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The objective of the graphical shortest path problem is to discover the least cost path in a weighted graph between a given source vertex and one or more destinations. This problem class has numerous practical applications including data network routing and speech recognition. This paper discusses the hardware realization of a recurrent spatiotemporal neural network for single source multiple-destination graphical shortest path problems. The network exhibits a regular interconnect structure and uses simple processing units in a combination which is well suited for VLSI implementation with a standard fabrication process.