[Paper Review] Fast, Responsive Decentralised Graph Colouring

Alessandro Checco, Douglas J. Leith|arXiv (Cornell University)|May 27, 2014

Optimization and Search Problems19 references3 citations

TL;DR

This paper proposes a novel, fully decentralized graph coloring algorithm called Fast Communication-Free Learning (FCFL) that achieves fast convergence to a proper coloring without message passing. It uses local sensing and stochastic learning with a novel memory mechanism to ensure responsiveness to topology changes, proving convergence in O(N log N) time with high probability when the number of colors exceeds the maximum degree Δ.

ABSTRACT

We solve, in a fully decentralised way (\ie with no message passing), the classic problem of colouring a graph. We propose a novel algorithm that is automatically responsive to topology changes, and we prove that it converges quickly to a proper colouring in $O(N\log{N})$ time with high probability for generic graphs (and in $O(\log{N})$ time if $Δ=O(1)$) when the number of available colours is greater than $Δ$, the maximum degree of the graph. We believe the proof techniques used in this work are of independent interest and provide new insight into the properties required to ensure fast convergence of decentralised algorithms.

Motivation & Objective

• To address the lack of provably fast decentralized graph coloring algorithms in dynamic, communication-constrained networks.
• To design a fully decentralized algorithm that automatically responds to topology changes without message passing.
• To provide theoretical convergence bounds for decentralized stochastic learning algorithms in graph coloring.
• To develop a lightweight algorithm suitable for implementation on resource-constrained devices like RFID tags.
• To identify key algorithmic components enabling fast convergence in decentralized settings.

Proposed method

• Proposes a novel decentralized graph coloring algorithm, FCFL, that uses local sensing and stochastic learning without message passing.
• Introduces a new type of 'memory' in the learning process to improve convergence speed.
• Employs a uniform random number generator for color selection and state transitions.
• Uses a superframe-based structure with periodic resets (via QueryAdjust commands) to enable responsiveness to topology changes.
• Applies a novel stochastic drift analysis to prove convergence bounds.
• Simplifies the algorithm for practical deployment on RFID tags, using only integer arithmetic and minimal memory.

Experimental results

Research questions

• RQ1 Can a decentralized graph coloring algorithm achieve provably fast convergence without message passing?
• RQ2 What algorithmic components are essential for ensuring fast convergence in decentralized stochastic learning?
• RQ3 How can a decentralized algorithm remain responsive to dynamic topology changes without centralized coordination?
• RQ4 What is the theoretical convergence time of such algorithms under realistic network constraints?
• RQ5 How do algorithm parameters affect convergence speed and stability?

Key findings

• The FCFL algorithm converges to a proper coloring in O(N log N) time with high probability for generic graphs when the number of available colors exceeds the maximum degree Δ.
• For graphs with bounded degree (∆ = O(1)), the algorithm converges in O(log N) time.
• The algorithm is provably fast, overcoming the exponential convergence time of prior Communication-Free Learning (CFL) approaches.
• The algorithm is highly efficient for resource-constrained devices, requiring only integer arithmetic, no floating-point operations, and minimal memory.
• In RFID applications, FCFL reduces tag reading time by up to 83% compared to classic slotted Aloha and 66% compared to state-of-the-art dynamic Aloha.
• Measured experiments show FCFL achieves full convergence in under 5 minutes for 1000 tags, with steady-state reading time of just 7 seconds versus 32 seconds for slotted Aloha.