多方指针跳跃的次线性通信协议及其下界
Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound
摘要：
研究了每层有$n$顶点的$k$层指针跳跃问题的单向前额数（NOF）通信复杂性.这个经典问题与复杂性理论的许多方面都有联系，最近出现了大量的研究活动，似乎为常数$k$的$Ω(n)$下限奠定了基础。我们的第一个结果是一个令人惊讶的次线性-即，$o(n)$ --适用于$k \ge 3$的问题的上限，打破了对这样一个下限的希望。仔细观察达到上限的协议，可以发现除了一个参与者之外，所有参与者都崩溃了，即，它们的消息仅取决于它们前面的层的组成。我们考虑协议的指针跳跃问题，所有的球员都崩溃。我们的第二个结果表明，一个强的$n - O(\log n)$下限在这种情况下举行。我们的第三个结果是另一个上界，表明非平凡的协议（非布尔版本）指针跳转是可能的，即使所有的球员都崩溃。我们的下界结果使用了一种新的证明技术，不同于那些早期的下界，有一个信息论的味道。我们希望这对进一步研究这一问题有所帮助。
英文摘要：
We study the one-way number-on-the-forehead (NOF) communication complexity of the $k$-layer pointer jumping problem with $n$ vertices per layer. This classic problem, which has connections to many aspects of complexity theory, has seen a recent burst of research activity, seemingly preparing the ground for an $Ω(n)$ lower bound, for constant $k$. Our first result is a surprising sublinear -- i.e., $o(n)$ -- upper bound for the problem that holds for $k \ge 3$, dashing hopes for such a lower bound. A closer look at the protocol achieving the upper bound shows that all but one of the players involved are collapsing, i.e., their messages depend only on the composition of the layers ahead of them. We consider protocols for the pointer jumping problem where all players are collapsing. Our second result shows that a strong $n - O(\log n)$ lower bound does hold in this case. Our third result is another upper bound showing that nontrivial protocols for (a non-Boolean version of) pointer jumping are possible even when all players are collapsing. Our lower bound result uses a novel proof technique, different from those of earlier lower bounds that had an information-theoretic flavor. We hope this is useful in further study of the problem.
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