It has been shown that one can design distributed algorithms that are (nearly) singularly optimal, meaning they simultaneously achieve optimal time and message complexity (within polylogarithmic factors), for several fundamental global problems such as broadcast, leader election, and spanning tree construction, under the $\text{KT}_0$ assumption. With this assumption, nodes have initial knowledge only of themselves, not their neighbors. In this case the time and message lower bounds are $\Omega(D)$ and $\Omega(m)$, respectively, where $D$ is the diameter of the network and $m$ is the number of edges, and there exist (even) deterministic algorithms that simultaneously match these bounds. On the other hand, under the $\text{KT}_1$ assumption, whereby each node has initial knowledge of itself and the identifiers of its neighbors, the situation is not clear. For the $\text{KT}_1$ CONGEST model (where messages are of small size), King, Kutten, and Thorup (KKT) showed that one can solve several fundamental global problems (with the notable exception of BFS tree construction) such as broadcast, leader election, and spanning tree construction with $\tilde{O}(n)$ message complexity ($n$ is the network size), which can be significantly smaller than $m$. Randomization is crucial in obtaining this result. While the message complexity of the KKT result is near-optimal, its time complexity is $\tilde{O}(n)$ rounds, which is far from the standard lower bound of $\Omega(D)$. In this paper, we show that in the $\text{KT}_1$ LOCAL model (where message sizes are not restricted), singular optimality is achievable. Our main result is that all global problems, including BFS tree construction, can be solved in $\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages, where both bounds are optimal up to polylogarithmic factors. Moreover, we show that this can be achieved deterministically.

翻译：已有研究表明，在$\text{KT}_0$假设下，可以为广播、领导者选举和生成树构造等若干基础全局问题设计（近似）奇点最优的分布式算法，即算法能同时达到最优时间复杂度和最优消息复杂度（在多项式对数因子内）。在此假设下，节点仅具有关于自身的初始知识，而不知其邻居信息。此时时间下界和消息下界分别为$\Omega(D)$和$\Omega(m)$，其中$D$为网络直径，$m$为边数，且存在（甚至确定性的）算法能同时匹配这些界。另一方面，在$\text{KT}_1$假设下（每个节点具有自身及其邻居标识符的初始知识），情况则不甚明确。对于$\text{KT}_1$ CONGEST模型（消息规模受限），King、Kutten和Thorup（KKT）证明了可以通过$\tilde{O}(n)$的消息复杂度（$n$为网络规模）解决若干基础全局问题（广度优先搜索树构造是显著例外），包括广播、领导者选举和生成树构造，该复杂度可能显著小于$m$。随机化对于该结果的获得至关重要。虽然KKT结果的消息复杂度近乎最优，但其时间复杂度为$\tilde{O}(n)$轮，远低于标准下界$\Omega(D)$。本文证明，在$\text{KT}_1$ LOCAL模型（消息规模不受限）中，奇点最优性是可实现的。我们的主要结果表明，所有全局问题（包括广度优先搜索树构造）均可在$\tilde{O}(D)$轮和$\tilde{O}(n)$条消息内解决，这两个界在多项式对数因子内均达到最优。此外，我们证明该结果可通过确定性算法实现。