[Paper Review] Fast Self-Stabilizing Minimum Spanning Tree Construction

Lélia Blin, Shlomi Dolev+2 more|arXiv (Cornell University)|Jun 16, 2010

Distributed systems and fault tolerance8 references4 citations

TL;DR

This paper presents a self-stabilizing minimum spanning tree (MST) algorithm that achieves $O(n^2)$ convergence time with $O(\log^2 n)$ bits of memory per node, improving upon prior self-stabilizing MST solutions by reducing convergence time by a factor of $\Theta(n)$ at the cost of a $O(\log n)$ increase in space complexity. The key innovation is a novel self-stabilizing nearest common ancestor (NCA) labeling scheme using $O(\log^2 n)$ bits, which enables efficient tree merging and correction in the GHS-inspired algorithm.

ABSTRACT

We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The space complexity of our solution is $O(\log^2n)$ bits and it converges in $O(n^2)$ rounds. Thus, this algorithm improves the convergence time of all previously known self-stabilizing asynchronous MST algorithms by a multiplicative factor $Θ(n)$, to the price of increasing the best known space complexity by a factor $O(\log n)$. The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with only $O(\log^2n)$ bits.

Motivation & Objective

• To design a self-stabilizing MST algorithm that is both time-efficient and space-compact for large-scale dynamic networks.
• To eliminate reliance on global parameters such as network size $n$ or diameter $D$, enhancing scalability.
• To extend the Gallager-Humblet-Spira (GHS) algorithm to self-stabilizing settings while preserving its decentralized and fast-converging properties.
• To develop a self-stabilizing nearest common ancestor (NCA) labeling scheme that uses only $O(\log^2 n)$ bits per node, enabling efficient tree operations in the MST construction.

Proposed method

• Leverages a novel self-stabilizing NCA labeling scheme that uses $O(\log^2 n)$ bits per node to compute nearest common ancestors in tree structures.
• Adapts the GHS algorithm for self-stabilizing settings by using the NCA labels to guide tree merging and edge correction phases.
• Employs a three-step merging phase: (1) bottom-up propagation of minimum outgoing edge information, (2) reversal of parent pointers from root to merging point, and (3) relabeling to propagate new tree structure information.
• Uses a correction phase that processes internal edges in order of increasing distance from the root’s NCA, ensuring incorrect edges are detected and removed in $O(h(T_i)^2)$ rounds per tree.
• Relies on the self-stabilizing property of the NCA labeling scheme, which ensures all nodes compute correct labels within $O(n^2)$ rounds from any initial configuration.
• Combines cycle removal and tree correction phases to ensure convergence to a legal MST configuration in $O(n^2)$ rounds.

Experimental results

Research questions

• RQ1 Can the Gallager-Humblet-Spira (GHS) MST algorithm be adapted to self-stabilizing settings without relying on global parameters?
• RQ2 Is it possible to design a self-stabilizing NCA labeling scheme that uses only $O(\log^2 n)$ bits per node?
• RQ3 Can a self-stabilizing MST algorithm achieve both fast convergence ($O(n^2)$ rounds) and compact memory usage ($O(\log^2 n)$ bits)?
• RQ4 How can tree merging and edge correction be efficiently coordinated in a self-stabilizing distributed system using compact labels?

Key findings

• The proposed algorithm converges to a correct minimum spanning tree in $O(n^2)$ rounds from any initial configuration, matching the best-known time complexity among self-stabilizing MST solutions.
• The algorithm uses only $O(\log^2 n)$ bits of memory per node, representing a $O(\log n)$ increase over prior $O(\log n)$-space solutions but enabling significantly faster convergence.
• The self-stabilizing NCA labeling scheme ensures that all nodes compute correct labels within $O(n^2)$ rounds, enabling correct tree operations during merging and correction.
• The merging phase completes in $O(n)$ rounds per merge, and since at most $n$ merges occur, the total time for MST construction is bounded by $O(n^2)$ rounds.
• The correction phase removes all incorrect tree edges in $O(h(T_i)^2)$ rounds per tree, with worst-case $O(n^2)$ rounds across all trees, ensuring correctness of the final MST.
• The solution is fully scalable and does not require knowledge of $n$, $D$, or unique edge weights, making it suitable for large-scale dynamic networks.