The Energy Complexity of Diameter and Minimum Cut Computation in Bounded-Genus Networks

Abstract

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 $$\varOmega (n)$$ 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.