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

TL;DR

This paper develops energy-efficient distributed algorithms for computing diameter and minimum cuts in bounded-genus graphs, significantly reducing energy consumption compared to general graphs by exploiting structural properties.

Contribution

It introduces algorithms with $ ilde{O}(\sqrt{n})$ energy complexity for diameter and minimum cut problems specifically in bounded-genus graphs, improving energy efficiency.

Findings

Algorithms achieve $ ilde{O}(\sqrt{n})$ energy for diameter and min cut.

Structural properties of bounded-genus graphs enable energy savings.

Significantly lower energy costs compared to general graph algorithms.

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 $\Omega(n)$ 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.