Log Diameter Rounds MST Verification and Sensitivity in MPC

TL;DR

This work studies two natural MST problems in the low-space MPC model: MST verification and MST sensitivity, relative to a candidate MST $T$ with diameter $D_T$. The authors develop a hierarchical clustering framework that contracts the input tree while preserving critical information, enabling $O(\log D_T)$ rounds with $s=O(n^\delta)$ local memory and linear global memory $g=O(m+n)$, and they also handle all-edge LCA computations to support necessary data flow. The MST verification algorithm demonstrates correctness and runs in $O(\log D_T)$ rounds with high probability, and the MST sensitivity algorithm extends the same round bound while preserving optimal global memory; both results are shown to be optimal up to the conditional 1-vs-2-cycle lower bound. Collectively, this work narrows the gap between achievable MPC round complexity and the known lower bounds for MST-related problems in the low-space regime, and introduces a reusable clustering approach that may generalize to other tree-structured computations on MPC.

Abstract

We consider two natural variants of the problem of minimum spanning tree (MST) of a graph in the parallel setting: MST verification (verifying if a given tree is an MST) and the sensitivity analysis of an MST (finding the lowest cost replacement edge for each edge of the MST). These two problems have been studied extensively for sequential algorithms and for parallel algorithms in the PRAM model of computation. In this paper, we extend the study to the standard model of Massive Parallel Computation (MPC). It is known that for graphs of diameter $D$, the connectivity problem can be solved in $O(\log D + \log\log n)$ rounds on an MPC with low local memory (each machine can store only $O(n^δ)$ words for an arbitrary constant $δ> 0$) and with linear global memory, that is, with optimal utilization. However, for the related task of finding an MST, we need $Ω(\log D_{\text{MST}})$ rounds, where $D_{\text{MST}}$ denotes the diameter of the minimum spanning tree. The state of the art upper bound for MST is $O(\log n)$ rounds; the result follows by simulating existing PRAM algorithms. While this bound may be optimal for general graphs, the benchmark of connectivity and lower bound for MST suggest the target bound of $O(\log D_{\text{MST}})$ rounds, or possibly $O(\log D_{\text{MST}} + \log\log n)$ rounds. As for now, we do not know if this bound is achievable for the MST problem on an MPC with low local memory and linear global memory. In this paper, we show that two natural variants of the MST problem: MST verification and sensitivity analysis of an MST, can be completed in $O(\log D_T)$ rounds on an MPC with low local memory and with linear global memory; here $D_T$ is the diameter of the input ``candidate MST'' $T$. The algorithms asymptotically match our lower bound, conditioned on the 1-vs-2-cycle conjecture.

Figures (2)

• Figure 1: A depiction of our hierarchical clustering.
• Figure 2: A depiction of cases 1, 2, 4, and 5 of the sensitivity contraction process (\ref{['def:sensitivity_contraction_process']}). (Case 3 omitted for brevity.) The orange clusters $c$ (senior) and $c_1$ (junior) are being contracted to form the blue cluster $c'$. The dashed edge is the relevant non-tree edge. In cases 4 and 5, we replace the non-tree edge with a root-to-leaf note (the dashed, green line) covering part of the non-tree edge inside $c'$, and shorten the remaining edge (the solid, green line).

Theorems & Definitions (62)

• Definition 1.1: $\textrm{MST}$ Verification
• Definition 1.2: $\textrm{MST}$ Sensitivity
• Definition 2.1
• Remark 2.2
• proof
• Remark 2.3
• proof
• Remark 2.4
• proof
• Definition 2.5: Clusters