Wavelet Analysis on Symbolic Sequences and Two-Fold de Bruijn Sequences

Abstract

The concept of symbolic sequences play important role in study of complex systems. In the work we are interested in ultrametric structure of the set of cyclic sequences naturally arising in theory of dynamical systems. Aimed at construction of analytic and numerical methods for investigation of clusters we introduce operator language on the space of symbolic sequences and propose an approach based on wavelet analysis for study of the cluster hierarchy. The analytic power of the approach is demonstrated by derivation of a formula for counting of two-fold de Bruijn sequences, the extension of the notion of de Bruijn sequences. Possible advantages of the developed description is also discussed in context of applied problem of construction of efficient DNA sequence assembly algorithms.

Appendix 1: Representation of the Set \(\mathcal {X}_n^{ \mathcal {A}_2}\) on a Regularly Branching Tree

Fig. 5

Graphical representation of the set $X_7^{ \mathcal {A}_2}$ by a binary tree. Only half of the tree (the sequences starting from 1) is plotted. The representatives of the factor sets $\mathcal {X}_n^{ \mathcal {A}_2}$ ($n=7,6,5,4,3,2,1$) are marked by the circles on corresponding level of the tree. The nubers inside denote period of the corresponding sequence.

On the Fig. 5 we graphically represent structure of the factor set $\mathcal {X}_7^{ \mathcal {A}_2}$ by a special choice of its representatives from the set $X_7^{ \mathcal {A}_2}$. One can compare it with the structure of the same set (see Fig. 2) revealed by the ultrametric distance (8).

The representatives of $\mathcal {X}_7^{ \mathcal {A}_2}$ are chosen to maximize the number $1+\sum _{k=1}^n a_k 2^{n-k}$, i.e. the index of the basis vector (1). For the case of prime n one can offer a synthetic algorithm based on the following principles:

• 1. Sequences are classified by $r=0,\dots ,n$, the lengths of the longest sub-string, $L=[1,\dots ,1]$, and by the frequency f of appearance L in the sequence. The integer partition of n, then has the form

  $$\begin{aligned} \sum _{j=1}^{s}\nu _j (r+m_j)=n,\qquad \sum _{j=1}^{s}\nu _j=f. \end{aligned}$$

  Let $L_j$ be the sequence L followed by arbitrary string of the length $m_j$, as on the scheme

  $$\begin{aligned} \underbrace{ \underbrace{1\dots 1}_{r} \underbrace{0\dots 0}_{m_1}}_{L_1} \underbrace{\underbrace{1\dots 1}_{r} \underbrace{0\dots 0}_{m_2}}_{L_2}\dots \end{aligned}$$

• 2. Since n is prime, there are at least two non-equal $\nu _j$ entering the integer partition of n. Let $\nu _{j_0}=\min _j\left\{ \nu _j\right\}$, then there is at least one $\nu _{j_1}>\nu _{j_0}$, such that $\nu _{j_1}$ is not divisible by $\nu _{j_0}$. Therefore one can organize a set of cyclic sequences constructed from two letters $L_{j_0}$ and $L_{j_1}$, such that they have no other periods except $\nu _{j_0}+\nu _{j_1}$, for instance

  $$\begin{aligned} \underbrace{L_{j_0}\dots L_{j_0}}_{\nu _{j_0}}\underbrace{L_{j_1}\dots L_{j_1}}_{\nu _{j_1}}. \end{aligned}$$

• 3. To obtain the representatives one should choose any suitable strategy of ordering of $\nu _{j_1}$ pieces between $\nu _{j_0}$ places, such that the resulting sequences of $L_{j_0}$’s and $L_{j_1}$’s cannot be obtained one from the other by any rotation and, finally, one has to and fill up the yet empty places within the strings $L_j$ by all possible ways.