[Paper Review] Fast and Simple Deterministic Algorithms for Highly-Dynamic Networks.

Keren Censor-Hillel, Neta Dafni+3 more|arXiv (Cornell University)|Jan 13, 2019

Distributed systems and fault tolerance4 citations

TL;DR

This paper introduces a simple, deterministic algorithm for highly-dynamic networks that maintains solutions to multiple fundamental problems—such as $(degree+1)$-coloring, maximal matching, and 2-approximate minimum weight vertex cover—in constant amortized time per round, even under arbitrary concurrent edge changes. The core contribution is the concept of locally-fixable labelings (LFLs), enabling efficient local repair using only small, logarithmic-sized messages.

ABSTRACT

This paper provides a surprisingly simple method for obtaining fast (constant amortized time) deterministic distributed algorithms for a highly-dynamic setting, in which arbitrarily many edge changes may occur in each round. Among the implications of our results are deterministic algorithms that maintain solutions to many problems, including $(degree+1)$-coloring, maximal matching, maximal independent set and the seemingly unrelated problem of a 2-approximation for minimum weight vertex cover (2-MWVC). These significantly improve upon prior work in various aspects, such as having $O(1)$ amortized round complexity, using message of logarithmic size only, handling arbitrarily many concurrent topology changes, being deterministic, having correctness guarantees for intermediate rounds, and more. The core of our work is in defining a subclass of locally-checkable labelings which we call locally-fixable labelings (LFLs). Very roughly speaking, these are labelings that allow a node to fix its neighborhood based solely on their old labels. We present a simple algorithm for LFLs with small labels, which handles multiple edge insertions/deletions while keeping the amortized round complexity at a small constant. We then extend it, for specific tasks, to handle the insertion and deletion of nodes. Moreover, we show that the same approach can also fix labeling with large labels, given that they can be made to behave as small labels.

Motivation & Objective

• To design deterministic distributed algorithms that maintain correct solutions in highly-dynamic networks with arbitrary, concurrent edge insertions and deletions.
• To achieve constant amortized round complexity for fundamental graph problems in dynamic settings where prior solutions were either randomized or had higher complexity.
• To introduce and formalize the class of locally-fixable labelings (LFLs), which enable efficient local repair of labels based solely on old neighborhood information.
• To extend the framework to handle node insertions and deletions, broadening applicability beyond edge-only changes.
• To show that even labelings with large labels can be efficiently managed if they can be reduced to behave like small-label systems.

Proposed method

• Introduce the class of locally-fixable labelings (LFLs), where a node can correct its local neighborhood using only its own and neighbors' previous labels.
• Design a simple, deterministic algorithm for LFLs that maintains correctness under multiple concurrent edge changes with $O(1)$ amortized round complexity.
• Use small, logarithmic-sized messages to propagate and update labels, ensuring low communication overhead.
• Extend the algorithm to handle node insertions and deletions by adapting the local repair mechanism to dynamic topology changes.
• Show that labelings with large labels can be transformed into equivalent small-label systems through appropriate encoding, preserving efficiency.
• Leverage the local fixability property to ensure correctness guarantees even during intermediate rounds, not just after full stabilization.

Experimental results

Research questions

• RQ1 Can deterministic distributed algorithms maintain solutions to fundamental graph problems in highly-dynamic networks with arbitrary concurrent edge changes?
• RQ2 What structural property of labeling problems enables constant-amortized-time maintenance in dynamic settings?
• RQ3 How can local repair mechanisms be designed to handle multiple simultaneous topology changes while preserving correctness and efficiency?
• RQ4 Can the framework be extended to support node insertions and deletions, not just edge updates?
• RQ5 Is it possible to manage large-label problems efficiently by reducing them to small-label systems?

Key findings

• The paper presents a deterministic algorithm for locally-fixable labelings (LFLs) with $O(1)$ amortized round complexity, enabling fast maintenance of solutions under arbitrary concurrent edge changes.
• The algorithm uses only logarithmic-sized messages, significantly reducing communication overhead compared to prior approaches.
• Correctness is guaranteed not only at stabilization but also during intermediate rounds, enhancing practical reliability.
• The framework supports both edge and node insertions/deletions, extending applicability to fully dynamic networks.
• The approach generalizes to problems with large labels, provided they can be encoded to behave like small-label systems.
• The method achieves deterministic, constant-amortized-time solutions for $(degree+1)$-coloring, maximal matching, maximal independent set, and 2-MWVC, outperforming prior work in multiple dimensions.