Główne pojęcia
The author formalizes the mathematical problem of colour-based graph contraction, which aims to efficiently process and analyze large networks by grouping vertices with the same colour into clusters and representing the graph using a smaller set of representative vertices.
Streszczenie
The content provides a formal mathematical definition and analysis of the colour-based graph contraction problem. The key points are:
Graph contraction is a useful technique to compact large graphs while preserving important connectivity information. It involves merging adjacent vertices that share the same colour into a single representative vertex.
The author introduces the concept of "colour-preserving contraction" and defines the "γ-contraction" problem, which aims to contract a vertex-coloured graph by iteratively merging vertices with the same colour.
The author proves that the colour-preserving contraction operation is commutative and associative, allowing for a variadic form that can contract an entire set of vertices at once.
The author introduces the "β-contraction" algorithm, which efficiently computes the γ-contraction of a graph. The algorithm works in two phases:
Evaluation of a colour sub-partition by constructing a forest-like structure where each tree spans a colour cluster.
Concurrent contraction of the colour clusters using the variadic form of the contraction operation.
The author provides a detailed analysis of the algorithm's data structures and implementation, as well as its computational complexity.
The proposed approach allows for a fast and parallelizable solution to the colour-based graph contraction problem, which is useful for managing and extracting insights from large networks.