Anti-Powers in Infinite Words

In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. From these results, we derive that at every position of an aperiodic uniformly recurrent word start anti-powers of any order. We further show that any infinite word avoiding anti-powers of order 3 is ultimately periodic, and that there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6, and leave open the question whether there exist aperiodic recurrent words avoiding anti-powers of order k for k=4,5

Finding the Anticover of a String

A k-anticover of a string x is a set of pairwise distinct factors of x of equal length k, such that every symbol of x is contained into an occurrence of at least one of those factors. The existence of a k-anticover can be seen as a notion of non-redundancy, which has application in computational biology, where they are associated with various non-regulatory mechanisms. In this paper we address the complexity of the problem of finding a k-anticover of a string x if it exists, showing that the decision problem is NP-complete on general strings for k ? 3. We also show that the problem admits a polynomial-time solution for k=2. For unbounded k, we provide an exact exponential algorithm to find a k-anticover of a string of length n (or determine that none exists), which runs in O*(min {3^{(n-k)/3)}, ((k(k+1))/2)^{n/(k+1)) time using polynomial space

Finding the Anticover of a String

A k-anticover of a string x is a set of pairwise distinct factors of x of equal length k, such that every symbol of x is contained into an occurrence of at least one of those factors. The existence of a k-anticover can be seen as a notion of non-redundancy, which has application in computational biology, where they are associated with various non-regulatory mechanisms. In this paper we address the complexity of the problem of finding a k-anticover of a string x if it exists, showing that the decision problem is NP-complete on general strings for k-3. We also show that the problem admits a polynomial-time solution for k = 2. For unbounded k, we provide an exact exponential algorithm to find a k-anticover of a string of length n (or determine that none exists), which runs in O(min{3 n-k 3 , ( k(k+1) 2 ) n k+1 }) time using polynomial space. 2012 ACM Subject Classification Mathematics of computing ! Combinatorics on words

String Factorizations Under Various Collision Constraints

In the NP-hard Equality-Free String Factorization problem, we are given a string S and ask whether S can be partitioned into k factors that are pairwise distinct. We describe a randomized algorithm for Equality-Free String Factorization with running time 2^k? k^{?(1)}+?(n) improving over previous algorithms with running time k^{?(k)}+?(n) [Schmid, TCS 2016; Mincu and Popa, Proc. SOFSEM 2020]. Our algorithm works for the generalization of Equality-Free String Factorization where equality can be replaced by an arbitrary polynomial-time computable equivalence relation on strings. We also consider two factorization problems to which this algorithm does not apply, namely Prefix-Free String Factorization where we ask for a factorization of size k such that no factor is a prefix of another factor and Substring-Free String Factorization where we ask for a factorization of size k such that no factor is a substring of another factor. We show that these two problems are NP-hard as well. Then, we show that Prefix-Free String Factorization with the prefix-free relation is fixed-parameter tractable with respect to k by providing a polynomial problem kernel. Finally, we show a generic ILP formulation for R-Free String Factorization where R is an arbitrary relation on strings. This formulation improves over a previous one for Equality-Free String Factorization in terms of the number of variables

Pattern Matching with Variables: Fast Algorithms and New Hardness Results

A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient algorithmsfootnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases

Pattern matching with variables: Efficient algorithms and complexity results

A pattern α (i. e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of α by terminal words. The respective matching problem, i. e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i. e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted clases of patterns