We consider the energy complexity of the leader election problem in the single-hop radio network model, where each device
v
has a unique identifier
\(\mathsf {ID}(v) \in \left\lbrace 1, 2, \ldots, N\right\rbrace \)
. Energy is a scarce resource for small battery-powered devices. For such devices, most of the energy is often spent on communication, not on computation. To approximate the actual energy cost, the energy complexity of an algorithm is defined as the maximum over all devices of the number of time slots where the device transmits or listens.
Much progress has been made in understanding the energy complexity of leader election in radio networks, but very little is known about the trade-off between time and energy. Chang et al. [STOC 2017] showed that the optimal deterministic energy complexity of leader election is
Θ
(log log
N
) if each device can simultaneously transmit and listen, but still leaving the problem of determining the optimal time complexity under any given energy constraint.
Time-energy trade-off:
For any
k
≥ log log
N
, we show that a leader among at most
n
devices can be elected deterministically in
O
(
k
·
n
1 + ϵ
) +
O
(
k
·
N
1/
k
) time and
O
(
k
) energy if each device can simultaneously transmit and listen, where ϵ > 0 is any small constant. This improves upon the previous
O
(
N
)-time
O
(log log
N
)-energy algorithm by Chang et al. [STOC 2017]. We provide lower bounds to show that the time-energy trade-off of our algorithm is near-optimal.
Dense instances:
For the dense instances where the number of devices is
n
=
Θ
(
N
), we design a deterministic leader election algorithm using only
O
(1) energy. This improves upon the
O
(log
*
N
)-energy algorithm by Jurdziński, Kutyłowski, and Zatopiański [PODC 2002] and the
O
(
α
(
N
))-energy algorithm by Chang et al. [STOC 2017]. More specifically, we show that the optimal deterministic energy complexity of leader election is
\(\Theta \left(\max \left\lbrace 1, \log \frac{N}{n}\right\rbrace \right) \)
if each device cannot simultaneously transmit and listen, and it is
\(\Theta \left(\max \left\lbrace 1, \log \log \frac{N}{n}\right\rbrace \right) \)
if each device can simultaneously transmit and listen.
