This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only $\text{poly} \log n$ energy. However, there exist some problems, such as computing the diameter of the graph, that require $\Omega(n)$ energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a $(1 \pm\epsilon)$-approximate $s$-$t$ minimum cut size with $\tilde{O}(\sqrt{n})$ energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

翻译：本文研究了多跳无线电网络中分布式图问题的能量复杂度，其中算法的能量成本由顶点最大唤醒轮数衡量。近期研究表明，部分问题（如广播、广度优先搜索和最大匹配）可通过能量高效算法解决，仅需消耗$\text{poly} \log n$的能量。然而，某些问题（如图的直径计算）需要$\Omega(n)$的能量才能求解。为提升这类问题的能量效率，我们聚焦于特殊图类——有界亏格图。针对有界亏格图，我们提出了精确直径、精确全局最小割大小以及$(1 \pm\epsilon)$-近似$s$-$t$最小割大小的算法，其能量消耗为$\tilde{O}(\sqrt{n})$。我们的方法基于一个通用框架，将顶点集划分为高度数与低度数两部分，并利用有界亏格图的结构性质，控制由低度数部分诱导子图中特定连通分量的数量。