|
| 1 | +using LightGraphs |
| 2 | + |
| 3 | +""" |
| 4 | + color_graph(g::LightGraphs, ::BacktrackingColor) |
| 5 | +
|
| 6 | +Returns a tight, distance-1 coloring of graph g |
| 7 | +using the minimum number of colors possible (i.e. |
| 8 | +the chromatic number of graph, χ(g)) |
| 9 | +""" |
| 10 | +function color_graph(g::LightGraphs.AbstractGraph, ::BacktrackingColor) |
| 11 | + v = nv(g) |
| 12 | + |
| 13 | + #A is list of vertices in non-increasing order of degree |
| 14 | + A = sort_by_degree(g) |
| 15 | + |
| 16 | + #F is the coloring of vertices, 0 means uncolored |
| 17 | + #Fopt is the optimal coloring of the graph |
| 18 | + F = zeros(Int, v) |
| 19 | + Fopt= zeros(Int, v) |
| 20 | + |
| 21 | + start = 1 |
| 22 | + |
| 23 | + #optimal color number |
| 24 | + opt = v + 1 |
| 25 | + |
| 26 | + #current vertex to be colored |
| 27 | + x = A[1] |
| 28 | + |
| 29 | + #colors[j] = number of colors in A[0]...A[j] |
| 30 | + #assume colors[0] = 1 |
| 31 | + colors = zeros(Int, v) |
| 32 | + |
| 33 | + #set of free colors |
| 34 | + U = zeros(Int, 0) |
| 35 | + push!(U, 1) |
| 36 | + |
| 37 | + #set of free colors of x |
| 38 | + freeColors = [Vector{Int}() for _ in 1:v] |
| 39 | + freeColors[x] = copy(U) |
| 40 | + |
| 41 | + while (start >= 1) |
| 42 | + |
| 43 | + back = false |
| 44 | + for i = start:v |
| 45 | + if i > start |
| 46 | + x = uncolored_vertex_of_maximal_degree(A,F) |
| 47 | + U = free_colors(x, A, colors, F, g, opt) |
| 48 | + sort!(U) |
| 49 | + end |
| 50 | + if length(U) > 0 |
| 51 | + k = U[1] |
| 52 | + F[x] = k |
| 53 | + cp = F[x] |
| 54 | + deleteat!(U, 1) |
| 55 | + freeColors[x] = copy(U) |
| 56 | + if i==1 |
| 57 | + l = 0 |
| 58 | + else |
| 59 | + l = colors[i-1] |
| 60 | + end |
| 61 | + colors[i] = max(k, l) |
| 62 | + else |
| 63 | + start = i-1 |
| 64 | + back = true |
| 65 | + break |
| 66 | + end |
| 67 | + end |
| 68 | + |
| 69 | + if back |
| 70 | + if start >= 1 |
| 71 | + x = A[start] |
| 72 | + F[x] = 0 |
| 73 | + U = freeColors[x] |
| 74 | + end |
| 75 | + else |
| 76 | + Fopt = copy(F) |
| 77 | + opt = colors[v-1] |
| 78 | + i = least_index(F,A,opt) |
| 79 | + start = i-1 |
| 80 | + if start < 1 |
| 81 | + break |
| 82 | + end |
| 83 | + |
| 84 | + #uncolor all vertices A[i] with i >= start |
| 85 | + uncolor_all!(F, A, start) |
| 86 | + |
| 87 | + for i = 1:start+1 |
| 88 | + x = A[i] |
| 89 | + U = freeColors[x] |
| 90 | + |
| 91 | + #remove colors >= opt from U |
| 92 | + U = remove_higher_colors(U, opt) |
| 93 | + freeColors[x] = copy(U) |
| 94 | + end |
| 95 | + end |
| 96 | + end |
| 97 | + return Fopt |
| 98 | +end |
| 99 | + |
| 100 | +""" |
| 101 | + sort_by_degree(g::LightGraphs.AbstractGraph) |
| 102 | +
|
| 103 | +Returns a list of the vertices of graph g sorted |
| 104 | +in non-increasing order of their degrees |
| 105 | +""" |
| 106 | +function sort_by_degree(g::LightGraphs.AbstractGraph) |
| 107 | + vs = vertices(g) |
| 108 | + degrees = (LightGraphs.degree(g, v) for v in vs) |
| 109 | + vertex_pairs = collect(zip(vs, degrees)) |
| 110 | + sort!(vertex_pairs, by = p -> p[2], rev = true) |
| 111 | + return [v[1] for v in vertex_pairs] |
| 112 | +end |
| 113 | + |
| 114 | +""" |
| 115 | + uncolored_vertex_of_maximal_degree(A::AbstractVector{<:Integer},F::AbstractVector{<:Integer}) |
| 116 | +
|
| 117 | +Returns an uncolored vertex from the partially |
| 118 | +colored graph which has the highest degree |
| 119 | +""" |
| 120 | +function uncolored_vertex_of_maximal_degree(A::AbstractVector{<:Integer},F::AbstractVector{<:Integer}) |
| 121 | + for v in A |
| 122 | + if F[v] == 0 |
| 123 | + return v |
| 124 | + end |
| 125 | + end |
| 126 | +end |
| 127 | + |
| 128 | + |
| 129 | +""" |
| 130 | + free_colors(x::Integer, |
| 131 | + A::AbstractVector{<:Integer}, |
| 132 | + colors::AbstractVector{<:Integer}, |
| 133 | + F::Array{Integer,1}, |
| 134 | + g::LightGraphs.AbstractGraph, |
| 135 | + opt::Integer) |
| 136 | +
|
| 137 | +Returns set of free colors of x which are less |
| 138 | +than optimal color number (opt) |
| 139 | +
|
| 140 | +Arguments: |
| 141 | +
|
| 142 | +x: Vertex who's set of free colors is to be calculated |
| 143 | +A: List of vertices of graph g sorted in non-increasing order of degree |
| 144 | +colors: colors[i] stores the number of distinct colors used in the |
| 145 | + coloring of vertices A[0], A[1]... A[i-1] |
| 146 | +F: F[i] stores the color of vertex i |
| 147 | +g: Graph to be colored |
| 148 | +opt: Current optimal number of colors to be used in the coloring of graph g |
| 149 | +""" |
| 150 | +function free_colors(x::Integer, |
| 151 | + A::AbstractVector{<:Integer}, |
| 152 | + colors::AbstractVector{<:Integer}, |
| 153 | + F::Array{Integer,1}, |
| 154 | + g::LightGraphs.AbstractGraph, |
| 155 | + opt::Integer) |
| 156 | + index = -1 |
| 157 | + |
| 158 | + freecolors = zeros(Int, 0) |
| 159 | + |
| 160 | + for i in eachindex(A) |
| 161 | + if A[i] == x |
| 162 | + index = i |
| 163 | + break |
| 164 | + end |
| 165 | + end |
| 166 | + |
| 167 | + if index == 1 |
| 168 | + colors_used = 0 |
| 169 | + else |
| 170 | + colors_used = colors[index-1] |
| 171 | + end |
| 172 | + |
| 173 | + colors_used += 1 |
| 174 | + for c = 1:colors_used |
| 175 | + c_allowed = true |
| 176 | + for w in inneighbors(g, x) |
| 177 | + if F[w] == c |
| 178 | + c_allowed = false |
| 179 | + break |
| 180 | + end |
| 181 | + end |
| 182 | + if c_allowed && c < opt |
| 183 | + push!(freecolors, c) |
| 184 | + end |
| 185 | + end |
| 186 | + |
| 187 | + return freecolors |
| 188 | + |
| 189 | +end |
| 190 | + |
| 191 | +""" |
| 192 | + least_index(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, opt::Integer) |
| 193 | +
|
| 194 | +Returns least index i such that color of vertex |
| 195 | +A[i] is equal to `opt` (optimal color number) |
| 196 | +""" |
| 197 | +function least_index(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, opt::Integer) |
| 198 | + for i in eachindex(A) |
| 199 | + if F[A[i]] == opt |
| 200 | + return i |
| 201 | + end |
| 202 | + end |
| 203 | +end |
| 204 | + |
| 205 | +""" |
| 206 | + uncolor_all(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, start::Integer) |
| 207 | +
|
| 208 | +Uncolors all vertices A[i] where i is |
| 209 | +greater than or equal to start |
| 210 | +""" |
| 211 | +function uncolor_all!(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, start::Integer) |
| 212 | + for i = start:length(A) |
| 213 | + F[A[i]] = 0 |
| 214 | + end |
| 215 | +end |
| 216 | + |
| 217 | +""" |
| 218 | + remove_higher_colors(U::AbstractVector{<:Integer}, opt::Integer) |
| 219 | +
|
| 220 | +Remove all the colors which are greater than or |
| 221 | +equal to the `opt` (optimal color number) from |
| 222 | +the set of colors U |
| 223 | +""" |
| 224 | +function remove_higher_colors(U::AbstractVector{<:Integer}, opt::Integer) |
| 225 | + if length(U) == 0 |
| 226 | + return U |
| 227 | + end |
| 228 | + u = zeros(Int, 0) |
| 229 | + for color in U |
| 230 | + if color < opt |
| 231 | + push!(u, color) |
| 232 | + end |
| 233 | + end |
| 234 | + return u |
| 235 | +end |
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