Exponential separations in the energy complexity of leader election

Abstract

Energy is often the most constrained resource for battery-powered wireless devices and the lion's share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the energy complexity of Leader Election and Approximate Counting in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions. We consider four collision-detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD and Receiver-CD (in which only transmitters or only listeners detect collisions), and No-CD (in which no one detects collisions.)The take-away message of our results is quite surprising. For randomized Leader Election algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD:Randomized: No-CD = Sender-CD Receiver-CD = Strong-CD and for deterministic Leader Election algorithms, there is another exponential gap in energy complexity, but in the reverse direction:In particular, the randomized energy complexity of Leader Election is Θ(log * n) in Sender-CD but Θ(log(log * n)) in Receiver-CD, where n is the (unknown) number of devices. Its deterministic complexity is Θ(log N ) in Receiver-CD but Θ(log log N ) in Sender-CD, where N is the (known) size of the devices' ID space.There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized Leader Election and Approximate Counting. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n = Θ(N ), with inverse-Ackermann-type (O(α(N ))) energy complexity.

Fast and Resource Competitive Broadcast in Multi-channel Radio Networks

“…exp (−Θ(i · lg n)) = n −Ω (1) In above, the second inequality is due to the fact that the length of epochs increases geometrically; the last inequality is due to Lemma Pr F > γβ · lg T · (T /n) Fig. 4.…”

“…Classical results often rely on variants of the DECAY procedure [10], while recent ones (see, e.g., [11]- [14]) tend to employ more advanced techniques (such as network decomposition) to improve performance. Besides time complexity, energy cost has also been taken into consideration when building communication primitives (see, e.g., [1], [2], [15], [16]), but usually without assuming the existence of a jamming adversary.…”

“…This setting motivates the notation of resource competitive algorithms which focus on optimizing relative cost. In particular, assuming for each node the cost of sending or listening on one channel for one slot is one unit of energy (while idling is free), 1 can we design algorithms (for, e.g., broadcasting) that ensure each node's cost is only o(T )? Effectively, such results would imply Eve cannot efficiently stop nodes from accomplishing the distributed computing task in concern.…”

“…Clearly, Nu can be written as the sum of R indicator random variables Nu = R i=1 N (i) u , where N (i) u = 1 iff u hears silence in the i th slot. To enforce correctness, we often need to argue Nu and Nv are close for any pair of nodes u and v. If Eve is oblivious (i.e., an offline adversary), then {N (1) u , · · · , N (R) u } are mutually independent. Together with E[Nu] = E[Nv], we can use powerful concentration inequalities like Chernoff bounds (see, e.g., [8]) to show Nu and Nv must be close.…”

“…Together with E[Nu] = E[Nv], we can use powerful concentration inequalities like Chernoff bounds (see, e.g., [8]) to show Nu and Nv must be close. Unfortunately, once Eves becomes adaptive (i.e., she can use past execution history to determine future behavior), channel feedback like {N (1) u , · · · , N (R) u } are not independent any more. To resolve this issue, we apply the coupling technique (see, e.g., [9]).…”