Computing a Maximal Independent Set Using Beeps

We consider the problem of finding a maximal independent set (MIS) in the discrete beeping model introduced in DISC 2010. At each time, a node in the network can either beep (i.e., emit a signal) or be silent. Silent nodes can only differentiate between no neighbor beeping, or at least one neighbor beeping. This basic communication model relies only on carrier-sensing. Furthermore, we assume nothing about the underlying communication graph and allow nodes to wake up (and crash) arbitrarily. We show that if a polynomial upper bound on the size of the network n is known, then with high probability every node becomes stable inO(log n) time after it is woken up. To contrast this, we establish a polynomial lower bound when no a priori upper bound on the network size is known. This holds even in the much stronger model of local message broadcast with collision detection. Finally, if we assume nodes have access to synchronized clocks or we consider a somewhat restricted wake up, we can solve the MIS problem in O(log n) time without requiring an upper bound on the size of the network, thereby achieving the same bit complexity as Luby’s MIS algorithm. ∗[email protected], Massachusetts Institute of Technology (MIT) †[email protected], Massachusetts Institute of Technology (MIT) ‡[email protected], University of Lugano (USI), Switzerland 1 ar X iv :1 10 8. 19 26 v1 [ cs .D C ] 9 A ug 2 01 1