The case

In this case instead of we will use . For a rainbow word, we will denote the number of -subwords which finish at the position . For

For simplicity, let us denote by . The -complexity of a rainbow word can be obtained by the formula

Because of (26.1) we can write in the case of

Denoting

we get

and the sequence is one of Fibonacci-type. For any we have and from this results.

Therefore the numbers are defined by the following recurrence equations:

for , for .

These numbers can be generated by the following generating function:

The -complexity can be expressed with these numbers by the following formula:

and

Complexity of Words

or

If then

where is the generating function of the Fibonacci numbers (with , ). Then, from this formula we have

and

Figure 26.13 contains the values of for and .

Figure 26.13. The -complexity of words of length .

The following theorem gives the value of in the case : Theorem 26.23 For we have

The main step in the proof is based on the formula

The value of can be also obtained by computing the number of sequences of length of 's and 's, with no more than adjacent zeros. In such a sequence one 1 represents the presence, one 0 does the absence of a letter of the word in a given -subword. Let denote the number of -length sequences of zeros and ones, in which the first and last position is 1, and the number of adjacent zeros is at most . Then it can be proved easily that

, for ,

,

, for all ,

because any such sequence of length ( ) can be continued in order to obtain a similar sequence of length in only one way (by adding a sequence of the form on the right). For the following formula also can be derived:

If we add one 1 or 0 at an internal position (e.g at the position) of each sequences, then we obtain sequences of length , but from these, sequences will have adjacent zeros.

The generating function corresponding to is

By adding zeros on the left and/or on the right to these sequences, we can obtain the number , as the number of all these sequences. Thus

( zeros can be added in ways to these sequences: on the left and on the right, on the left and on the right, and so on).

From the above formula, the generating function corresponding to the complexities can be obtained as a

product of the two generating functions and , thus:

2.1.2. The case

In the sequel instead of we will use . In this case the distance between two letters picked up to be neighbours in a subword is at least .

Let us denote by the number of -subwords which begin at the position in a rainbow word of length . Using our previous example (abcdef), we can see that , , ,

, , and .

The following formula immediately results:

for , and ,

For simplicity, will be denoted by .

The -complexity of rainbow words can be computed by the formula:

This can be expressed also as

Complexity of Words

because of the formula

Figure 26.14. Values of .

In the case the complexity can be computed easily: .

From (26.2) we get the following algorithm for the computation of . The numbers

for a given and are obtained in the array . Initially all these elements are equal to . The call for the given and and the desired is:

Input

FOR TO DO B Array is a global one.

Output

The recursive algorithm is the following:

B( )

1 2 FOR TO 3 DO IF 4 THEN B 5 6 7 RETURN

This algorithm is a linear one.

If the call is , the elements will be obtained in the following order: , , , ,

, , and .

Lemma 26.24 , where is the th Fibonacci number.

Proof. Let us consider a rainbow word and let us count all its -subwords which begin with . If we change for in each -subword which begin with , we obtain -subwords too. If we add in front of each -subword which begin with , we obtain -subwords too. Thus

So is a Fibonacci number, and because , we obtain that .

Theorem 26.25 , where is the th Fibonacci number.

Proof. From equation (26.4) and Lemma 26.24 [103]:

If we use the notation , because of the formula

a generalized middle sequence will be obtained:

Let us call this sequence -middle sequence. Because of the equality , the -middle sequence can be considered as a generalization of the Fibonacci sequence.

Then next linear algorithm computes , by using an array to store the necessary previous elements:

Middle( )

1 2 FOR TO 3 DO 4 FOR TO 5 DO 6 print 7 RETURN

Using the generating function , the following closed formula can be obtained:

This can be used to compute the sum , which is the coefficient of in the expansion of the function

So . Therefore

Theorem 26.26 , where and is the th elements of -middle sequence.

Proof. The proof is similar to that in Theorem 26.25 [103] taking into account the equation (26.7).

Theorem 26.27 , for , .

Proof. Let us consider the generating function . Then, taking into account the

equation (26.6) we obtain . The

general term in this expansion is equal to

Complexity of Words

and the coefficient of is equal to

The coefficient of is

By Theorem 26.26 [104] , and an easy computation yields

2.2. 26.2.2 General words

The algorithm Warshall-Latin can be used for nonrainbow words too, with the remark that repeating

subwords must be eliminated. For the word and , the result is:

, and with and we have .

In document Table of Contents (Pldal 112-117)