We study the design of energy-efficient algorithms for the LOCAL and CONGEST models. Specifically, as a measure of complexity, we consider the maximum, taken over all the edges, or over all the nodes, of the number of rounds at which an edge, or a node, is active in the algorithm. We first show that every Turing-computable problem has a CONGEST algorithm with constant node-activation complexity, and therefore constant edge-activation complexity as well. That is, every node (resp., edge) is active in sending (resp., transmitting) messages for only $O(1)$ rounds during the whole execution of the algorithm. In other words, every Turing-computable problem can be solved by an algorithm consuming the least possible energy. In the LOCAL model, the same holds obviously, but with the additional feature that the algorithm runs in $O(\mbox{poly}(n))$ rounds in $n$-node networks. However, we show that insisting on algorithms running in $O(\mbox{poly}(n))$ rounds in the CONGEST model comes with a severe cost in terms of energy. Namely, there are problems requiring $\Omega(\mbox{poly}(n))$ edge-activations (and thus $\Omega(\mbox{poly}(n))$ node-activations as well) in the CONGEST model whenever solved by algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Finally, we demonstrate the existence of a sharp separation between the edge-activation complexity and the node-activation complexity in the CONGEST model, for algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Specifically, under this constraint, there is a problem with $O(1)$ edge-activation complexity but $\tilde{\Omega}(n^{1/4})$ node-activation complexity.

翻译：我们研究LOCAL和CONGEST模型中能效算法的设计。具体而言，作为复杂度度量，我们考虑算法执行过程中边或节点处于活跃状态的轮次数在所有边或所有节点上的最大值。首先证明：每个图灵可计算问题均存在具有常数节点激活复杂度的CONGEST算法，因而边激活复杂度也为常数。这意味着每个节点（或边）在整个算法执行过程中仅需活跃$O(1)$轮来发送（或传输）消息。换言之，每个图灵可计算问题都能通过消耗最低可能能量的算法求解。在LOCAL模型中，该结论显然成立，且额外具有在$n$节点网络上以$O(\mbox{poly}(n))$轮运行的特性。然而，我们证明在CONGEST模型中坚持使用$O(\mbox{poly}(n))$轮运行的算法将导致严重的能量代价：存在某些问题在CONGEST模型中需要$\Omega(\mbox{poly}(n))$次边激活（因而也需要$\Omega(\mbox{poly}(n))$次节点激活），才能以$O(\mbox{poly}(n))$轮运行的算法求解。最后，我们证明在限制为$O(\mbox{poly}(n))$轮运行的CONGEST模型中，边激活复杂度与节点激活复杂度之间存在显著分离：在该约束下，存在一个具有$O(1)$边激活复杂度但具有$\tilde{\Omega}(n^{1/4})$节点激活复杂度的问题。