Super-d-complexity of finite words

In this paper we introduce and study a new complexity measure for finite words. For positive integer d special scattered subwords, called super-d-subwords, in which the gaps are of length at least (d−1), are defined. We give methods to compute super-d-complexity (the total number of different super-d-subwords) in the case of rainbow words (with pairwise different letters) by recursive algorithms, by mahematical formulas and by graph algorithms. In the case of general words, with letters from a given alphabet without any restriction, the problem of the maximum value of the super-d-complexity of all words of length n is presented. Subject Classifications: MSC2010: 68R15 CCS1998: G.2.1, F.2.2 1 A new complexity measure: the super-d-complexity Sequences of characters called words or strings are widely studied in combinatorics, and used in various fields of sciences (e.g. chemistry, physics, social sciences, biology [3, 4, 5, 9] etc.). The elements of a word are called letters. A contiguous part of a word (obtained by erasing a prefix or/and a suffix) is a subword or factor. If we erase arbitrary letters from a word, what is obtained is a scattered subword. Special scattered subwords, in which the consecutive letters are at distance at most d (d ≥ 1) in the original word, are called dsubwords [6, 7]. In this paper we define another kind of scattered subwords, in which the original distance between two letters which are consecutive in the subword, is at least d (d ≥ 1), these will be called super-d-subwords. One can easily observe that in any given word, the 1-subwords are exactly the (ordinary) subwords, and the super-1-subwords are exactly the scattered subwords. The complexity of a word is defined as the total number of its different subwords. The definitions of d-complexity and super-d-complexity are similar. For a (finite) alphabet Σ, as usual, Σ and Σ∗ are the sets of all words of length n, and of all finite words, respectively, over Σ. In order to formalize the above, we introduce the following two definitions. 252 8th Joint Conf. Math. & Comp. Sci. July 14–17, 2010, Komárno, Slovakia Definition 1 Let n, d and s be positive integers, and u = x1x2 . . . xn ∈ Σ . A super-d-subword of length s of u is defined as v = xi1xi2 . . . xis where i1 ≥ 1, d ≤ ij+1 − ij < n for j = 1, 2, . . . , s − 1, is ≤ n. Definition 2 The super-d-complexity of a word is the total number of its different super-d-subwords. The super-2-subwords of the word abcdef are the following: a, ac, ad, ae, af, ace, acf, adf, b, bd, be, bf, bdf, c, ce, cf, d, df, e, f, therefore the super-2complexity of this word is 20. 2 Computing the super-d-complexity of rainbow words Words with pairwise different letters are called rainbow words. The super-dcomplexity of a rainbow word of length n does not depends on what letters it contains, and is denoted by S(n, d). Let us denote by bn,d(i) the number of super-d-subwords which begin at the i-th position in a rainbow word of length n. Using our previous example (abcdef ), we can see that b6,2(1) = 8, b6,2(2) = 5, b6,2(3) = 3, b6,2(4) = 2, b6,2(5) = 1, and b6,2(6) = 1. We immediately get the following formula: bn,d(i) = 1 + bn,d(i+d) + bn,d(i+d+1) +· · ·+ bn,d(n), (1) for n > d, 1 ≤ i ≤ n− d, and bn,d(1) = 1 for n ≤ d. The super-d-complexity of rainbow words can be computed by the following formula: S(n, d) = n ∑ i=1 bn,d(i). (2) This can be expressed also as S(n, d) = n ∑ k=1 bk,d(1), (3) because of the formula S(n+ 1, d) = S(n, d) + bn+1,d(1). 8th Joint Conf. Math. & Comp. Sci. July 14–17, 2010, Komárno, Slovakia 253 ❍ ❍ ❍ n d 1 2 3 4 5 6 7 8 9 10 11 1 1 1 1 1 1 1 1 1 1 1 1 2 3 2 2 2 2 2 2 2 2 2 2 3 7 4 3 3 3 3 3 3 3 3 3 4 15 7 5 4 4 4 4 4 4 4 4 5 31 12 8 6 5 5 5 5 5 5 5 6 63 20 12 9 7 6 6 6 6 6 6 7 127 33 18 13 10 8 7 7 7 7 7 8 255 54 27 18 14 11 9 8 8 8 8 9 511 88 40 25 19 15 12 10 9 9 9 1