The research of combinatorics on phrases is a comparatively new study sector within the fields of discrete and algorithmic arithmetic. that includes an easy, available kind, Algorithmic Combinatorics on Partial phrases provides combinatorial and algorithmic ideas within the rising box of phrases and partial phrases. This e-book features a wealth of routines and difficulties that assists with quite a few set of rules tracing, set of rules layout, mathematical proofs, and software implementation. it is also a variety of labored instance and diagrams, making this a priceless textual content for college kids, researchers, and practitioners looking to comprehend this advanced topic the place many difficulties stay unexplored.

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Extra resources for Algorithmic Combinatorics on Partial Words

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4 If x and y are nonempty conjugate partial words, then there exists a partial word z satisfying the conjugacy equation xz ↑ zy. Moreover, in this case there exist partial words u, v such that x ⊂ uv, y ⊂ vu, and z ⊂ (uv)n u for some integer n ≥ 0. True or false? 5 If k = 4 and l = 10, then determine whether the following partial S words are (4, 10)-special or not? 6 Find k, l such that z = acbca cbc cac is (k, l)-special. 7 Let x = ab a a b and y = a babba a b. Show that xy ↑ yx and that xy is not (|x|, |y|)-special.

3. 3 Let u and v be partial words. Prove that if v is primitive and v ⊂ u, then u is primitive as well. 4 S Let u be a partial word of length p, where p is a prime number. Prove that u is not primitive if and only if α(u) ≤ 1. 5 Construct a partial word with one hole of length 12 over the alphabet {a, b} that is weakly 5-periodic, weakly 8-periodic but not 1-periodic. 6 Let u be a word over an alphabet A, and let v = ua for any letter a in A. Prove that p(u) ≤ p(v). 7 For partial words u and v, does u ↑ v imply u ⊂ v.

So although the partial words a b and a b are not equal by our definition, they may very well be equal, if we only had more information. To sharpen our understanding of this possibility, we introduce and discuss two alternative methods of relating partial words: containment and compatibility. 9 If u and v are two partial words of equal length, then u is said to be contained in v, denoted by u ⊂ v, if all elements in D(u) are in D(v) and u(i) = v(i) for all i ∈ D(u). We sometimes write u ❁ v if u ⊂ v but u = v.