basic csetautomorphisms code

ext/NautyACSetsExt.jl

@@ -4,8 +4,12 @@ using ACSets
 using nauty_jll
 
 """Compute CSetNautyRes from an ACSet."""
-function ACSets.call_nauty(g::ACSet)::CSetNautyRes
-  ACSets.NautyInterface.parse_res(nauty_res(g), g)
+function ACSets.call_nauty(g::ACSet; use_nauty=true)::CSetNautyRes
+  if Sys.iswindows() || !use_nauty
+      CSetAutomorphisms.to_nauty_res(g)
+  else 
+    ACSets.NautyInterface.parse_res(nauty_res(g), g)
+  end
 end
 
 """Make shell command to dreadnaut (nauty) and collect stdout text."""

src/ACSets.jl

@@ -14,7 +14,7 @@ include("DenseACSets.jl")
 include("intertypes/InterTypes.jl")
 include("serialization/Serialization.jl")
 include("ADTs.jl")
-include("NautyInterface.jl")
+include("nauty/Nauty.jl")
 
 @reexport using .ColumnImplementations: AttrVar
 @reexport using .Schemas
@@ -23,6 +23,6 @@ include("NautyInterface.jl")
 @reexport using .InterTypes
 @reexport using .ACSetSerialization
 using .ADTs
-@reexport using .NautyInterface
+@reexport using .Nauty
 
 end

src/nauty/CSetAutomorphisms.jl

@@ -0,0 +1,275 @@
+"""
+This is code which is required because nauty_jll is not supported on Windows: 
+thus we need a makeshift implementation of Nauty within Julia. There is 
+potential to do this in a much cleaner and more efficient way with a virtual 
+machine, but for now performance is not a priority. 
+"""
+module CSetAutomorphisms
+
+using ...ACSetInterface, ...DenseACSets, ...Schemas
+using ..NautyInterface
+using Permutations 
+
+
+# Color assigned to each elem of each component
+const CDict = Dict{Symbol, Vector{Int}}
+
+"""Construct permutation σ⁻¹ such that σσ⁻¹=id"""
+invert_perms(x::CDict) = Dict([k=>Base.invperm(v) for (k, v) in collect(x)])
+
+check_auto(x::CDict)::Bool = all(Base.isperm, values(x))
+
+# Sequence of keys, to index a Tree
+const VPSI = Vector{Pair{Symbol, Int}}
+
+max0(x::Vector{Int})::Int = isempty(x) ? 0 : maximum(x)
+
+include("ColorRefine.jl")
+
+# CSets with attributes replaced w/ combinatorial representatives
+#################################################################
+
+"""
+To compute automorphisms of Attributed CSets, we create a pseudo CSet which has
+additional components for each data type.
+
+This is inefficient for attributes which have a total order on them
+(e.g. integers/strings) since we solve for a canonical permutation of the
+attributes. Future work could address this by initializing the coloring with
+the 'correct' canonical order.
+"""
+function pseudo_cset(g::ACSet)::Tuple{ACSet, Dict{Symbol,Vector{Any}}} 
+  # Create copy of schema (+ an extra component for each datatype)
+  S = acset_schema(g)
+  pres = deepcopy(S)
+  append!(pres.obs, pres.attrtypes)
+  append!(pres.homs, pres.attrs)
+  empty!.([pres.attrtypes, pres.attrs])
+
+  # Use Julia ordering to give each value an index
+  attrvals = Dict(map(attrtypes(S)) do at
+    vals = Set{Any}()
+    [union!(vals, g[a]) for a in attrs(S; just_names=true, to=at)] 
+    at => vcat(filter(x->!(x isa AttrVar), vals) |> collect |> sort, 
+               AttrVar.(parts(g, at)))
+  end)
+
+  # Create and populate pseudo-cset
+  res = AnonACSet(pres, index=arrows(S; just_names=true))
+
+  copy_parts!(res, g)
+
+  # Replace data value with an index for each attribute
+  for t in attrtypes(S)
+    add_parts!(res, t, length(attrvals[t]) - nparts(g, t))
+    for (a,d,_) in attrs(S; to=t)
+      for p in parts(g, d)
+        res[p, a] = findfirst(==(g[p, a]), attrvals[t])
+      end
+    end
+  end
+
+  (res, attrvals)
+end
+
+"""
+Inverse of pseudo_cset. Requires mapping (generated by `pseudo_cset`) of indices
+for each Data to the actual data values.
+"""
+function pseudo_cset_inv(g::ACSet, orig::ACSet, attrvals::AbstractDict) 
+  S = acset_schema(orig)
+  orig = deepcopy(orig)
+  for arr in hom(S)
+    orig[arr] = g[arr]
+  end
+  for (darr,_,tgt) in attrs(S)
+    orig[darr] = attrvals[tgt][g[darr]]
+  end
+  orig
+end
+
+# Results
+#########
+
+"""Apply a coloring to a C-set to get an isomorphic cset"""
+function apply_automorphism(c::ACSet, d::CDict)
+  check_auto(d) || error("received coloring that is not an automorphism: $d")
+  new = deepcopy(c)
+  for (arr, src, tgt) in homs(acset_schema(c))
+    new[d[src], arr] = d[tgt][c[arr]]
+  end
+  for (arr, src, _) in attrs(acset_schema(c))
+    new[d[src], arr] = c[arr]
+  end
+  new
+end
+
+function to_nauty_res(g::ACSet)
+  p, avals = pseudo_cset(g)
+  c, m = [pseudo_cset_inv(apply_automorphism(p, Dict(a)), g, avals) => a 
+          for a in autos(p)[1]] |> sort |> first
+  strhsh = string(c)
+  orbits = Dict{Symbol, Vector{Int}}() # todo
+  generators = Pair{Int, Vector{Permutation}}[] # todo
+  CSetNautyRes(strhsh, orbits, generators, 0, m, c)
+end
+
+# Trees
+#######
+
+"""Find index at which two vectors diverge (used in `search_tree`)"""
+function common(v1::Vector{T}, v2::Vector{T})::Int where {T}
+  for (i, (x, y)) in enumerate(zip(v1, v2))
+    x == y || return i - 1
+  end
+  min(length(v1), length(v2))
+end
+
+"""
+Search tree explored by Nauty. Each node has an input coloring, a refined 
+coloring, and a set of children indexed by which element (in the smallest 
+nontrivial orbit) has its symmetry artificially broken.
+"""
+struct Tree
+  coloring::CDict
+  saturated::CDict
+  children::Dict{Pair{Symbol, Int}, Tree}
+  Tree() = new(CDict(), CDict(), Dict{Pair{Symbol, Int}, Tree}())
+end
+
+"""Get a node via a sequence of edges from the root"""
+function Base.getindex(t::Tree, pth::VPSI)::Tree
+  ptr = t
+  for p in pth
+    ptr = ptr.children[p]
+  end
+  ptr
+end
+
+"""Automorphism based pruning when we've found a new leaf node (τ @ t)"""
+function compute_auto_prune(tree::Tree, t::VPSI, τ::CDict, leafnodes::Set{VPSI})::Set{VPSI}
+  skip = Set{VPSI}()
+  for p in filter(!=(t), leafnodes)
+    π = tree[p].saturated
+    i = common(p, t)
+    _, _, c = abc = p[1:i], p[1:i+1], t[1:i+1] # == t[1:i]
+    a_, b_, c_ = [tree[x].saturated for x in abc]
+    γ = compose_perms(π, invert_perms(τ))
+    if (compose_perms(γ, a_) == a_ && compose_perms(γ, b_) == c_)
+      # skip everything from c to a
+      for i in length(c):length(t)
+        push!(skip, t[1:i])
+      end
+      break
+    end
+  end
+  filter(!=(t), skip) # has something gone wrong if we need to do this?
+end
+
+"""
+Get vector listing nontrivial colors (which component and which color index) as
+well as how many elements have that color. E.g. for (V=[1,1,2], E=[1,2,2,2,3,3])
+we would get `[2=>(:V,1), 3=>(:E,2), 2=>(:E, 3)]`
+"""
+function get_colors_by_size(coloring::CDict)::Vector{Pair{Int,Tuple{Symbol, Int}}}
+  res = []
+  for (k, v) in coloring
+    for color in 1:max0(v)
+      n_c = count(==(color), v)
+      n_c > 1 && push!(res, n_c => (k, color)) # Store which table and which color
+    end
+  end
+  res
+end
+
+
+"""To reduce branching factor, split on the SMALLEST nontrivial partition"""
+function split_data(coloring::CDict)::Tuple{Symbol, Int, Vector{Int}}
+  colors_by_size = sort(get_colors_by_size(coloring), rev=false)
+  isempty(colors_by_size) && return :_nothing, 0, []
+  split_tab, split_color = colors_by_size[1][2]
+  colors = coloring[split_tab]
+  split_inds = findall(==(split_color), colors)
+  (split_tab, split_color, split_inds)
+end
+
+"""
+DFS tree of colorings, with edges being choices in how to break symmetry
+Goal is to acquire all leaf nodes.
+
+Algorithm from "McKay’s Canonical Graph Labeling Algorithm" by Hartke and
+Radcliffe (2009).
+
+McKay's "Practical Graph Isomorphism" (Section 2.29: "storage of identity
+nodes") warns that it's not a good idea to check for every possible automorphism
+pruning (for memory and time concerns). To do: look into doing this in a more
+balanced way. Profiling code will probably reveal that checking for automorphism
+pruning is a bottleneck.
+
+Inputs:
+ - g: our structure that we are computing automorphisms for
+ - res: all automorphisms found so far
+ - split_seq: sequence of edges (our current location in the tree)
+ - tree: all information known so far - this gets modified
+ - leafnodes: coordinates of all automorphisms found so far
+ - skip: flagged coordinates which have been pruned
+"""
+function search_tree!(g::ACSet, init_coloring::CDict, split_seq::VPSI,
+                      tree::Tree, leafnodes::Set{VPSI}, skip::Set{VPSI})
+  curr_tree = tree[split_seq]
+  # Perform color saturation
+  coloring = color_saturate(g; init_color=init_coloring)
+  for (k, v) in pairs(coloring)
+    curr_tree.coloring[k] = init_coloring[k]
+    curr_tree.saturated[k] = v
+  end
+
+  split_tab, _, split_inds = split_data(coloring)
+
+  # Check if we are now at a leaf node
+  if isempty(split_inds)
+    # Add result to list of results
+    push!(leafnodes, split_seq)
+    check_auto(coloring) # fail if not a perm
+
+    # Prune with automorphisms
+    pruned = compute_auto_prune(tree, split_seq, coloring, leafnodes)
+    isempty(pruned) || union!(skip, pruned)
+  else 
+    # Branch on this leaf
+    for split_ind in split_inds
+      if split_ind == split_inds[1]
+        # Construct arguments for recursive call to child
+        new_coloring = deepcopy(coloring)
+        new_seq = vcat(split_seq, [split_tab => split_ind])
+        new_coloring[split_tab][split_ind] = maximum(coloring[split_tab]) + 1
+        curr_tree.children[split_tab => split_ind] = Tree()
+        search_tree!(g, new_coloring, new_seq, tree, leafnodes, skip)
+      end
+    end
+  end
+end
+
+"""Get coordinates of all nodes in a tree that have no children"""
+function get_leaves(t::Tree)::Vector{VPSI}
+  if isempty(t.children)
+    [Pair{Symbol,Int}[]]
+  else
+    res = []
+    for (kv, c) in t.children
+      for pth in get_leaves(c)
+        push!(res, vcat([kv],pth))
+      end
+    end
+    res
+  end
+end
+
+"""Compute the automorphisms of a CSet"""
+function autos(g::ACSet)::Tuple{Set{CDict}, Tree}
+  tree, leafnodes = Tree(), Set{VPSI}()
+  search_tree!(g, nocolor(g), VPSI(), tree, leafnodes,Set{VPSI}())
+  Set([tree[ln].saturated for ln in leafnodes]), tree
+end
+
+end # module