AI 中文总结
本研究修正并扩展了DragonFly+网络超额订阅证明，证明2:1比率适用于所有置换模式，并实验发现全局布线在一般情况下的影响。
AI 中文摘要
Max-Host Dragonfly+拓扑的原始论文证明，对于置换流量模式，最坏情况下的超额订阅比率在期望上为2:1。我们表明该证明仅覆盖了置换模式的一个子集，具体而言，是所有主机对位于不同组且对于任何接收组至少有两个发送主机位于不同组的模式。该证明包含两个错误，这两个错误相互抵消从而产生了正确的结果。此外，该证明隐含地使用了一个没有支撑论证的重要观察：离开间接组的流量，现在被迫沿最小成本路径前往接收组，仍然可以几乎分散到所有出站全局链路上，这与来自发送组的流量（可以自然选择任何出站组）的分散方式形成对比。我们证明了对于更大的模式集合（包括任何置换），2:1的比率在期望上成立。我们只需要知道任何主机发送和接收的流量至多为线路速率。这一约束可以被修改以获得任何模式上预期FCT界限的近似值。我们还攻击了没有期望假设的超额订阅问题，并找到了在小交换机端口数下对置换以高概率成立的界限。虽然在期望上全局层布线不影响超额订阅率，但我们通过实验发现，在一般情况下它确实有影响。我们找到了相对于原始论文中的默认全局布线能够获得明显加速的拓扑，在端口数最多为8时加速比低于7%。
英文摘要
The Max-Host Dragonfly+ topology's original paper proves that there is a 2:1 worst-case oversubscription ratio in expectation for the permutation traffic pattern. We show that the proof only covers a subset of permutation patterns, specifically those in which all host pairs are in different groups, and for any receiver group there are at least two sending hosts in distinct groups. The proof contains two mistakes that cancel out to produce the correct result. Furthermore, the proof implicitly uses an important observation without a backing argument: traffic leaving an indirect group, now forced to follow min-cost paths to receiver groups, can still be split across almost all outgoing global links, in contrast with splitting traffic from a sender group that can naturally select any outgoing group. We prove that the 2:1 ratio holds in expectation for a larger set of patterns, including any permutation. We only need to know that any host sends and receives at most line rate traffic. This constraint can be altered to obtain approximations for expected FCT bounds on any pattern. We also attack the oversubscription problem without the expectation assumption, and find bounds that hold with high probability on permutations for small switch radixes. While in expectation the global layer wiring doesn't affect the oversubscription rate, we experimentally find that it matters in the general case. We find topologies that obtain visible speedups against the default global wiring from the original paper, under $7\%$ for radixes at most $8$.
