Multi-Hop Network Localization in Multi-Radius Unit Disk Graph Model

Consider localizing a large number of wireless nodes, none of which has direct connectivity to 3 or more reference terminals. Since no node has enough distance measures to perform trilateration, they cannot determine their locations individually, but must rely on multihop network localization approaches and graph rigidity to get localized altogether. Previous work has modeled this problem as a unit disk graph, where the radius gamma sets a clear cut on node connectivity - within gamma two nodes have absolute connection and hence reliable distance measure, and beyond gamma they have none. Such a black-and-white model is idealistic, and this paper considers a more practical scenario where two nodes, in addition to being absolutely connected or disconnected, may fall in a gray area of random connectivity, and subsequently their distance measure may be available or reliable with some probability p. We propose a network localization scheme that captures this situation via a tree-search algorithm based on a multi-radius unit graph model. Leveraging an interesting philosophy of "no news is good news," the implicit but useful knowledge that two nodes have no connectivity is also exploited to substantially reduce the search space and expedite the localization speed. Although the general problem of network localization is NP-Hard, extensive simulations on randomly-generated graph geometry show that the proposed search algorithm has a very fast average computation time, far less than the worst-case (if the solution exists).