On the complexity of distributed BFS in ad hoc networks with spontaneous wake-up

We study time and message complexity of the problem of building a BFS tree by a spontaneously awaken node in ad hoc network. Computation is in synchronous rounds, and messages are sent via point-to-point bi-directional links. Network topology is modeled by a graph. Each node knows only its own id and the id's of its neighbors in the network and no pre-processing is allowed; therefore the solutions to the problem of spanning a BFS tree in this setting must be distributed. We deliver a deterministic distributed solution that trades time for messages, mainly, with time complexity O(D . min(D; n=f(n)) . logD . log n) and with the number of point-to-point messages sent O(n. (min(D; n=f(n))+f(n)) . logD. log n), for any n-node network with diameter D and for any monotonically non-decreasing sub-linear integer function f. Function f in the above formulas come from the threshold value on node degrees used by our algorithms, in the sense that nodes with degree at most f(n) are treated differently that the other nodes. This yields the first BFS-finding deterministic distributed algorithm in ad hoc networks working in time o(n) and with o(n2) message complexity, for some suitable functions f(n) = o(n= log2 n), provided D = o(n= log4 n).