[Paper Review] Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Mohsen Ghaffari, Julian Portmann|arXiv (Cornell University)|Aug 9, 2019

Complexity and Algorithms in Graphs9 citations

TL;DR

This paper presents a deterministic distributed algorithm for constructing network decompositions in the CONGEST model using small messages, achieving improved round complexity and enabling faster randomized and deterministic algorithms for key problems like Maximal Independent Set (MIS), spanners, and neighborhood covers. The key contribution is a novel recursive clustering approach with meta-node abstraction that reduces dependency on large-message assumptions, yielding sub-polynomial round complexity with high probability.

ABSTRACT

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].

Motivation & Objective

• To develop a deterministic distributed algorithm for network decomposition in the CONGEST model that uses only small messages (O(log n)-bit per round).
• To improve the round complexity of existing deterministic network decomposition algorithms while maintaining the same quality of decomposition.
• To enable faster randomized and deterministic algorithms for fundamental distributed problems such as MIS, spanner construction, and neighborhood covers.
• To resolve the open problem of deterministic sparse neighborhood cover construction with small messages.

Proposed method

• Introduce a recursive clustering technique using meta-nodes to represent clusters of nodes, reducing the problem size at each level.
• Use a hierarchical decomposition where meta-nodes are grouped into super-clusters based on ruling sets and local clustering rules.
• Apply a randomized execution strategy with multiple parallel runs per cluster, using one-bit messages to check success conditions.
• Use convergecast and local verification to detect successful runs and coordinate cluster formation across the network.
• Leverage the network decomposition to process components in parallel across color classes, ensuring non-adjacency between clusters of the same color.
• Combine the decomposition with existing MIS algorithms (e.g., Ghaffari [Gha16]) in a pipelined fashion across color classes to achieve high-probability success.

Experimental results

Research questions

• RQ1 Can we construct a deterministic network decomposition in the CONGEST model with small messages and improved round complexity compared to prior work?
• RQ2 Can we achieve a randomized network decomposition with exponentially small failure probability using only small messages?
• RQ3 Can the improved network decomposition be used to accelerate known distributed algorithms for MIS, spanners, and dominating set approximation?
• RQ4 Can we resolve the open problem of deterministic sparse neighborhood cover construction in the CONGEST model with small messages?
• RQ5 Does the recursive meta-node abstraction allow for better scalability and lower round complexity in distributed graph algorithms?

Key findings

• The paper presents a deterministic network decomposition algorithm with round complexity 2^O(√(log log n)) for (O(√(log log n)), 2^O(√(log log n))) decompositions, improving over prior work.
• The randomized variant achieves success probability 1 - 1/poly(n) with the same round complexity, outperforming Elkin and Neiman [PODC’16] in success probability.
• A faster randomized MIS algorithm is obtained with round complexity O(log Δ · √(log log n)) + 2^O(√(log log n)), improving on Ghaffari [SODA’19].
• The first deterministic algorithm for sparse neighborhood cover construction in the CONGEST model is presented, resolving an open question from Elkin [SODA’04].
• Improved deterministic algorithms for spanner construction and dominating set approximation are achieved, matching or improving upon prior bounds from Ghaffari and Kuhn [DISC’18] and Deurer et al. [PODC’19].
• The method enables efficient parallel processing across color classes, with each cluster’s MIS computation running independently and in parallel, ensuring correctness and high success probability.