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In this paper, we propose the greedy smallest-cost-rate path first (GRASP) algorithm to route power from sources to loads in a digital microgrid (DMG). Routing of power from distributed energy resources (DERs) to loads of a DMG comprises matching loads to DERs and the selection of the smallest-cost-rate path from a load to its supplying DERs. In such a microgrid, one DER may supply power to one or many loads, and one or many DERs may supply the power requested by a load. Because the optimal method is NP-hard, GRASP addresses this high complexity by using heuristics to match sources and loads and to select the smallest-cost-rate paths in the DMG. We compare the cost achieved by GRASP and an optimal method based on integer linear programming on different IEEE test feeders and other test networks. The comparison shows the trade-offs between lowering complexity and achieving optimal-cost paths. The results show that the cost incurred by GRASP approaches that of the optimal solution by small margins. In the adopted networks, GRASP trades its lower complexity for up to 18% higher costs than those achieved by the optimal solution.