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模型中高能效算法的设计。具体而言，作为复杂度度量，我们考虑算法运行中边或节点处于活跃状态的轮次最大值，该最大值取遍所有边或所有节点。首先证明，每个图灵可计算问题都存在常数节点激活复杂度的CONGEST算法，因而也具有常数边激活复杂度。即，在算法整个执行过程中，每个节点（相应地，每条边）仅活跃$O(1)$轮发送（相应地，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，同样结论显然成立，且额外特性在于算法在$n$节点网络中运行$O(\mbox{poly}(n))$轮。然而，我们表明在CONGEST模型中坚持算法运行$O(\mbox{poly}(n))$轮会带来严重的能量代价。具体地，当求解被限制在$O(\mbox{poly}(n))$轮内运行的问题时，存在需要$\Omega(\mbox{poly}(n))$次边激活（相应地也需$\Omega(\mbox{poly}(n))$次节点激活）的问题。最后，我们展示在CONGEST模型中，对于被限制在$O(\mbox{poly}(n))$轮内运行的算法，边激活复杂度与节点激活复杂度之间存在显著分化。特别地，在此约束下，存在一个问题具有$O(1)$边激活复杂度但节点激活复杂度为$\tilde{\Omega}(n^{1/4})$。