

• 6K. Iwama (1983). In: Proc. 15th Annual ACM Symposium on Theory
• f Computing (STOC), Boston, Massachusetts, USA
• 7S. Buss and M. Soltys (2013). In: Journal of Computer and System

Sciences

Stack Exchange discussion board

Turning words into (linear) graphs

Let u = u1 u2 . . . un ∈ An be a word on some alphabet A.

◮ The graph associated to u, denoted V(Gu), is defined by

V(Gu) = {u1, u2, . . . , un}, and E(Gu) = {{ui, uj} : i = j ∧ ui = uj}. We write (ui, uj) for an edge of E(Gu) if it is clear from the context that i < j.

◮ The structure of this underlying graph is linear, i.e., the

set of vertices is equipped with a natural total order < defined by ui < uj if and only if i < j. In other words, the vertices of Gu correspond to the letters

• f u in the left to right order and there is an edge between

any two identical distinct letter of u.

◮ Clearly, Gu is the disjoint union of cliques, one clique for

each distinct letter of u.

Turning words into (linear) graphs

u = ababbbaa Gu a b a b b b a a

Words, linear graphs, and inclusion-free matchings

Let u = u1 u2 . . . un ∈ An be a word on some alphabet A, and let Gu be the linear graph associated to u.

◮ A matching is perfect if it covers all the vertices of the

graph.

◮ In case the set of vertices is equipped with a total order, a

matching M is said to be inclusion-free if there do not exist (independent) edges (ui, uj) and (uk, uℓ) in M such that ui < uk < uℓ < uj or uk < ui < uj < uℓ.

Lemma

Let u ∈ A∗ for some alphabet A, and Gu be the corresponding linear graph. Then, there exists v ∈ A∗ such that u ∈ v ✁ v if and only if there exists an inclusion-free perfect matching in Gu.

Words, linear graphs, and inclusion-free matchings

Let u = u1 u2 . . . un ∈ An be a word on some alphabet A, and let Gu be the linear graph associated to u.

◮ A matching is perfect if it covers all the vertices of the

graph.

◮ In case the set of vertices is equipped with a total order, a

matching M is said to be inclusion-free if there do not exist (independent) edges (ui, uj) and (uk, uℓ) in M such that ui < uk < uℓ < uj or uk < ui < uj < uℓ.

Lemma

Let u ∈ A∗ for some alphabet A, and Gu be the corresponding linear graph. Then, there exists v ∈ A∗ such that u ∈ v ✁ v if and only if there exists an inclusion-free perfect matching in Gu.

Linear graph and inclusion-free perfect matching

u = a b b a a b b a Gu a b a b b b a a M a b a b b b a a

A direct consequence

Theorem

Let u ∈ A∗ be such that |u|a ≤ 4 for every letter a ∈ A. It can be decided in polynomial-time whether or not u is the shuffle of some word v ∈ A∗ with itself.

A direct consequence

Theorem

Let u ∈ A∗ be such that |u|a ≤ 4 for every letter a ∈ A. It can be decided in polynomial-time whether or not u is the shuffle of some word v ∈ A∗ with itself.

A direct consequence

Theorem

Let u ∈ A∗ be such that |u|a ≤ 4 for every letter a ∈ A. It can be decided in polynomial-time whether or not u is the shuffle of some word v ∈ A∗ with itself.

Being a square for the shuffle product

Theorem

Given u ∈ A∗, it is NP-complete to decide whether or not u is the shuffle of some word v ∈ A∗ with itself.

◮ We propose a polynomial-time reduction from the

NP-complete Longest Common Subsequence for binary words8: ”Given a collection of words U = {u1, u2, . . . , um} such that ui ∈ {0, 1}∗ for 1 ≤ i ≤ m, and a positive integer k, decide whether there exists a subsequence of size k common to all sequences of U? ”

◮ Without loss of generality, we may assume that |ui| = |uj|

for 1 ≤ i < j ≤ m, and that that we are looking for a common subsequence with p letters 0 and q letters 1, k = p + q.

• 8D. Maier (1978). In: Journal of the ACM 25.2

2-interval pattern matching 9

|

SIGN IN SIGN UP

On two open problems of 2-interval patterns

Authors: Shuai Cheng Li David R. Cheriton School of Computer Science, University of

Waterloo, Waterloo ON N2L 3G1, Canada

Ming Li

David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON N2L 3G1, Canada

Published in: · Journal Theoretical Computer Science archive Volume 410 Issue 24-25, May, 2009 Pages 2410-2423

Elsevier Science Publishers Ltd. Essex, UK table of contents doi>10.1016/j.tcs.2009.02.033

2009 Article Bibliometrics

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2-interval pattern bioinformatics contact map np-hard

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Abstract Authors References Cited By Index Terms Publication Reviews Comments Table of Contents

The 2-interval pattern problem, introduced in [Stephane Vialette, On the computational complexity of 2-interval pattern matching problems Theoret. Comput. Sci. 312 (2-3) (2004) 223-249], models general problems with biological structures such as protein contact maps and macroscopic describers of secondary structures of ribonucleic acids. Given a set of 2-intervals D and a model R, the problem is to find a maximum cardinality subset D^' of D such that any two 2-intervals in D^' satisfy R, where R is a subset of relations on disjoint 2-intervals: precedence (<), nest (@?), and cross (@?). The problem left unanswered at present is that of whether there is a polynomial time solution for the 2-interval pattern problem, when R={<,@?} and all the support intervals of D are disjoint. In this paper, we present a reduction from the clique problem to show that, in this case, the problem is NP-hard. The disjoint 2-interval pattern matching problem is to decide whether a disjoint 2-interval pattern (called the pattern) is a substructure of another disjoint 2-interval pattern (called the target). In general, the problem is NP-hard, but when there are restrictions on the form of the pattern, the problem can, in some cases, be solved in polynomial time. In particular, a polynomial time algorithm has been proposed (Gramm, WABI 2004 and IEEE/ACM TCBB 2004) for the case where the patterns are so-called crossing contact maps. In this paper we show that the problem is actually NP-hard and point out an error in the analysis of the above algorithm.

9S.C. Li and M. Li (2009). In: Theoretical Computer Science 410.24-25

Lossless transmission

It will convenient to see the reduction as a flow-like procedure, where some piece of information (the common subsequence) emitted from gadget Ws (the source) propagates lossless to gadget Wt (the sink) going through all gadgets Wi, 1 ≤ i ≤ m (every such gadget being associated to an input word

• f our input instance of 01-LCS).

Being a square for the shuffle product - the big picture

SOURCE TARGET LOSSLESS TRANSMISSION

Being a square for the shuffle product - the big picture

TARGET Second word Last word First word SOURCE Emit word Transmit word Transmit word Transmit word Transmit word

LOSSLESS TRANSMISSION LINE

Construction

The string of interest is defined by: w = Ws W1 W2 . . . Wm Wt, where Ws, W1, W2, . . ., Wm and Wt are words in A∗.

◮ Ws is the source. ◮ Each Wi is associated to an input binary string. ◮ Wt is the target.

Designing the source and the target

◮ Source

Ws = s′ 0pq s′ s (0p 1)q 0p s

◮ Target

Wt = t (0p 1)q 0q t t′ 0pq t′

◮ Recall that we are looking for a common subsequence of

lenth k with p letters 0 and q letters 1, k = p + q.

◮ Letters s, s′, t and t′ do not occur in any other gadget

word.

Designing the source : the big picture

Designing a string gadget

For any input binary string ui = ui,1ui,2 . . . ui,n, define Wi = xi W ′

i xi

yi W ′

i yi,

where W ′

i = ui,1 zi ui,2 zi . . . ui,n−1 zi ui,n. ◮ Letters xi, yi and zi only occur in the gadget word Wi.

Designing a string gadget: the big picture

Forward direction

◮ Suppose that there exists a common subsequence v of the

words u1, u2, . . . , um with p occurrences of the letter 0 and q occurrence of the letter 1.

◮ The solution v propagates lossless from source gadget Ws

to target gadget Wt going through all string gadgets Wi, 1 ≤ i ≤ m (every such gadget being associated to an input word of our input instance of 01-LCS).

Reverse direction

◮ Suppose that w is a square for the shuffle product. ◮ According to our previous lemma, this amount to saying

that Gw has an inclusion-free perfect matching M.

Reverse direction: Easy observations for the source

Reverse direction: Transmitting from the source

Reverse direction: Internal transmission

Reverse direction: Internal transmission

Iterated shuffle

Theorem

It is NP-complete to decide whether or not a word u ∈ A∗ is in the iterated shuffle of some word v ∈ A∗ with u = v.

Proof.

◮ In the previous reduction, some letters occur exactly two

times in the constructed word w.

◮ In this case, w cannot be the shuffle of k ≥ 3 identical

copies of some word v ∈ A∗.

◮ In other words, if w is in the iterated shuffle of some

distinct word v ∈ A∗, then w is a square for the shuffle product.

Turning the problem into an optimisation task

Let u ∈ A∗. Find the largest v u that is a square for the shuffle product. Define f (u) := max{|v| : v u and ∃w v such that v ∈ w ✁ w} g(n, k) := min{f (u) : u ∈ An and |A| = k}

Theorem (10)

g(n, 2) = n − o(n).

◮ Any binary word of length n can be split into two

identical subwords and, perhaps, a remaining subword of length o(n).

• 10M. Axenovich, Y. Person, and S. Puzynina (2013). In: J. Comb.

Theory, Ser. A 120.4

Turning the problem into an optimisation task

Let u ∈ A∗. Find the largest v u that is a square for the shuffle product. Define f (u) := max{|v| : v u and ∃w v such that v ∈ w ✁ w} g(n, k) := min{f (u) : u ∈ An and |A| = k}

Theorem (11)

There is polynomial-time approximation scheme (PTAS) for computing f (u).

• 11M. Jiang (2007). In: 1st Annual International Conference on

Combinatorial Optimization and Applications (COCOA’07), Xi’an, Shaanxi, China. Vol. 4616. Lecture Notes in Computer Science

Being the shuffle of a word with its reverse

Given u ∈ A∗, does there exist v ∈ A∗ such that u ∈ v ✁ v?

Theorem (12)

Let u be some word over some binary alphabet A. It is polynomial-time solvable to determine whether or not there exists v ∈ A∗ such that u ∈ v ✁ v.

• 12D. Henshall N. Rampersad and J. Shallit (2011).

Being the shuffle of a word with its reverse

Given u ∈ A∗, does there exist v ∈ A∗ such that u ∈ v ✁ v?

Key facts

◮ If there exists v ∈ A∗ such that u ∈ v ✁

v, then u is an Abelian square (i.e., u = v v′, where v′ is a permutation

• f v).

◮ If u is a binary abelian square, then there exists v ∈ A∗

such that u ∈ v ✁ v.

◮ Notice that the equivalence is no longer true for larger

alphabets. The word 012012 is an example of a ternary Abelian square that cannot be written as an element of v ✁ v for any word v.

Words and linear graphs

Let u = u1 u2 . . . un ∈ An be a word on some alphabet A, and let Gu be the linear graph associated to u.

◮ A matching is perfect if it covers all the vertices of the

graph.

◮ In case the set of vertices is equipped with a total order, a

matching M is said to be 2-nested if the following two conditions hold:

◮ For every edge (u, v) in M, one endpoint is in the first half

• f the nodes of Gu and the other is in the second half;

◮ M can be partitioned into two disjoint matchings M1 and

M2 such that both M1 and M2 are nested matchings

Lemma

Let u ∈ A∗ for some alphabet A, and Gu be the corresponding linear graph. Then, there exists v ∈ A∗ such that u ∈ v ✁ v if and only if there exists a 2-nested perfect matching in Gu.

Words and linear graphs

Let u = u1 u2 . . . un ∈ An be a word on some alphabet A, and let Gu be the linear graph associated to u.

◮ A matching is perfect if it covers all the vertices of the

graph.

◮ In case the set of vertices is equipped with a total order, a

matching M is said to be 2-nested if the following two conditions hold:

◮ For every edge (u, v) in M, one endpoint is in the first half

• f the nodes of Gu and the other is in the second half;

◮ M can be partitioned into two disjoint matchings M1 and

M2 such that both M1 and M2 are nested matchings

Lemma

Let u ∈ A∗ for some alphabet A, and Gu be the corresponding linear graph. Then, there exists v ∈ A∗ such that u ∈ v ✁ v if and only if there exists a 2-nested perfect matching in Gu.

Being the shuffle of a word with its reverse

ABACBACA ∈ ABCA ✁ ABCD

Hardness

Theorem

Given u ∈ A∗, it is NP-complete to decide whether or not u is the shuffle of some word v ∈ A∗ with its reverse. (i.e., does there exist some v ∈ A∗ such that u ∈ v ✁ v?).

◮ Reduction from our 01-LCS problem. ◮ The snail reduction is, however, much more complicated.

A pattern avoidance point of view

Pattern containment / involvement / avoidance:

◮ A permutation π is said to contain another permutation

σ, in symbols σ π, if there exists a subsequence of entries

• f π that has the same relative order as σ, and in this case

σ is said to be a pattern of π.

◮ Otherwise, π is said to avoid the permutation σ. ◮ The subsequence of π need not consist of consecutive

entries. For example, permutation π = 391867452 (written in one-line notation) contains the pattern σ = 51342, as can be seen by considering the subsequence 91672.

A pattern avoidance point of view

Theorem

Let u ∈ A2n. The two following statements are equivalent.

• 1. There exists v ∈ An such that u ∈ v ✁

v.

• 2. There exist v, w ∈ A∗ such that u ∈ (v ✁ w) (

v ✁ w).

• 3. There exist v ∈ An and 123-avoiding permutation π ∈ Sn

such that u = v π(v).

A pattern avoidance point of view

Theorem

Let u ∈ A2n be an Abelian square. The following statements are equivalent.

• 1. There exist v, w ∈ A∗ such that u ∈ (v ✁ w) (v ✁ w).
• 2. There exist v ∈ An and a 321-avoiding permutation π ∈ Sn

such that u = v π(v).

A pattern avoidance point of view

Theorem

Let u ∈ A2n. The two following statements are equivalent.

• 1. There exists v ∈ An such that u ∈ v ✁ v.
• 2. There exist v, w ∈ An, |v| + |w| = k, and π ∈ Sk such that

u ∈ (vw ✁ v) (π(v) ✁ w π(v)).

(So many) Problems left open

How hard is the problem of detecting squares for the shuffle product for bounded alphabet words?

◮ It is proved in 13 that the problem is NP-complete for an

alphabet with 9 letters (it is claimed that this can be improved to 7 letters).

◮ It is claimed without proof in 14 (Fact 2 Subsection 2.2)

that detecting squares for the shuffle product is NP-complete for binary words! This result – that would be an important improvement

• ver Buss and Soltys’ proof – is yet to be confirmed.
• 13S. Buss and M. Soltys (2013). In: Journal of Computer and System

Sciences

• 14H. Aoki, R. Uehara, and K. Yamazaki (2001). Tech. rep. 1185. RIMS

Kokyuroku

(So many) Problems left open

Determine a simple closed form for ak(n) := #

  

• u∈{0,1...,k−1}n

u ✁ u

   .

From Henshall, Rampersad and Shallit:

n 1 2 3 4 5 6 7 8 9 a2(n) 1 2 6 22 82 320 1268 5102 20632 83972 a3(n) 1 3 15 93 621 4425 32703 258901 a4(n) 1 4 28 244 2332 23848 254416 a5(n) 1 5 45 505 6265 83225 a6(n) 1 6 66 906 13806 225336

• 15D. Henshall N. Rampersad and J. Shallit (2011).

(So many) Problems left open

Determine a simple closed form for ak(n) := #

  

• u∈{0,1...,k−1}n

u ✁ u

   .

Conjecture:15 ai(n) =

2n

n

• n + 1in −
• 2n − 1

n + 1

• in−1 + O(in−2).

(Fun) Fact: #(ιn ✁ ιn) is the n-th Catalan number.

• 15D. Henshall N. Rampersad and J. Shallit (2011).

(So many) Problems left open

Determine a simple closed form for ak(n) := #

  

• u∈{0,1...,k−1}n

u ✁ u

   .

Conjecture: Let u ∈ {0, 1}2n, n ≥ 4 with |u|0 = 2k, k ∈ N. If k < n − k and |u|0110 + |u|1001 < (2n − 2k − 1)

k+1

2

, the u is a

square for the shuffle product. Fact: 0k−110k+112n−2k−1 is a not a square for the shuffle product.

• 15D. Henshall N. Rampersad and J. Shallit (2011).

(So many) Problems left open

Let u, v1, v2, . . . , vk ∈ A∗. Decide whether or not u ∈ ✁k

i=1vi. Where does fall the problem of deciding

whether or not u ∈ ✁k

i=1vi in the W-hierarchy for

parameter k?

◮ Parameterized complexity.

In particular, what about the (parameterized) complexity

• f deciding whether or not u ∈ ✁k

i=1vi for parameter k? ◮ Given words u, v1, v2, . . . , vk ∈ A∗, it is NP-complete to

decide whether or not u ∈ ✁k

i=1vi 16 and 17. ◮ The question which lies at the heart of this is: How hard is

this problem compared to LCS?

• 16A. Mansfield (1983). In: Discrete Applied Mathematics 5

17M.K. Warmuth and D. Haussler (1984). In: Journal of Computer and

System Sciences 28.3

(So many) Problems left open

How to extend recognition of squares for the shuffle product to larger permutation classes?

(So many) Problems left open

How to extend recognition of squares for the shuffle product to larger permutation classes? Gu a b a b b b a a M a b a b b b a a u ∈ abba ✁ abba

(So many) Problems left open

How to extend recognition of squares for the shuffle product to larger permutation classes? Gu a b a b b b a a M a b a b b b a a u ∈ baba ✁ abba

(So many) Problems left open

How to extend recognition of squares for the shuffle product to larger permutation classes? Let u ∈ A2n. Does there exist a characterization of some X ⊂ Sn such that the two following statements are equivalent:

• 1. There exists v ∈ An such that u ∈ v ✁ v.
• 2. There exists v ∈ An and π ∈ X such that u ∈ v ✁ π(v)

Remarks

◮ Statement 1. may be rephrased as follows:

There exists v ∈ An such that u ∈ v ✁ ι(v).

◮ Let u ∈ A2n for some alphabet Σ and π ∈ Sn There exists

v ∈ An such that u ∈ v ✁ π(v) if and only if there exists w ∈ An such that u ∈ w ✁ π−1(w).

(So many) Problems left open

Let u ∈ A2n and M a perfect crossing-free matching in

• Gu. What can be said about u?

◮ If there exists a perfect crossing-free matching in Gu, then

there exist v ∈ An and a (2413, 3142, 3412)-avoiding permutation π ∈ Sn such that u ∈ v ✁ π(v).

◮ (2413, 3142)-avoiding permutations form the class of

separable permutations.

Allauzen, C. (2000). Calcul efficace du shuffle de k mots.

• Tech. rep. Institut Gaspard Monge,

Universit´ e Marne-la-Vall´ ee. Aoki, H., R. Uehara, and K. Yamazaki (2001). Expected Length of Longest Common Subsequences of Two Biased Random Strings and Its Application.

• Tech. rep. 1185. RIMS

Kokyuroku, pp. 1–10. Axenovich, M., Y. Person, and S. Puzynina (2013). “A regularity lemma and twins in words”. In: J. Comb. Theory, Ser. A 120.4, pp. 733–743. Buss, S. and M. Soltys (2013). “Unshuffling a Square is NP-Hard”. In: Journal of Computer and System Sciences. To appear. Iwama, K. (1983). “Unique Decomposability of Shuffled Strings: A Formal Treatment of Asynchronous Time-Multiplexed Communication”. In: Proc. 15th Annual ACM Symposium

• n Theory of Computing (STOC), Boston, Massachusetts, USA. ACM, pp. 374–381.

Jiang, M. (2007). “A PTAS for the weighted 2-interval pattern problem over the preceding-and-crossing model”. In: 1st Annual International Conference on Combinatorial Optimization and Applications (COCOA’07), Xi’an, Shaanxi, China. Ed. by A. Dress, Y. Xu, and B. Zhu. Vol. 4616. Lecture Notes in Computer Science, pp. 378–387. Leeuwen, J. van and M. Nivat (1982). “Efficient Recognition of Rational Relations”. In: Information Processing Letters 14.1, pp. 34–38. Li, S.C. and M. Li (2009). “On two open problems of 2-interval patterns”. In: Theoretical Computer Science 410.24-25, pp. 2410–2423. Maier, D. (1978). “The Complexity of Some Problems on Subsequences and Supersequences”. In: Journal of the ACM 25.2, pp. 322–336. Mansfield, A. (1983). “On the computational complexity of a merge recognition problem”. In: Discrete Applied Mathematics 5, pp. 119–122. Rampersad, D. Henshall N. and J. Shallit (2011). “Shuffling and Unshuffling”. http://arxiv.org/abs/1106.5767.

Spehner, J.-C. (1986). “Le Calcul Rapide des Melanges de Deux Mots”. In: Theoretical Computer Science, pp. 171–203. Warmuth, M.K. and D. Haussler (1984). “On the complexity of iterated shuffle”. In: Journal of Computer and System Sciences 28.3, pp. 345–358.

Lemma

Let u ∈ {0, 1}2n be a binary string of length 2n with 2k

• ccurrences of 0 for some 0 ≤ k ≤ n. If

|u|0110 + |u|1001 < k(n − k) (k + 1)(n − k + 1)

2k

k

• 2k−2

k−1

• 2n−2k

n−k

• 2n−2k−2

n−k−1

• the u is a square for the shuffle product.