On asymptotically optimal routing in large wireless networks and Geometrical Optics analogy

Abstract

This work is based on the observation that, as the number of nodes in a wireless network approaches infinity, minimum cost routes become smooth curves that observe the same laws followed by rays of light in properly defined optical media. Accordingly, in this paper an analogy between optimal routing in large wireless networks and Geometrical Optics is first formally defined. The analogy is based on the concept of the cost function, which plays the role of the refractive index in the networking context. Then, the relevance of the principle of Fermat and the eikonal equation in routing problems is shown, and a methodology for calculating the cost function is proposed and applied in two cases of special interest, i.e., bandwidth limited and energy limited networks. The applicability of the Optics-Networking analogy is also discussed in the case of networks with large but finite numbers of nodes. Finally, novel, distributed route discovery protocols that make use of the analogy are outlined.