Balls and Funnels: Energy Efficient Group-to-Group Anycasts

TL;DR

This paper introduces the g2g-anycast problem, a versatile network design challenge modeling various communication systems, and provides approximation algorithms with theoretical guarantees for energy-efficient group-to-group communication.

Contribution

The paper formulates the g2g-anycast problem, demonstrates its generality, and develops approximation algorithms with provable bounds for energy-efficient network design.

Findings

An $O( ext{log}^4 n)$ approximation algorithm for general weights.

A scalable $O( ext{log} n)$ approximation algorithm for Euclidean space.

Hardness of approximation results matching the algorithms' bounds.

Abstract

We introduce group-to-group anycast (g2g-anycast), a network design problem of substantial practical importance and considerable generality. Given a collection of groups and requirements for directed connectivity from source groups to destination groups, the solution network must contain, for each requirement, an omni-directional down-link broadcast, centered at any node of the source group, called the ball; the ball must contain some node from the destination group in the requirement and all such destination nodes in the ball must aggregate into a tree directed towards the source, called the funnel-tree. The solution network is a collection of balls along with the funnel-trees they contain. g2g-anycast models DBS (Digital Broadcast Satellite), Cable TV systems and drone swarms. It generalizes several well known network design problems including minimum energy unicast, multicast, broadcast, Steiner-tree, Steiner-forest and Group-Steiner tree. Our main achievement is an $O(\log^4 n)$ approximation, counterbalanced by an $\log^{(2-\epsilon)}n$ hardness of approximation, for general weights. Given the applicability to wireless communication, we present a scalable and easily implemented $O(\log n)$ approximation algorithm, Cover-and-Grow for fixed-dimensional Euclidean space with path-loss exponent at least 2.