Vol 9 No1 - Journal of Cell and Molecular Biology - Haliç Üniversitesi

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denotes the ith nonempty suffix of the string S$, 0 ≤ i < n. The space and time complexity of this suffix tree is O (n). Consider the suffix tree as an example shown in Figure 1 for the string S = acaaacatat. Using the suffix tree of S1 # S2, MUMs(Maximum Unique Match) (Delcher et al., 1999) can be computed in O (n) time and space, where n = [S1 # S2] and # is a symbol neither occurring in S1 nor in S2. However, the space consumption of the suffix tree is a major problem when comparing large genomes. Figure 1. The suffix tree for S = acaaacatat. Suffix array Suffix array is designed for efficient searching of a large text. It requires only 4n bytes (4 bytes per input character) in its basic form. Searching a text can be performed by binary search using the suffix array. Suffix trees can be constructed in O (n) time in the worst case, versus O (nlogn) time for suffix arrays. Suffix arrays will prove to be better than suffix trees for many applications. The suffix array (denoted by suftab) of the string S is an array of integers in the range 0 to n, specifying the lexicographic ordering of the n + 1 suffixes of the string S$. That is, S (suftab [0]), S (suftab [1] … S (suftab[n]) is the sequence of suffixes of S$ in ascending lexicographic order as shown in Table 1. Suffix array generation Creating suffix array requires time O (nlogn) and searching for a pattern in it requires time O (nlogm), where n is the length of the pattern and m is the length of the string. In (Abouelhoda et al., 2002), the authors provides a systematic way of replacing string processing algorithm based on a bottom-up traversal of a suffix tree by a corresponding algorithm that is based on an enhanced suffix array. The advantages of this approach are better search, more space efficient Bio-database compression using enhanced suffix array 47 than suffix tree since the implementation of the enhanced suffix array requires only 5n bytes and the running times of the algorithms are much better than those based on the suffix tree and the algorithm is easier to implement on enhanced suffix arrays than on suffix trees. Table 1. The Lcp-interval table of S = acaaacatat$ s.no Suftab Lcptab S(suftab[i]) 0 2 0 aaacatat$ 1 3 2 aacatat$ 2 0 1 acaaacatat$ 3 4 3 acatat$ 4 6 1 atat$ 5 8 2 at$ 6 1 0 caaacatat$ 7 5 2 catat$ 8 7 0 tat$ 9 9 1 t$ 10 10 0 $ A pair of substrings R = ((i1, j1); (i2, j2)) is a repeated pair if and only if = ((i1, j1) ≠ (i2, j2)) and S [(i1… j1)] = S [(i2 …j2)]. The length of R is j1 - i1+1. A repeated ((i1, j1); (i2, j2)) is called left maximal if S[i1 ─ 1] ≠ S[i2 ─ 1] and right maximal if S[j1 ─ 1] ≠ S[j2 ─ 1]. A repeated pair is called maximal if it is left and right maximal. A substring ω of S is a (maximal) repeat if there is a (maximal) repeated pair ((i1, j1); (i2, j2)) such that ω = S [i1...j1]. A super maximal repeat is a maximal repeat that never occurs as a substring of any other maximal repeat. The suffix array suftab is an array of integers in the range 0 to n, specifying the lexicographic ordering of the n+1 suffixes of the string S$. That is, Ssuftab [0]; Ssuftab [1] … Ssuftab[n] is the sequence of suffixes of S$ in ascending lexicographic order, where Si = S [i ... n -1 $ denotes the ith nonempty suffix of the string S$ , 0 ≤ i ≤ n. The suffix array requires 4n bytes. The Lcp-interval table Lcptab is an array of integers in the range 0 to n and Lcptab [0] = 0 shown table 1. Lcptab[i] is the length of the longest common prefix of Ssuftab [i−1] and Ssuftab[i], for 1 ≤ i ≤ n. Since Ssuftab[n] = $, it is always have Lcptab[n] = 0. The Lcp-table can be computed as a by-product during the construction of the suffix
48 Arumugam KUNTHAVAI and Somasundaram VASANTHA RATHNA array. The lcp-interval tree of S = acaaacatat$ is shown in Figure 2. Figure 2. The Lcp-interval tree of S = acaaacatat$ An interval [i...j], where 0 ≤ i ≤ j ─ n, in an Lcparray is called an Lcp-interval of Lcp-value ℓ (denoted by ℓ-[i...j]) if Lcptab[i] < ℓ Lcptab[k] ≤ ℓ for all k with i+1 ≤ k ≤ j Lcptab[k] = ℓ for at least one k with i+1 ≤ k ≤ j Lcptab [j + 1] < ℓ Every index k, i+1 ≤ k ≤ j, with Lcptab[k] = Ssuftab is called ℓ index. The set of all ℓ indices of an ℓ interval [i...j] will be denoted by ℓ Indices (i...j). If [i...j] is an ℓ-interval such that ω = S[suftab[i]..Suftab[i]+ ℓ -1] is the longest common prefix of the suffixes Ssuftab[i]; Ssuftab[i+1]; … ; Suftab[j], then [i...j] is also called ω-interval. Based on the analogy between the suffix array and the suffix tree, it is desirable to enhance the suffix array with additional information to determine, for any ℓ- interval [i..j], all its child intervals in constant time using enhancing the suffix array with two tables. Enhanced Suffix array The new data structure consists of the suffix array, the Lcp-interval table, and an additional table: the child-table cldtab shown in Table 2. The child-table is a table of size n+1 indexed from 0 to n and each entry contains three values: up, down, and nextℓIndex. Each of these three values requires 4 bytes in the worst case. The values of each cldtab-entry are defined as follows (it is assumed that min Φ = max Φ = 1): 1. cldtab[i].up = Min {q Є [0..i - 1] | Lcptab[q] > Lcptab[i] and for all k Є [q + 1..i - 1] : Lcptab[k] ≥ Lcptab[q]} 2. cldtab[i].down = Max {q Є [i + 1.. n] | Lcptab[q] > Lcptab[i] and for all k Є [i + 1..q - 1] : Lcptab[k] ≥ Lcptab[q]} 3. cldtab[i].next ℓ Index = Min {q Є [i + 1.. n] | Lcptab[q] = Lcptab[ i] and for all k Є [i + 1..q - 1] : Lcptab[k] > Lcptab[i]} The child-table stores the parent-child relationship of Lcp-intervals. For an ℓ -interval [i...j] whose ℓ -indices are i1 < i2
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Page 37 and 38: 32 Preeti SINGH et al. Introduction
Page 39 and 40: 34 Preeti SINGH et al. Discussion C
Page 41 and 42: 36 Preeti SINGH et al. Mukhopadhyay
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Page 47 and 48: 42 Jeyaraj ANBURAJ et al. Table 3.
Page 49 and 50: 44 Jeyaraj ANBURAJ et al. Gokhale M
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Page 59 and 60: 54 Sabira TAIPAKOVA et al. crystall
Page 61 and 62: 56 Sabira TAIPAKOVA et al. Results
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