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 ID(v) ∈{ 1, 2, ⋖ , N} . 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 tradeoff 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 tradeoff: For any k ≥ log log N, we show that a leader among at most n devices can be elected deterministically in O(k ċ n^1+ε) + O(k ċ 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 tradeoff 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 \(\Theta (\max \lbrace 1, \log \tfrac{N}{n}\rbrace)\) if each device cannot simultaneously transmit and listen, and it is \(Θ (\max \lbrace 1, \log \log \tfrac{N}{n}\rbrace)\) if each device can simultaneously transmit and listen.

我们考虑单跳无线网络模型中领导者选举问题的能量复杂性，其中每个设备v都有一个唯一的标识符ID( v ) ∈{ 1, 2, ⋖ , N } 。对于小型电池供电设备来说，能源是一种稀缺资源。对于此类设备，大部分能量通常花费在通信上，而不是计算上。为了近似实际的能量成本，算法的能量复杂度被定义为设备传输或侦听的时隙数量在所有设备上的最大值。

在理解无线电网络中领导人选举的能量复杂性方面已经取得了很大进展，但对于时间和能量之间的权衡却知之甚少。张等人。[STOC 2017] 表明，如果每个设备可以同时传输和监听，但仍然存在在任何给定能量约束下确定最佳时间复杂度的问题，则 领导者选举的最佳确定性能量复杂度为 θ (log log N )。

• 时间-能量权衡：对于任何k ≥ log log N，我们证明最多n 个设备中的领导者可以在O ( k ċ n ^1+ε ) + O ( k ċ N ^1/k ) 时间和O内确定性地选出( k ) 能量（如果每个设备可以同时传输和监听），其中 ε > 0 是任何小常数。这比之前的O ( N ) 时间O (log log N)-能量算法，作者：Chang 等人。[STOC 2017]。我们提供下限来表明我们算法的时间-能量权衡接近最优。

• 密集实例：对于设备数量为n = θ ( N ) 的密集实例，我们设计了仅使用O (1) 能量的确定性领导者选举算法。这改进了Jurdziński、Kutyłowski 和 Zatopiański [PODC 2002] 的O (log* N )-能量算法以及O (α ( N))-Chang 等人的能量算法。[STOC 2017]。更具体地说，我们表明，如果每个设备不能同时传输和监听，则领导者选举的最佳确定性能量复杂度为 \(\Theta (\max \lbrace 1, \log \tfrac{N}{n}\rbrace)\)，如果每个设备可以同时传输和监听，则为 \(\max \lbrace 1, \log \log \tfrac{N}{n}\rbrace)\)。