The Smith-Waterman algorithm

The SW algorithm is used to identify the similarity between two sequences by computing the maximum local alignment score. Given two sequences S₁ and S₂ of lengths l₁ and l₂ respectively, the SW algorithm computes the similarity score H(i, j) of two sequences ending at position i and j of S₁ and S₂, respectively. H(i, j) is computed as in equation (1), for 1 ≤ i ≤ l₁, 1 ≤ j ≤ l₂:

(1)

where sbt is the substitution matrix, ρ is the gap open penalty and σ is the gap extension penalty. A substitution matrix sbt gives the substitution rates of amino acids in proteins, derived from alignments of protein sequences. The recurrences are initialized as H(i, 0) = H(0, j) = E(i, 0) = F(0, j) = 0 for 0 ≤ i ≤ l₁ and 0 ≤ j ≤ l₂. The maximum local alignment score is defined as the maximum score in H. The computation of each cell in H depends on its left, upper, and upper-left neighbors, as shown by the three arrows in Additional file 1. In addition, this data dependency implies that all cells on the same minor diagonal in the alignment matrix are independent, and can be computed in parallel. Thus, the alignment can be computed in minor-diagonal order from the top-left corner to the bottom-right corner in the alignment matrix, where the computation of minor diagonal i only needs the results of minor diagonals i-1 and i-2.

CUDA programming model

CUDA is an extension of C/C++ with a minimalist set of abstractions for expressing parallelism, enabling users to write scalable multi-threaded parallel code for CUDA-enabled GPUs [18]. A CUDA program consist of two parts: a host program running on the host CPU, and one or more parallel kernels which can be executed on GPUs with NVIDIA's Tesla unified graphics and computing architecture [19].

A kernel is written in conventional scalar C-code, which represents the operations to be performed by a single thread and is invoked as a set of concurrently executing threads. These threads are organized into a grid of thread blocks, where a thread block is a set of concurrent threads. This hierarchical organization has implications for thread communication and synchronization. Threads in a thread block are allowed to synchronize with each other using barriers, and can communicate through a per-block shared memory (PBSM). However, threads located in different thread blocks cannot communicate or synchronize directly. To write efficient CUDA programs, besides the PBSM, it is important to understand the other memory spaces in more detail: non-cached global and local memory, cached texture and constant memory as well as on-chip registers.

The Tesla architecture is built around a fully programmable scalable processor array, organized into a number of streaming multiprocessors (SMs). Each SM contains eight scalar processors (SPs), sharing a PBSM of size 16 KB. All threads of a thread block are executed concurrently on a single SM. The SM executes threads in small groups of 32 threads, called warps, in an SIMT fashion. When one thread blocks is scheduled to execute on an SM, threads in the thread block are split into warps that get scheduled by the SIMT unit. A warp executes one common instruction at a time, but allows for instruction divergence. When divergence occurs, the warp serially executes each branch path. Thus, parallel performance is generally penalized by data-dependent conditional branches and improved if all threads in a warp follow the same execution path. Branch divergence occurs only in a warp, and different warps run independently regardless of common or disjointed code paths they are executing.

Virtualized SIMD vector programming model

Because a warp executes one common instruction at a time, all threads in a warp are implicitly synchronized after executing any instruction. This execution manner is very similar to the characteristic of SIMD vector organizations that a single instruction controls multiple processing elements. Therefore, it is viable to virtualize a warp as an SIMD vector with each thread as a vector element. An alternative virtualization at the warp level is to divide a warp into several thread groups of equal size and then virtualize each thread group as a vector with each thread in the group as an element. However, for the current CUDA-enabled GPU technologies, this warp-level virtualization limits the virtualized vector length to 32. To support longer vector lengths, vectors can be virtualized at the thread-block level, where a thread block is considered as a large vector with each thread in the thread block as an element. In this case, the intrinsic function __syncthreads() has to be used to explicitly synchronize all threads at specific synchronization points in the kernel to keep the correctness of the virtualized vector computations.

In this paper, we refer to the virtualized vector as virtualized SIMD vector and its corresponding programming model as virtualized SIMD model to differentiate from the real SIMD vector organizations. Since this virtualization is based on the SIMT model, the virtualized SIMD model shares all the features of the SIMT model with an additional ability to conduct vector computations. We define VL to denote the length of a virtualized vector, i.e. the number of data lanes of the vector. For the convenience of discussion, we assume that the first element (indexed by 0) is on the rightmost and the last element (indexed by VL - 1) on the leftmost of the vector. Each thread comprising a virtualized vector is assigned a vector ID vtid that is equal to the position index of its corresponding vector element in the vector of length VL, where 0 ≤ vtid < VL. In this paper, we use warp-level virtualization to implement vectorized SW algorithms.

Query profile

To calculate H(i, j), the substitution score sbt(S₁[i], S₂[j]), from the substitution matrix, is added to H(i-1, j-1). Due to the huge number of iterations in the SW algorithm calculation, reducing the number of instructions needed to perform one cell calculation has a significant impact on the execution time. In this regard, Rognes et al. [10] and Farrar [11] suggested the use of a query profile parallel to the query sequence for each possible residue. A query profile is pre-calculated just once before database searches, and can be calculated in two layouts: a sequential layout [10] and a striped layout [11].

Given a query sequence Q of length l defined over an alphabet Σ, a query profile is defined as a numerical string set P = {P _r | r ∈ Σ}, where P _r is a numeric string consisting of substitution scores that are required to compute a complete column (or row) of the alignment matrix, and the values of P _r depend on whether it uses a sequential or a striped layout. For a sequential query profile (see Additional file 2), P _r (i), the i-th value of P _r , is defined as

(2)

Even though a sequential query profile is initially designed for SIMD vector computation of the SW algorithm, it is also suitable for scalar computation of the algorithm. For SIMD vector computation, it generally aligns l according to vector length VL for performance consideration and pads Q with dummy residues that have a substitution score of zero between itself and any residue.

A striped query profile (see Additional file 3) is designed for SIMD vector computation. To construct a striped query profile, given a vector length VL, the query sequence Q is divided into a set of equal length query segments QSEG = {QSEG₁, QSEG₂, ..., QSEG _VL } of VL elements. The length T of each query segment is equal to (l + VL - 1)/VL. If l is not a multiple of VL, Q is first padded with dummy residues. For simplicity, we assume l is a multiple of VL. Correspondingly, each numerical string P _r of a striped query profile can be considered as a set of non-overlapping, consecutive VL-length vector segments VSEG = {VSEG₁, VSEG₂, ..., VSEG _T } of T elements, where the i-th element of VSEG _j maps the j-th element of QSEG _i . Hence, P _r (i) of a striped query profile is defined as

(3)

Optimized SIMT Smith-Waterman algorithm using CUDA

The SIMT SW algorithm used by CUDASW++ 1.0 is designed based on the SIMT abstraction of CUDA-enabled GPUs, which enables thread-level parallelism for independent scalar threads as well as data parallelism for coordinated threads. It uses two stages to complete the database searches: the first stage uses inter-task parallelization using thread-level parallelism, and the second stage uses intra-task parallelization using data parallelism. Since the first stage dominates the total runtime when searching large database, the optimizations of CUDASW++ 2.0 are focused on this stage. The performance of CUDASW++ 2.0 is significantly improved due to the following optimizations: introducing a sequential query profile and using a packed data format.

A sequential query profile stored in texture memory is used to replace random access to the substitution matrix in shared memory. Inspired by the fact that texture instructions output filtered samples, typically a four-component (RGBA) color [19], the sequential query profile is re-organized using a packed data format, where each numerical string P _r is packed and represented using the char 4 vector data type, instead of the char scalar data type. In this way, four substitution scores are realized using only one texture fetch, thus significantly improving texture memory throughput. Like the query profile, each subject sequence is also re-organized using a packed data format, where four successive residues of each subject sequence are packed together and represented using the uchar4 vector data type. In this case, when using the cell block division method, the four residues loaded by one texture fetch are further stored in shared memory for the use of the inner loop (see the pseudocode of the CUDA kernel shown in Figure 1). In addition, some redundant operations are removed to improve instruction pipeline throughput.

Basic vectorized Smith-Waterman algorithm using CUDA

The basic vectorized SW algorithm is designed by directly mapping the striped SW algorithm [11] onto CUDA-enabled GPUs using CUDA, based on the virtualized SIMD vector programming model. For the computation of each column of the alignment matrix, the striped SW algorithm consists of two loops: an inner loop calculating local alignment scores postulating that F values do not contribute to the corresponding H values, and a lazy-F loop correcting any errors introduced from the calculations of the inner loop. The basic vectorized algorithm uses a striped query profile. In the alignment matrix, for a specific column, the inner loop is completed in T iterations by moving SIMD vectors sequentially through all vector segments of P _r corresponding to this column. For the convenience of discussion, define vecH(i, j), vecE(i, j) and vecF to process the H, E and F values of the cells corresponding to VSEG _i of P _r , where 1 ≤ i ≤ T, for the j-th column of the alignment matrix. Using virtualized SIMD vectors, several technical issues have to be addressed for this CUDA algorithm, including saturation algorithmic operations, shift operations and predicate operations on virtualized vectors.

Saturation additions and subtractions are required to calculate alignment scores. Since CUDA-enabled graphics hardware lacks support for these operations, maximum and minimum operations are used to artificially implement them. The integer functions max(x, y) and min(x, y), in the CUDA runtime library, are used to avoid divergence. Shift operations on vectors are required both for the inner and lazy-F loops. We implement these operations using shared memory, where all threads comprising a virtualized vector writes their original values to a share memory buffer and then reads their resulting values from the buffer as per the number of shift elements. Additional file 4 gives the CUDA pseudocode for shifting a virtualized vector by n elements to the left. As can be seen from the pseudocode, one shift operation is time-consuming as compared with vector register operations in a real SIMD vector architectures, even though access to shared memory without bank conflicts has a much lower latency than device memory [20].

The lazy-F loop requires predicate operations on virtualized vectors when determining whether to continue or exit the loop by checking vecF against the values of vecH(i, j). An approach is to use shared memory to simulate these operations. Although this approach is effective, it is inefficient due to the overhead incurred by the accesses to shared memory. Fortunately, CUDA-enabled GPU devices with compute capability 1.2 and higher provide the support for two warp vote functions __all(int) and __any(int), providing an indispensible capability to perform fast predicate operations across all threads within a warp. We use the __all(int) warp vote function to implement the predicate operations on virtualized vectors for the lazy-F loop.

The striped query profile is stored in texture memory to exploit the texture cache. Subject sequences and the query profile are stored using the scalar data type in an unpacked fashion because the inner loop is a for loop without manually unrolling. The intermediate element values of vecH(i, j) and vecE(i, j) are stored in global memory, with vecF stored in registers, to support long query sequences. To improve global memory access efficiency, we use the unsigned half-word data type to store the H and E values in global memory.

Partitioned vectorized Smith-Waterman algorithm using CUDA

To gain higher performance, we have investigated a partitioned vectorized SW algorithm using CUDA. This algorithm first divides a query sequence into a series of non-overlapping, consecutive small partitions as per a specified partition length (PL), and then aligns the query sequence to a subject sequence partition by partition. For the partitioned vectorized algorithm, PL must be a multiple of VL. The alignment between one partition of the query sequence and the subject sequence is performed using the basic vectorized algorithm. In this case, because PL is usually set to be relatively smaller, shared memory or registers can be used to store the alignment scores.

In this algorithm, it considers each partition as a new query sequence and constructs a striped query profile for each partition. However, this partitioned vectorized algorithm makes the alignment of one partition depend on the alignment scores of the previous partition in the alignment matrix (see Figure 2). More specifically, the computation of the H and F values of the first row of one partition depends on the H and F values of the last row of the previous partition (note that the positions of the first and the last rows are kept unchanged for a striped query profile regardless of the specific values of PL and VL). In this case, after having completed the computation of one column of a partition, the H and F values of the last cell in this column have to be saved for the future use of the next partition. For performance consideration, instead of directly storing F value of the last cell of the partition, we store the F value of the first cell of the next partition, calculated from the H and F values of the last cell in the current partition. However, there is a problem with the striped algorithm in that for a specific column, after exiting from the lazy-F loop, it makes sure that the H and E values of all cells are correct, but provides no guarantee that vecF stores the correct F value of the last cell. This is because the lazy-F loop is likely to quit, with high probability, with no need to re-calculate all the cell values. Since the H values of all cells in the last row of the previous partition are always correct, the correctness of our partitioned vectorized algorithm will be proved if we could prove that the correctness of the F values of all cells in the last row of the previous partition does not affect the correct computation of F values of all cells in the first row of the current partition. In the following, we will prove that the F values of all cells in the first row of the current partition can always be correctly calculated.

Theorem 1. For the partitioned vectorized SW algorithm, the F values of all cells in the first row of the current partition are correctly computed regardless of the correctness of the F values of all cells in the last row of its previous partition.

Proof. Taking cells B and C in Figure 2 as an example, define C _F to denote the F value of C, B _H to denote the H value of B, and B _F to denote the F value of B, where C _F = max(B _H - ρ - σ, B _F - σ) according to equation (1). For the striped SW algorithm, the correctness of the F value of the last cell for a specific column j in the alignment matrix depends on two possible conditions.

Case 1: the lazy-F loop does not stop until the F values of all cells in column j have been corrected because the F value of each cell contributes to its H value. In this case, due to the recalculation of all cells, vecF stores the correct F value of the last cell. Since both B _H and B _F are correct, C _F is definitely correctly calculated.

Case 2: the lazy-F loop stops after some iterations with no need to recalculate all cells. This means that the F values of the remaining cells will not contribute to their corresponding H values, but might not be equal to their correct F values directly calculated using equation (1). In this case, because B _F - σ ≤ B _H - ρ - σ, C _F is equal to B _H - ρ - σ so that C _F has no relationship to B _F .

From the above discussion, a conclusion can be drawn that C _F can always be correctly calculated regardless of whether B _F is correct or not. Therefore, the theorem is proven.

The partitioned vectorized algorithm stores the values of vecH and vecE of one column for a partition in registers to achieve peak performance. Using registers, the inner loop and the lazy-F loops are manually unrolled according to the number of vector segments. Since the computation of the inner loop is fully unrolled, the access to the striped query profile can be optimized using the packed data format. Like the optimized SIMT algorithm, each numerical string P _r of this query profile is packed and represented using the char 4 vector data type according to the access order of threads comprising a virtualized vector. However, the subject sequences are not packed because we failed to find performance improvement using the packed data format. Figure 3 shows the pseudocode of the CUDA kernel of the partitioned vectorized algorithm.