BDD-Based Algorithm for SCC Decomposition of Edge-Coloured Graphs

TL;DR

This paper introduces a novel BDD-based symbolic algorithm for efficiently detecting monochromatic strongly connected components in large edge-coloured graphs, addressing scalability issues in complex system analysis.

Contribution

The paper presents a new symbolic algorithm for SCC detection in edge-coloured graphs, optimized for large-scale graphs with many colours, and demonstrates its effectiveness through experimental evaluation.

Findings

Algorithm performs $O(p imes n imes log~n)$ symbolic steps in worst case.

Successfully analyzes graphs with up to $2^{48}$ states.

Effective in systems biology applications involving Boolean networks.

Abstract

Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs. The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical. This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p \cdot n \cdot log~n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices. We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a modelling framework commonly appearing in systems biology.