Broadcast Throughput in Radio Networks: Routing vs. Network Coding
  The broadcast throughput in a network is defined as the average number of messages that can be transmitted per unit time from a given source to all other nodes when time goes to infinity.   Classical broadcast algorithms treat messages as atomic tokens and route them from the source to the receivers by making intermediate nodes store and forward messages. The more recent network coding approach, in contrast, prompts intermediate nodes to mix and code together messages. It has been shown that certain wired networks have an asymptotic network coding gap, that is, they have asymptotically higher broadcast throughput when using network coding compared to routing. Whether such a gap exists for wireless networks has been an open question of great interest. We approach this question by studying the broadcast throughput of the radio network model which has been a standard mathematical model to study wireless communication.   We show that there is a family of radio networks with a tight $\Theta(\log \log n)$ network coding gap, that is, networks in which the asymptotic throughput achievable via routing messages is a $\Theta(\log \log n)$ factor smaller than that of the optimal network coding algorithm. We also provide new tight upper and lower bounds that show that the asymptotic worst-case broadcast throughput over all networks with $n$ nodes is $\Theta(1 / \log n)$ messages-per-round for both routing and network coding. 
Noga Alon, Mohsen Ghaffari, Bernhard Haeupler, Majid Khabbazian
cs.DS cs.DC math.CO