Traversals and coloring
Graphs.jl includes various routines for exploring graphs.
Index
Graphs.Coloring
Graphs.all_simple_paths
Graphs.bfs_parents
Graphs.bfs_tree
Graphs.bipartite_map
Graphs.degree_greedy_color
Graphs.dfs_parents
Graphs.dfs_tree
Graphs.diffusion
Graphs.diffusion_rate
Graphs.eulerian
Graphs.gdistances
Graphs.gdistances!
Graphs.greedy_color
Graphs.has_path
Graphs.is_bipartite
Graphs.is_cyclic
Graphs.maximum_adjacency_visit
Graphs.mincut
Graphs.non_backtracking_randomwalk
Graphs.perm_greedy_color
Graphs.random_greedy_color
Graphs.randomwalk
Graphs.self_avoiding_walk
Graphs.topological_sort
Graphs.topological_sort_by_dfs
Graphs.tree
Full docs
Graphs.bfs_parents
— Methodbfs_parents(g, s[; dir=:out])
Perform a breadth-first search of graph g
starting from vertex s
. Return a vector of parent vertices indexed by vertex. If dir
is specified, use the corresponding edge direction (:in
and :out
are acceptable values).
Performance
This implementation is designed to perform well on large graphs. There are implementations which are marginally faster in practice for smaller graphs, but the performance improvements using this implementation on large graphs can be significant.
Graphs.bfs_tree
— Methodbfs_tree(g, s[; dir=:out])
Provide a breadth-first traversal of the graph g
starting with source vertex s
, and return a directed acyclic graph of vertices in the order they were discovered. If dir
is specified, use the corresponding edge direction (:in
and :out
are acceptable values).
Graphs.gdistances!
— Methodgdistances!(g, source, dists; sort_alg=QuickSort)
Fill dists
with the geodesic distances of vertices in g
from source vertex (or collection of vertices) source
. dists
should be a vector of length nv(g)
filled with typemax(T)
. Return dists
.
For vertices in disconnected components the default distance is typemax(T)
.
An optional sorting algorithm may be specified (see Performance section).
Performance
gdistances
uses QuickSort
internally for its default sorting algorithm, since it performs the best of the algorithms built into Julia Base. However, passing a RadixSort
(available via SortingAlgorithms.jl) will provide significant performance improvements on larger graphs.
Graphs.gdistances
— Methodgdistances(g, source; sort_alg=QuickSort)
Return a vector filled with the geodesic distances of vertices in g
from source
. If source
is a collection of vertices each element should be unique. For vertices in disconnected components the default distance is typemax(T)
.
An optional sorting algorithm may be specified (see Performance section).
Performance
gdistances
uses QuickSort
internally for its default sorting algorithm, since it performs the best of the algorithms built into Julia Base. However, passing a RadixSort
(available via SortingAlgorithms.jl) will provide significant performance improvements on larger graphs.
Graphs.has_path
— Methodhas_path(g::AbstractGraph, u, v; exclude_vertices=Vector())
Return true
if there is a path from u
to v
in g
(while avoiding vertices in exclude_vertices
) or u == v
. Return false if there is no such path or if u
or v
is in excluded_vertices
.
Graphs.tree
— Methodtree(parents)
Convert a parents array into a directed graph.
Graphs.bipartite_map
— Methodbipartite_map(g) -> Vector{UInt8}
For a bipartite graph g
, return a vector c
of size $|V|$ containing the assignment of each vertex to one of the two sets ($c_i == 1$ or $c_i == 2$). If g
is not bipartite, return an empty vector.
Implementation Notes
Note that an empty vector does not necessarily indicate non-bipartiteness. An empty graph will return an empty vector but is bipartite.
Examples
julia> using Graphs
julia> g = SimpleGraph(3);
julia> bipartite_map(g)
3-element Vector{UInt8}:
0x01
0x01
0x01
julia> add_vertices!(g, 3);
julia> add_edge!(g, 1, 2);
julia> add_edge!(g, 2, 3);
julia> bipartite_map(g)
6-element Vector{UInt8}:
0x01
0x02
0x01
0x01
0x01
0x01
Graphs.is_bipartite
— Methodis_bipartite(g)
Return true
if graph g
is bipartite.
Examples
julia> using Graphs
julia> g = SimpleGraph(3);
julia> add_edge!(g, 1, 2);
julia> add_edge!(g, 2, 3);
julia> is_bipartite(g)
true
julia> add_edge!(g, 1, 3);
julia> is_bipartite(g)
false
Graphs.dfs_parents
— Methoddfs_parents(g, s[; dir=:out])
Perform a depth-first search of graph g
starting from vertex s
. Return a vector of parent vertices indexed by vertex. If dir
is specified, use the corresponding edge direction (:in
and :out
are acceptable values).
Implementation Notes
This version of DFS is iterative.
Graphs.dfs_tree
— Methoddfs_tree(g, s[;dir=:out])
Provide a depth-first traversal of the graph g
starting with source vertex s
, and return a directed acyclic graph of vertices in the order they were discovered. If dir
is specified, use the corresponding edge direction (:in
and :out
are acceptable values).
Graphs.is_cyclic
— Functionis_cyclic(g)
Return true
if graph g
contains a cycle.
Implementation Notes
The algorithm uses a DFS. Self-loops are counted as cycles.
Graphs.topological_sort
— Functiontopological_sort(g)
Return a topological sort of a directed graph g
as a vector of vertices in topological order.
Implementation Notes
This is currently just an alias for topological_sort_by_dfs
Graphs.topological_sort_by_dfs
— Functiontopological_sort_by_dfs(g)
Return a topological sort of a directed graph g
as a vector of vertices in topological order.
Graphs.diffusion
— Methoddiffusion(g, p, n)
Run diffusion simulation on g
for n
steps with spread probabilities based on p
. Return a vector with the set of new vertices reached at each step of the simulation.
Optional Arguments
initial_infections=sample(vertices(g), 1)
: A list of vertices that
are infected at the start of the simulation.
watch=Vector()
: While simulation is always run on the full graph,
specifying watch
limits reporting to a specific set of vertices reached during the simulation. If left empty, all vertices will be watched.
normalize=false
: iffalse
, set the probability of spread from a vertex $i$ to
each of the outneighbors of $i$ to $p$. If true
, set the probability of spread from a vertex $i$ to each of the outneighbors
of $i$ to $\frac{p}{outdegreee(g, i)}$.
rng=nothing
: A random generator to sample from.
Graphs.diffusion_rate
— Methoddiffusion_rate(results)
diffusion_rate(g, p, n; ...)
Given the results of a diffusion
output or the parameters to the diffusion
simulation itself, (run and) return the rate of diffusion as a vector representing the cumulative number of vertices infected at each simulation step, restricted to vertices included in watch
, if specified.
Graphs.Coloring
— Typestruct Coloring{T}
Store the number of colors used and mapping from vertex to color
Graphs.degree_greedy_color
— Methoddegree_greedy_color(g)
Color graph g
iteratively in the descending order of the degree of the vertices.
Graphs.greedy_color
— Methodgreedy_color(g; sort_degree=false, reps = 1)
Color graph g
based on Greedy Coloring Heuristics
The heuristics can be described as choosing a permutation of the vertices and assigning the lowest color index available iteratively in that order.
If sort_degree
is true then the permutation is chosen in reverse sorted order of the degree of the vertices. parallel
and reps
are irrelevant in this case.
If sort_degree
is false then reps
colorings are obtained based on random permutations and the one using least colors is chosen.
Graphs.perm_greedy_color
— Methodperm_greedy_color(g, seq)
Color graph g
according to an order specified by seq
using a greedy heuristic. seq[i] = v
implies that vertex v is the $i^{th}$ vertex to be colored.
Graphs.random_greedy_color
— Methodrandom_greedy_color(g, reps)
Color the graph g
iteratively in a random order using a greedy heuristic and choose the best coloring out of reps
such random colorings.
Graphs.maximum_adjacency_visit
— Methodmaximum_adjacency_visit(g[, distmx][, log][, io][, s])
maximum_adjacency_visit(g[, s])
Return the vertices in g
traversed by maximum adjacency search, optionally starting from vertex s
(default 1
). An optional distmx
matrix may be specified; if omitted, edge distances are assumed to be 1. If log
(default false
) is true
, visitor events will be printed to io
, which defaults to STDOUT
; otherwise, no event information will be displayed.
Graphs.mincut
— Methodmincut(g, distmx=weights(g))
Return a tuple (parity, bestcut)
, where parity
is a vector of integer values that determines the partition in g
(1 or 2) and bestcut
is the weight of the cut that makes this partition. An optional distmx
matrix of non-negative weights may be specified; if omitted, edge distances are assumed to be 1.
Graphs.non_backtracking_randomwalk
— Functionnon_backtracking_randomwalk(g, s, niter; rng=nothing, seed=nothing)
Perform a non-backtracking random walk on directed graph g
starting at vertex s
and continuing for a maximum of niter
steps. Return a vector of vertices visited in order.
Graphs.randomwalk
— Methodrandomwalk(g, s, niter; rng=nothing, seed=nothing)
Perform a random walk on graph g
starting at vertex s
and continuing for a maximum of niter
steps. Return a vector of vertices visited in order.
Graphs.self_avoiding_walk
— Methodself_avoiding_walk(g, s, niter; rng=nothing, seed=nothing)
Perform a self-avoiding walk on graph g
starting at vertex s
and continuing for a maximum of niter
steps. Return a vector of vertices visited in order.
Graphs.eulerian
— Methodeulerian(g::AbstractSimpleGraph{T}[, u::T]) --> T[]
Returns a Eulerian trail or cycle through an undirected graph g
, starting at vertex u
, returning a vector listing the vertices of g
in the order that they are traversed. If no such trail or cycle exists, throws an error.
A Eulerian trail or cycle is a path that visits every edge of g
exactly once; for a cycle, the path starts and ends at vertex u
.
Optional arguments
- If
u
is omitted, a Eulerian trail or cycle is computed withu = first(vertices(g))
.
Graphs.all_simple_paths
— Methodall_simple_paths(g, u, v; cutoff) --> Graphs.SimplePathIterator
all_simple_paths(g, u, vs; cutoff) --> Graphs.SimplePathIterator
Returns an iterator that generates all simple paths in the graph g
from a source vertex u
to a target vertex v
or iterable of target vertices vs
. A simple path has no repeated vertices.
The iterator's elements (i.e., the paths) can be materialized via collect
or iterate
. Paths are iterated in the order of a depth-first search.
If the requested path has identical source and target vertices, i.e., if u = v
, a zero-length path [u]
is included among the iterates.
Keyword arguments
The maximum path length (i.e., number of edges) is limited by the keyword argument cutoff
(default, nv(g)-1
). If a path's path length is greater than cutoff
, it is omitted.
Examples
julia> g = complete_graph(4);
julia> spi = all_simple_paths(g, 1, 4)
SimplePathIterator{SimpleGraph{Int64}}(1 → 4)
julia> collect(spi)
5-element Vector{Vector{Int64}}:
[1, 2, 3, 4]
[1, 2, 4]
[1, 3, 2, 4]
[1, 3, 4]
[1, 4]
We can restrict the search to path lengths less than or equal to a specified cut-off (here, 2 edges):
julia> collect(all_simple_paths(g, 1, 4; cutoff=2))
3-element Vector{Vector{Int64}}:
[1, 2, 4]
[1, 3, 4]
[1, 4]