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This paper proposes a fast method for computing the costs of all-pairs shortest paths (APSPs) on the graphics processing unit (GPU). The proposed method is implemented using compute unified device architecture (CUDA), which offers us a development environment for performing general-purpose computation on the GPU. Our method is based on Harish's iterative algorithm that computes the cost of the single-source shortest path (SSSP) for every source vertex. We present that exploiting task parallelism in the APSP problem allows us to efficiently use on-chip memory in the GPU, reducing the amount of data being transferred from relatively slower off-chip memory. Furthermore, our task parallel scheme is useful to exploit a higher parallelism, increasing the efficiency with highly threaded code. As a result, our method is 3.4--15 times faster than the prior method. Using on-chip memory, our method eliminates approximately 20% of data loads from off-chip memory.