What Can We Compute in a Single Round of the Congested Clique?

TL;DR

The paper proves a fundamental bandwidth limitation for computing an MST in a single round of the unicast congested clique: any randomized one-round MST algorithm that succeeds with probability at least 3/4 must use at least $\\Omega(\\log^3 n)$ bits on each link, under the KT_1 and shared randomness assumptions. Consequently, MST computation requires at least two rounds when per-message bandwidth is restricted to $O(\\log n)$ bits. The proof reduces the one-round problem to a SKETCH-model connectivity problem via a specially constructed hard graph $H$ containing a subgraph $G$ built from blocks, enabling a tight lower bound of $\\Omega(\\log^3 n)$ on the per-link bandwidth. This work highlights a clear tradeoff between rounds and bandwidth in the congested clique, and motivates further study of two-round algorithms and connectivity with partial outputs.

Abstract

We show that any one-round algorithm that computes a minimum spanning tree (MST) in the unicast congested clique must use a link bandwidth of $Ω(\log^3 n)$ bits in the worst case. Consequently, computing an MST under the standard assumption of $O(\log n)$-size messages requires at least $2$ rounds. This is the first round complexity lower bound in the unicast congested clique for a problem where the output size is small, i.e., $O(n\log n)$ bits. Our lower bound holds as long as every edge of the MST is output by an incident node. To the best of our knowledge, all prior lower bounds for the unicast congested clique either considered problems with large output sizes (e.g., triangle enumeration) or required every node to learn the entire output.

Figures (1)

• Figure 1: The hard input graph $H$ on which computing an MST requires messages of $\Omega(\log^3 n)$ bits for one-round algorithms in the unicast congested clique. The green-shaded regions are the blocks $G_1$, $G_2$, and $G_3$, which form the lower bound graph $G$ of Y-soda21. (Note that the figure shows a simplified version of this graph where we omitted the parts that are not relevant for understanding our proof.) A snake-shaped line corresponds to a path of $0$-weight edges. The orange vertices and edges are added to $G$ to form graph $H$. In each block $G_i$, vertex $s_1^{(i)}$ is either connected to $t_1^{(i)}$ or $t_2^{(i)}$. As shown in Y-soda21, the graph induced by these blocks (i.e. without $a$ and $b$) is connected if and only if the latter case happens in an odd number of blocks. Since all edges except $\{ a,b \}$ have weight $0$, the MST of $H$ will contain $\{ a,b \}$ if and only if the subgraph induced by $G_1$, $G_2$, and $G_3$ is disconnected.

Theorems & Definitions (2)

• theorem 1
• lemma 1: Y-soda21