MST-Based Construction Algorithm

• MST-based construction algorithm is a graph method that leverages Minimum Spanning Trees to ensure globally minimal connectivity and decompose complex structures.
• It is applied in distributed and parallel computation to reduce network diameter and communication overhead through strategic sparsification and asynchronous merging.
• Key contributions include achieving linear time performance and sublinear message complexity in dense graphs by using two-phase sparsification techniques.

A Minimum Spanning Tree (MST)-based construction algorithm is any algorithm whose core workflow, structural decomposition, or application leverages the existence or properties of an MST within a graph or a metric space. Across distributed computation, self-stabilizing protocols, parallel and approximate methods, and machine learning, MST-based construction is a central design primitive for ensuring globally minimal connectivity, decomposing complex structures, and enabling efficient hierarchical analysis.

1. Distributed MST Construction: Sublinear Communication and Time

In distributed networks, particularly under asynchronous and bandwidth-limited models such as CONGEST, MST-based construction algorithms focus on minimizing both time and message complexity. The key modern approach is to reduce the effective density or diameter of the network prior to executing MST routines, exploiting the geometric and combinatorial structure found in the MST subgraph.

A seminal result in this area is the use of two-phase sparsification and MST computation. The algorithm in "Faster asynchronous MST and low diameter tree construction with sublinear communication" constructs an MST in the asynchronous KT$_1$ CONGEST model via the following main stages (Mashreghi et al., 2019):

• Phase A: Sparsification
  • Nodes are partitioned into “high-degree” and “low-degree” by a threshold $\Delta=n^{1/2+\epsilon}$.
  • “Star” nodes are sampled with probability $p=\Theta((n^{1/2-\epsilon}\log n)^{-1})$ such that, with high probability, every high-degree node is adjacent to at least one star.
  • The subgraph $G'$ induced by high-degree nodes and stars undergoes an asynchronous Borůvka-style merging process to create a forest of bounded total diameter.
  • The final sparse subgraph $S$ includes all edges in this forest and all edges incident to any low-degree node.
• Phase B: MST on Sparse Subgraph
  • Any asynchronous MST construction algorithm can be applied to $S$.
  • Running time: $$O(n^{1-2\epsilon} + T(n, \tilde O(n^{3/2+\epsilon})))$$, where $T(n, m)$ is the time complexity of the chosen MST routine.
  • Messages: $$\tilde O(n^{3/2+\epsilon} + M(n, \tilde O(n^{3/2+\epsilon})))$$.

By setting $\epsilon=0$, one obtains $O(n)$ time and $\tilde O(n^{3/2})$ messages, the first asynchronous MST algorithm achieving both linear time and sublinear ($o(m)$) communication cost for sufficiently dense graphs. All steps leverage the backbone provided by the MST structure, reducing the search and communication space for subsequent merging and fragment identification.

Table 1: Time–Message Tradeoffs for Asynchronous MST Construction

$\epsilon$	Time Complexity	Message Complexity
$0$	$O(n)$	$\tilde O(n^{3/2})$
$>0$ (if possible)	$O(n^{1-2\epsilon})$ (potentially $o