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Solving Dynamic Programming Problem by Pipeline Implementation on GPU

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 $n \times n$ 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 pipeline-stages 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.