Manal Mohamed

We consider the problem of preprocessing a text $T$ of length $n$ and a dictionary $\mathcal{D}$ in order to be able to efficiently answer queries $CountDistinct(i,j)$, that is, given $i$ and $j$ return the number of patterns from $\mathcal{D}$ that occur in the fragment $T[i \mathinner{.\,.} j]$. The dictionary is internal in the sense that each pattern in $\mathcal{D}$ is given as a fragment of $T$. This way, the dictionary takes space proportional to the number of patterns $d=|\mathcal{D}|$ rather than their total length, which could be $\Theta(n\cdot d)$. An $\tilde{\mathcal{O}}(n+d)$-size data structure that answers $CountDistinct(i,j)$ queries $\mathcal{O}(\log n)$-approximately in $\tilde{\mathcal{O}}(1)$ time was recently proposed in a work that introduced internal dictionary matching [ISAAC 2019]. Here we present an $\tilde{\mathcal{O}}(n+d)$-size data structure that answers $CountDistinct(i,j)$ queries $2$-approximately in $\tilde{\mathcal{O}}(1)$ time. Using range queries, for any $m$, we give an $\tilde{\mathcal{O}}(\min(nd/m,n^2/m^2)+d)$-size data structure that answers $CountDistinct(i,j)$ queries exactly in $\tilde{\mathcal{O}}(m)$ time. We also consider the special case when the dictionary consists of all square factors of the string. We design an $\mathcal{O}(n \log^2 n)$-size data structure that allows us to count distinct squares in a text fragment $T[i \mathinner{.\,.} j]$ in $\mathcal{O}(\log n)$ time.

The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word $w$ in a given sequence $x$ can be used for classifying $w$ as avoided or overabundant. The definitions used for the expectation and deviation of $w$ in this statistical model were described and biologically justified by Brendel et al. (J Biomol Struct Dyn 1986). We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (Algorithms Mol Biol 2017). In this article, we extend this study by presenting an $\mathcal{O}(n)$-time and $\mathcal{O}(n)$-space algorithm for computing all overabundant words in a sequence $x$ of length $n$ over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree $\mathcal{T}$ of $x$: the number of distinct factors of $x$ whose longest infix is the label of an explicit node of $\mathcal{T}$ is no more than $3n-4$. We further show that the presented algorithm is time-optimal by proving that $\mathcal{O}(n)$ is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms.

The deviation of the observed frequency of a word w from its expected frequency in a given sequence x is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of the deviation of w, denoted by $$\textit{dev}(w)$$ dev ( w ) , effectively characterises the extent of a word by its edge contrast in the context in which it occurs. A word w of length $$k>2$$ k > 2 is a $$\rho $$ ρ -avoided word in x if $$\textit{dev}(w) \le \rho $$ dev ( w ) ≤ ρ , for a given threshold $$\rho < 0$$ ρ < 0 . Notice that such a word may be completely absent from x. Hence, computing all such words naïvely can be a very time-consuming procedure, in particular for large k. In this article, we propose an $$\mathcal {O}(n)$$ O ( n ) -time and $$\mathcal {O}(n)$$ O ( n ) -space algorithm to compute all $$\rho $$ ρ -avoided words of length k in a given sequence of length n over a fixed-sized alphabet. We also present a time-optimal $$\mathcal {O}(\sigma n)$$ O ( σ n ) -time algorithm to compute all $$\rho $$ ρ -avoided words (of any length) in a sequence of length n over an integer alphabet of size $$\sigma $$ σ . In addition, we provide a tight asymptotic upper bound for the number of $$\rho $$ ρ -avoided words over an integer alphabet and the expected length of the longest one. We make available an implementation of our algorithm. Experimental results, using both real and synthetic data, show the efficiency and applicability of our implementation in biological sequence analysis. The systematic search for avoided words is particularly useful for biological sequence analysis. We present a linear-time and linear-space algorithm for the computation of avoided words of length k in a given sequence x. We suggest a modification to this algorithm so that it computes all avoided words of x, irrespective of their length, within the same time complexity. We also present combinatorial results with regards to avoided words and absent words.

The deviation of the observed frequency of a word $w$ from its expected frequency in a given sequence $x$ is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of the standard deviation of $w$, denoted by $std(w)$, effectively characterises the extent of a word by its edge contrast in the context in which it occurs. A word $w$ of length $k>2$ is a $\rho$-avoided word in $x$ if $std(w) \leq \rho$, for a given threshold $\rho < 0$. Notice that such a word may be completely absent from $x$. Hence computing all such words na\"{\i}vely can be a very time-consuming procedure, in particular for large $k$. In this article, we propose an $O(n)$-time and $O(n)$-space algorithm to compute all $\rho$-avoided words of length $k$ in a given sequence $x$ of length $n$ over a fixed-sized alphabet. We also present a time-optimal $O(\sigma n)$-time and $O(\sigma n)$-space algorithm to compute all $\rho$-avoided words (of any length) in a sequence of length $n$ over an alphabet of size $\sigma$. Furthermore, we provide a tight asymptotic upper bound for the number of $\rho$-avoided words and the expected length of the longest one. We make available an open-source implementation of our algorithm. Experimental results, using both real and synthetic data, show the efficiency of our implementation.