Different geographic routing protocols have different requirements on routing metric designs to ensure proper operation. Combining a wrong type of routing metrics with a geographic routing protocol may produce unexpected results, such as geographic routing loops and unreachable nodes. In this paper, we propose a novel routing algebra system to investigate the compatibilities between routing metrics and three geographic routing protocols including greedy, face, and combined greedy-face routing. Five important algebraic properties, respectively, named odd symmetry, transitivity, strict order, source independence, and local minimum freeness, are defined in this algebra system. Based on these algebraic properties, the necessary and sufficient conditions for loop-free, delivery-guaranteed, and consistent routing are derived when greedy, face, and combined greedy-face routing serve as packet forwarding schemes or as path discovery algorithms, respectively. Our work provides essential criteria for evaluating and designing geographic routing protocols.

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