To determine similarity between two nucleotide or protein, the Smith-Waterman algorithm compares all possible segments and assigns a score. It returns segment with the highest score. For example, consider s and t two sequences to be compared. The algorithm begins by creating a matrix M of dimensions equal to the lengths of the sequences s and t. Then the cell values of matrix M are calculated starting from the cell in the upper left corner to the cell at the bottom right corner. Formula (1) gives the expression of .

Where:

-S is a “Blosum Scoring matrix”³.

-d is a fixed constant corresponding to the alignment of a letter and an empty score (-).

-t [j] s [i] means that t [j] and s [i] are animo acide.

-s[i] and t[j]―correspond respectively to the alignement of the―a animo acide (animo acide with-).

-M [i] [j] is intuitively the score of an alignment ending with t [i] s [j]. At each step, when a maximum value is calculated, it is stored along the direction in which it is obtained: on the diagonal (i − 1, j − 1), just above (i − 1, j) or to the left (i, j − 1). Computing the matrix M and the information regarding the directions in which the highest values are obtained, require much time and memory space.

To restore the best alignment, we proceed as follows:

• Find the maximum value in the matrix M. This is the end of the local alignment with the best score.

• Go to (the) cell (s) adjacent(s) Maximum score:

-movement on the diagonal shows an alignment of the letters t [i] and s [j];

-an horizontal movement means of a bias t [i] and a blank (−) between s [j − 1] and s [j];

-an vertical movement is analogous to the horizontal displacement;

-when the maximum score is 0, it means that the optimal local alignment starts at M [i + 1] [j + 1].

• The optimum score is given by the equation:

Here are two sequences to be compared:

t: CGGGTATC

s: CCCTAGGT

Figure 1 shows the values in the matrix of scores at the end of the execution of the Smith Waterman algorithm:

Once all the cells of the matrix scores are calculated to find the best local alignment, we start with the cell where the maximum score has been identified, then back to the cell that was used to determine the score and so on. And the optimal local alignment in the Smith-Waterman algorithm is:

C G G G --A T

---C T A G G T

As shown in Figure 2, calculations are done in parallel according to the anti-diagonal.

At time T1, a single cell is calculated, at time T2, two cells are calculated, at time T3, three cells are calculated, etc.

Generally, the cell M(i,j) is computed at time T_ij = i + j ? 1.

The number of cells calculated at each iteration T_i is given as above.

T1 at first, then T2 and T3, and so on. We obtain the representation in Figure 3.

This representation of the scoring matrix clearly shows the tasks that can be performed simultaneously.

Suppose we have two sequences of identical size N. The number of computable Nb_i cells at the same time T_i is given by the Formula (2).

We assume Nb the sum of Nb_i. Formula (3) permits to verify that the new approach takes into account the N² cells of the scoring matrix.

Thus without taking into account the dependancies between the cells during the computation, the number of iterations to calculate the N² cells is 2N − 1.

So, if represents the average number of cells computable per passage we have in (4),

M represents the matrix of scores, i the line number and j the column number.

We define an application as follows:

For each cell referenced (i, j):

is an bijective application. It transforms the matrix in Figure 2 to the matrix in Figure 3.

In fact is a change of reference from (i, j) ® (i, i + j − 1).

The function f represents a rotation of the scoring matrix cells which transforms antidiagonals to columns. However, we must keep in mind the treatments which must be applied on sequences to be aligned.

In the original representation, M [i] [j] depends on the values of cells M [i] [j − 1], M [i − 1] [j] and M [i − 1] [j − 1], as shown in Figure 4.

In the new representation, computing m [i] [j] depends on m [i-1] [j − 2], m [i − 1] [j − 1] and m [i] [j − 1] as shown in Figure 5.

The change of representation permits to calculate values of cells on the same column. So, the genomic sequence located on the column reference changes as we have shown in Figure 5. It follows a refitting of the value S [s [i]] [t [j]] used in the research of the value of m [i] [j] which becomes S [s [i’]] [t [j’ − i’]] for m [i’] [j’].

Once all the values of the matrix scores are calculated, we must produce the results. The score is obtained in the same way as in the original form, i.e. the maximum value of cells, but acids are obtained using function.

In Figure 6, we have compared the sequential computation time of the matrix’s scoring in its initial representation and the new reorganization. We note that the new representation has almost the same performance as the initial representation for sequences of length less to fourteen thousand (14,000) nucleotides. This experiment aims to show that the two representations of the matrix’s scoring are equivalent, sequentially in the range of sequences that we study: no spare time. This enables us to guarantee for the continuation of the study that considering the new representation of the matrix’s coring did not induce additional execution time. For the rest, we will use the same interval of sequences.

Note that, we have been limited in trying to go beyond this size of sequence due to our calculation capabilities.

OpenMP is based on the principle of shared memory [12] -[15] . The computation to be performed is decomposed into multiple tasks. Tasks are performed by the available computational units. The treatment to be performed, and data variables can be stored in a location accessible to all processing units. Shared variables are declared in the Shared() option while thread-specific variables are in the Private() optional list().

Experiments were performed on DNA sequences of various lengths using OpenMP. As we shall see, the optimum dosages of the block size depend mainly on the size N of the sequences to align.

We propose two opportunities for parallelization of the calculation of the values of cells in the matrix scoring by the Smith-Waterman algorithm.

The scoring matrix is reorganized: all cells in a column can be calculated simultaneously. Thus at each iteration, the cells of a single column are calculated. Each thread will calculate elements of one or more cells.

A thread can read the updated content elements from another thread to calculate its own elements.

This solution has the advantage of providing the list of computable elements in an iteration but has the disadvantage of combining expectations.

It should also be noted that at this level, it is possible to calculate in multiple loops. At each loop, threads are created.

As there’s no extra time outside access for reading or writing to the different cells of the matrix, the runtime in both cases are the same. So, we will treat one case.

Initial form of the Smith Waterman algorithm has many implementation on GP-GPU as in [16] -[20] . To perform the calculation on GP-GPU, the scoring matrix is represented in vector form. Each ring launched calculates the elements of a column of the new representation. These elements are identified from the parameters (i, j). In total 2N iterations are launched.

Figure 10 represents the results obtained.

For OpenMP we examine three (3) cases. The first case uses two (2) threads.

The second uses also two (2) threads with a chunk of fifty (50) and the last one two (2) threads with a chunk of two hundred and fifty (250).

The best case is the second one. We notice also that beyond 14,000 acids per sequence, the three cases have equivalent results.

The implementation on GP-GPU gives better acceleration compared to OpenMP. For sequences used, the performance is improved more than 25 times.