Cost for new method

There are now $\log_2 N$ pairwise communication phases in which packets of size $n u / N$ are exchanged.

$$\tau_{\rm new} = \log_2 N \left[ \alpha + {n u \over \beta N} \right]$$

• Fewer packets exchanged $\Rightarrow$ lower latency cost.
• Larger packets exchanged $\Rightarrow$ higher transmission cost.

Expect the new method to be advantageous in the limits of small $n$ and large $N$.

Exact cross-over depends upon the machine: the product $\alpha \beta$ determines the packet size which costs as much in latency as transmission to send.