API

matrix_colors(A, alg::ColoringAlgorithm = GreedyD1Color())

Return the colorvec vector for the matrix A using the chosen coloring algorithm. If a known analytical solution exists, that is used instead. The coloring defaults to a greedy distance-1 coloring.

Note that if A isa SparseMatrixCSC, the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

    _cols_by_rows(rows_index,cols_index)

Returns a vector of rows where each row contains a vector of its column indices.

    _rows_by_cols(rows_index,cols_index)

Returns a vector of columns where each column contains a vector of its row indices.

    color_graph(g::Graphs.AbstractGraphs, ::AcyclicColoring)

Returns a coloring vector following the acyclic coloring rules (1) the coloring corresponds to a distance-1 coloring, and (2) vertices in every cycle of the graph are assigned at least three distinct colors. This variant of coloring is called acyclic since every subgraph induced by vertices assigned any two colors is a collection of trees—and hence is acyclic.

Reference: Gebremedhin AH, Manne F, Pothen A. New Acyclic and Star Coloring Algorithms with Application to Computing Hessians

color_graph(g::Graphs.AbstractGraph, ::BacktrackingColor)

Return a tight, distance-1 coloring of graph g using the minimum number of colors possible (i.e. the chromatic number of graph, χ(g))

    greedy_star1_coloring

Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

1. every pair of adjacent vertices receives distinct colors

(a distance-1 coloring)

1. For any vertex v, any color that leads to a two-colored path

involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

    greedy_star2_coloring

Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

1. every pair of adjacent vertices receives distinct colors

(a distance-1 coloring)

1. For any vertex v, any color that leads to a two-colored path

involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

TODO: add text explaining the difference between star1 and star2

color_graph(G::VSafeGraph,::ContractionColor)

Find a coloring of the graph g such that no two vertices connected by an edge have the same color.

color_graph(g::VSafeGraph, alg::GreedyD1Color)

Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

contract!(g, y, x)

Contract the vertex y to x, both of which belong to graph G, that is delete vertex y and join x with the neighbors of y if they are not already connected with an edge.

    find(w::Integer,
         x::Integer,
         g::Graphs.AbstractGraph,
         two_colored_forest::DisjointSets{<:Integer})

Returns the root of the disjoint set to which the edge connecting vertices w and x in the graph g belongs to

    find_edge(g::Graphs.AbstractGraph, v::Integer, w::Integer)

Returns an integer equivalent to the index of the edge connecting the vertices v and w in the graph g

free_colors(x::Integer,
            A::AbstractVector{<:Integer},
            colors::AbstractVector{<:Integer},
            F::Array{Integer,1},
            g::Graphs.AbstractGraph,
            opt::Integer)

Returns set of free colors of x which are less than optimal chromatic number (opt)

Arguments:

x: Vertex who's set of free colors is to be calculated A: List of vertices of graph g sorted in non-increasing order of degree colors: colors[i] stores the number of distinct colors used in the coloring of vertices A[0], A[1]... A[i-1] F: F[i] stores the color of vertex i g: Graph to be colored opt: Current optimal number of colors to be used in the coloring of graph g

    grow_star!(two_colored_forest::DisjointSets{<:Integer},
                first_neighbor::AbstractVector{<: Tuple{Integer,Integer}},
                v::Integer,
                w::Integer,
                g::Graphs.AbstractGraph,
                color::AbstractVector{<:Integer})

Grow a 2-colored star after assigning a new color to the previously uncolored vertex v, by comparing it with the adjacent vertex w. Disjoint set is used to store stars in sets, which are identified through key edges present in g.

    insert_new_tree!(two_colored_forest::DisjointSets{<:Integer},
                      v::Integer,
                      w::Integer,
                      g::Graphs.AbstractGraph)

creates a new singleton set in the disjoint set 'twocoloredforest' consisting of the edge connecting v and w in the graph g

least_index(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, opt::Integer)

Returns least index i such that color of vertex A[i] is equal to opt (optimal chromatic number)

length_common_neighbor(g, z, x)

Find the number of vertices that share an edge with both the vertices z and x belonging to the graph g.

    matrix2graph(sparse_matrix, [partition_by_rows::Bool=true])

A utility function to generate a graph from input sparse matrix, columns are represented with vertices and 2 vertices are connected with an edge only if the two columns are mutually orthogonal.

Note that the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

max_degree_vertex(G, nn)

Find the vertex in the group nn of vertices belonging to the graph G which has the highest degree.

max_degree_vertex(G)

Find the vertex in graph with highest degree.

    merge_trees!(two_colored_forest::DisjointSets{<:Integer},
                  v::Integer,
                  w::Integer,
                  x::Integer,
                  g::Graphs.AbstractGraph)

Subroutine to merge trees present in the disjoint set which have a common edge.

    min_index(forbidden_colors::AbstractVector{<:Integer}, v::Integer)

Returns min{i > 0 such that forbidden_colors[i] != v}

non_neighbors(G, x)

Find the set of vertices belonging to the graph G which do not share an edge with the vertex x.

    prevent_cycle!(first_visit_to_tree::AbstractVector{<:Tuple{Integer,Integer}},
                    forbidden_colors::AbstractVector{<:Integer},
                    v::Integer,
                    w::Integer,
                    x::Integer,
                    g::Graphs.AbstractGraph,
                    two_colored_forest::DisjointSets{<:Integer},
                    color::AbstractVector{<:Integer})

Subroutine to avoid generation of 2-colored cycle due to coloring of vertex v, which is adjacent to vertices w and x in graph g. Disjoint set is used to store the induced 2-colored subgraphs/trees where the id of set is an integer representing an edge of graph 'g'

remove_higher_colors(U::AbstractVector{<:Integer}, opt::Integer)

Remove all the colors which are greater than or equal to the opt (optimal chromatic number) from the set of colors U

sort_by_degree(g::Graphs.AbstractGraph)

Returns a list of the vertices of graph g sorted in non-increasing order of their degrees

uncolor_all(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, start::Integer)

Uncolors all vertices A[i] where i is greater than or equal to start

uncolored_vertex_of_maximal_degree(A::AbstractVector{<:Integer},F::AbstractVector{<:Integer})

Returns an uncolored vertex from the partially colored graph which has the highest degree

vertex_degree(g, z)

Find the degree of the vertex z which belongs to the graph g.