On the geometry of wireless network multicast in 2-D

TL;DR

This paper presents a geometric approach to optimally position a relay in a low-SNR wireless multicast network to maximize rate, decoupling flow optimization from relay placement and achieving near cut-set bound performance.

Contribution

It introduces a geometric solution for relay placement that simplifies multicast rate maximization in low-SNR networks, with algorithms for optimal positioning.

Findings

Relay position depends on a single path, regardless of receiver count.

Optimal relay placement nearly achieves the cut-set bound.

The method applies across various power levels and path loss exponents.

Abstract

We provide a geometric solution to the problem of optimal relay positioning to maximize the multicast rate for low-SNR networks. The networks we consider, consist of a single source, multiple receivers and the only intermediate and locatable node as the relay. We construct network the hypergraph of the system nodes from the underlying information theoretic model of low-SNR regime that operates using superposition coding and FDMA in conjunction (which we call the "achievable hypergraph model"). We make the following contributions. 1) We show that the problem of optimal relay positioning maximizing the multicast rate can be completely decoupled from the flow optimization by noticing and exploiting geometric properties of multicast flow. 2) All the flow maximizing the multicast rate is sent over at most two paths, in succession. The relay position is dependent only on one path (out of the two), irrespective of the number of receiver nodes in the system. Subsequently, we propose simple and efficient geometric algorithms to compute the optimal relay position. 3) Finally, we show that in our model at the optimal relay position, the difference between the maximized multicast rate and the cut-set bound is minimum. We solve the problem for all (Ps,Pr) pairs of source and relay transmit powers and the path loss exponent \alpha greater than 2.