The Knowledge Till rho CONGEST model is a variant of the classical CONGEST model of distributed computing in which each vertex v has initial knowledge of the radius-rho ball centered at v. The most commonly studied variants of the CONGEST model are KT0 CONGEST in which nodes initially know nothing about their neighbors and KT1 CONGEST in which nodes initially know the IDs of all their neighbors. It has been shown that having access to neighbors' IDs (as in the KT1 CONGEST model) can substantially reduce the message complexity of algorithms for fundamental problems such as BROADCAST and MST. For example, King, Kutten, and Thorup (PODC 2015) show how to construct an MST using just Otilde(n) messages in the KT1 CONGEST model, whereas there is an Omega(m) message lower bound for MST in the KT0 CONGEST model. Building on this result, Gmyr and Pandurangen (DISC 2018) present a family of distributed randomized algorithms for various global problems that exhibit a trade-off between message and round complexity. These algorithms are based on constructing a sparse, spanning subgraph called a danner. Specifically, given a graph G and any delta in [0,1], their algorithm constructs (with high probability) a danner that has diameter Otilde(D + n^{1-delta}) and Otilde(min{m,n^{1+delta}}) edges in Otilde(n^{1-delta}) rounds while using Otilde(min{m,n^{1+\delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. In the main result of this paper, we show that if we assume the KT2 CONGEST model, it is possible to substantially improve the time-message trade-off in constructing a danner. Specifically, we show in the KT2 CONGEST model, how to construct a danner that has diameter Otilde(D + n^{1-2delta}) and Otilde(min{m,n^{1+delta}}) edges in Otilde(n^{1-2delta}) rounds while using Otilde(min{m,n^{1+\delta}}) messages for any delta in [0,1/2].

翻译：知识半径ρ的CONGEST模型是经典分布式计算CONGEST模型的变体，其中每个顶点v具有以v为中心、半径为ρ的球的初始知识。CONGEST模型最常研究的变体是KT0 CONGEST（节点初始时对邻居一无所知）和KT1 CONGEST（节点初始时已知所有邻居的ID）。研究表明，接入邻居ID（如KT1 CONGEST模型）可显著降低BROADCAST和MST等基本问题的算法消息复杂度。例如，King、Kutten与Thorup（PODC 2015）展示了如何在KT1 CONGEST模型中仅用Otilde(n)条消息构建MST，而KT0 CONGEST模型中MST存在Ω(m)消息下界。基于此，Gmyr与Pandurangen（DISC 2018）提出了一系列针对多种全局问题的分布式随机算法，这些算法在消息复杂度和轮复杂度之间呈现权衡关系，其核心是通过构建称为danner的稀疏生成子图实现。具体而言，给定图G和任意δ∈[0,1]，他们的算法能以高概率构建一个danner：该子图直径达Otilde(D + n^{1-δ})、边数达Otilde(min{m,n^{1+δ}})，在Otilde(n^{1-δ})轮内完成，使用Otilde(min{m,n^{1+δ}})条消息，其中n、m、D分别表示G的节点数、边数和直径。本文的主要结果表明，若采用KT2 CONGEST模型，danner构建的时-消息权衡可得到显著改善。具体地，我们在KT2 CONGEST模型中证明：对于任意δ∈[0,1/2]，可构建一个直径达Otilde(D + n^{1-2δ})、边数达Otilde(min{m,n^{1+δ}})的danner，其运行轮数为Otilde(n^{1-2δ})，消息使用量为Otilde(min{m,n^{1+δ}})。