Towards singular optimality in the presence of local initial knowledge

TL;DR

It is shown that if the authors assume the KT2 CONGEST model, it is possible to substantially improve the time-message trade-off in constructing a danner.

Abstract

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+δ}}) 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+δ}}) messages for any delta in [0,1/2].

Figures (2)

• Figure 2: This figure depicts the Phase 2 of the Cluster Growing algorithm. The left figure shows the situation before growth and the right figure shows the situation after growth. There are three clusters denoted by $C_1$ (top), $C_2$ (left), $C_3$ (right). A red disk marks the center within each cluster. Suppose that $C_1$ is sampled during Phase 2, whereas the remaining two clusters remain non-sampled. Further suppose that $C_2$ is a low-degree cluster and $C_3$ is a high-degree cluster. So we add edges from $C_2$ to all its neighbors (one edge per neighbor) to the danner; these are shown as brown thick edges. The cluster $C_3$ is adjacent to $C_1$ and hears from it via the green edge; this green edge is added both to the cluster $C_1$ and to the danner $H$. Furthermore, cluster $C_3$ joins cluster $C_1$.
• Figure 3: Given $n=16$, construct a graph $G$ with the 5 steps, where $A_1=\{1,2,3,4\},A_2=\{5,6,7,8\}, A_3=\{9,10,11,12\}\text{ and } A_4=\{13,14,15,16\}$. Given such a $G$, we have $\Bar{G}$ after applying AGPV sparsification algorithm awerbuch1990trade.

Theorems & Definitions (37)

• Lemma 3
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• Lemma 4
• proof
• Lemma 5
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• Lemma 6
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• Lemma 7
• proof