https://hal.inria.fr/hal-00991414Kowalski, Dariusz R.Dariusz R.KowalskiDepartment of Computer Science [Liverpool] - University of LiverpoolKrzywdziński, KrzysztofKrzysztofKrzywdzińskiFaculty of Mathematics and Computer Science [Poznan] - UAM - Adam Mickiewicz University in PoznańOn the complexity of distributed BFS in ad hoc networks with spontaneous wake-up.HAL CCSD2013BFS treead hoc networkdistributed algorithmmessage complexity[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-05-15 11:33:572018-01-15 11:43:262014-05-15 11:38:15enJournal articlesapplication/pdf1We 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 an undirected 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 $O(D\cdot\min(D,n/f(n)) \cdot \log D \cdot \log n)$ and with the number of point-to-point messages sent $O(n\cdot(\min(D,n/f(n)) + f(n)) \cdot \log D \cdot \log n)$, 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 $o(n^2)$ message complexity, for some suitable functions $f(n)=o(n/\log^2 n)$, provided $D=o(n/\log^4 n)$.