Solving Dynamic Programming Problem by Pipeline Implementation on GPU

Abstract

In this paper, we show the effectiveness of a pipeline implementation of Dynamic Programming (DP) on GPU. As an example, we explain how to solve a matrix-chain multiplication (MCM) problem by DP on GPU. This problem can be sequentially solved in O(n 3) steps by DP where n is the number of matrices, because its solution table is of size n × n and each element of the table can be computed in O(n) steps. A typical speedup strategy for this is to parallelize the O(n) step computation of each element, which can be easily achieved by parallel prefix computation, i.e., an O(log n) step computation with n threads in a tournament fashion. By such a standard parallelizing method, we can solve the MCM problem in O(n 2 log n) steps with n threads. In our approach, we solve the MCM problem on GPU in a pipeline fashion, i.e., we use GPU cores for supporting pipelinestages so that many elements of the solution table are partially computed in parallel at one time. Our implementation determines one output value per one computational step with n threads in a pipeline fashion and constructs the solution table totally in O(n 2) steps with n threads.

Accelerating pipeline implementation of dynamic programming on GPU

“…Even if a problem is solved by DP with two-dimensional solution table, the problem may be reduced to the S-DP problem of one-dimensional solution table. The interested reader should refer to [5], where the matrix-chain multiplication (MCM) problem (e.g., [6]) is discussed as an example.…”

“…First, we introduce the pipeline implementation technique shown in our previous studies [4,5]. In the next section, we will further accelerate the algorithm.…”

“…See the step 4 of Figure 3 as an example. In the step 4, ST [3], ST [5], and ST [6] are referred to, though the computation of ST [6] had just been completed in the preceding step 3. Hence, in the case shown in Figure 3, while step x being executed, the value of ST that had just been calculated in the preceding step x − 1 is immediately used for the step x, which means that multiple consecutive iterations of the outer loop of the pipeline algorithm cannot be executed at the same time.…”

“…In our previous studies [4,5], we solve the simplified DP problem on GPU in a pipeline fashion, i.e., we use GPU cores for supporting pipeline-stages so that several elements of the solution tables are respectively computed partially at one time. Our pipeline implementation determines one output value per one computational step with k threads in a pipeline fashion and solves the simplified DP problem on GPU in O(n) steps, which is k times faster than the O(nk) steps for the sequential algorithm.…”

“…Here in this paper, we try to further accelerate the pipeline implementation with larger parallelism. In our previous studies [4,5], the parallelism is up to k, the number of pipelinestages. Here, to accelerate the pipeline implementation, we propose a p-fold pipeline technique, and show that it enables larger parallelism more than k. We also discuss the effectiveness of our method with explaining the 0-1 knapsack problem by DP as an example.…”