def planar_graph_grammar():
"""Constructs the grammar for planar graphs.
Returns
-------
grammar : DecompositionGrammar
The grammar for sampling from G, G_dx and G_dx_dx.
"""
# Some shortcuts to make the grammar more readable.
Rule = pybo.AliasSampler
G_1 = Rule('G_1')
G_1_dx = Rule('G_1_dx')
G_1_dx_dx = Rule('G_1_dx_dx')
G_1_dx_dx_dx = Rule('G_1_dx_dx_dx')
G = Rule('G')
G_dx = Rule('G_dx')
G_dx_dx = Rule('G_dx')
Set = pybo.SetSampler
grammar = pybo.DecompositionGrammar()
grammar.rules = one_connected_graph_grammar().rules
EarlyRejectionControl.grammar = grammar
grammar.add_rules({
'G':
Set(0, G_1),
'G_dx':
G_1_dx * G,
'G_dx_dx':
G_1_dx_dx * G + G_1_dx * G_dx,
'G_dx_dx_dx':
G_1_dx_dx_dx * G + G_1_dx_dx * G_dx + G_1_dx_dx * G_dx +
G_1_dx * G_dx_dx
})
grammar.set_builder(['G', 'G_dx', 'G_dx_dx', 'G_dx_dx_dx'],
PlanarGraphBuilder())
return grammar
def irreducible_dissection_grammar():
"""Builds the dissection grammar. Must still be initialized with init().
Returns
-------
DecompositionGrammar
The grammar for sampling from J_a and J_a_dx.
"""
# Some shorthands to keep the grammar readable.
L = pybo.LAtomSampler
Rule = pybo.AliasSampler
K = Rule('K')
K_dx = Rule('K_dx')
K_dx_dx = Rule('K_dx_dx')
I = Rule('I')
I_dx = Rule('I_dx')
I_dx_dx = Rule('I_dx_dx')
J = Rule('J')
J_dx = Rule('J_dx')
J_dx_dx = Rule('J_dx_dx')
Bij = pybo.BijectionSampler
Rej = pybo.RejectionSampler
grammar = pybo.DecompositionGrammar()
# This grammar depends on the binary tree grammar so we add it.
grammar.rules = binary_tree_grammar().rules
EarlyRejectionControl.grammar = grammar
grammar.add_rules({
# Non-derived dissections (standard, rooted, admissible).
'I':
Bij(K, closure),
# We drop the 3*L*U factor here.
# This bijection does not preserve l-size/u-size.
'J':
Bij(I, add_random_root_edge),
'J_a':
Rej(J, is_admissible),
# Derived dissections.
# The result is not a derived class, the bijection does not preserve l-size.
'I_dx':
Bij(K_dx, closure),
# We drop the factor 3*U.
# This bijection does not preserve l-size/u-size.
'J_dx':
Bij(I + L() * I_dx, add_random_root_edge),
'J_a_dx':
Rej(J_dx, is_admissible),
# Bi-derived dissections.
# Does not preserve l-size, result is not a derived class.
'I_dx_dx':
Bij(K_dx_dx, closure),
# We dropped a factor.
'J_dx_dx':
Bij(2 * I_dx + L() * I_dx_dx, add_random_root_edge),
'J_a_dx_dx':
Rej(J_dx_dx, is_admissible)
})
return grammar
def one_connected_graph_grammar():
"""Constructs the grammar for connected planar graphs.
Returns
-------
DecompositionGrammar
The grammar for sampling from G_1_dx and G_1_dx_dx.
"""
# Some shortcuts to make the grammar more readable.
L = pybo.LAtomSampler
Rule = pybo.AliasSampler
G_2_dx = Rule('G_2_dx')
G_2_dx_dx = Rule('G_2_dx_dx')
G_2_dx_dx_dx = Rule('G_2_dx_dx_dx')
G_1_dx = Rule('G_1_dx')
G_1_dx_dx = Rule('G_1_dx_dx')
G_1_dx_dx_dx = Rule('G_1_dx_dx_dx')
Set = pybo.SetSampler
LSubs = pybo.LSubsSampler
Bij = pybo.BijectionSampler
Rej = pybo.RejectionSampler
grammar = pybo.DecompositionGrammar()
grammar.rules = two_connected_graph_grammar().rules
EarlyRejectionControl.grammar = grammar
grammar.add_rules({
'G_1':
Bij(
Rej(
G_1_dx,
rej_to_G_1 # See lemma 15.
),
underive
),
'G_1_dx':
Set(
0,
LSubs(
G_2_dx,
L() * G_1_dx
)
),
'G_1_dx_dx':
Bij(
Bij(
(G_1_dx + L() * G_1_dx_dx) * LSubs(G_2_dx_dx, L() * G_1_dx),
subs_marked_vertex
) * G_1_dx,
merge
),
'G_1_dx_dx_dx':
Bij(
Bij(
(2 * G_1_dx_dx + L() * G_1_dx_dx_dx) * LSubs(G_2_dx_dx, L() * G_1_dx),
subs_marked_vertex
) * G_1_dx,
merge
)
+ Bij(
Bij(
(G_1_dx + L() * G_1_dx_dx) ** 2 * LSubs(G_2_dx_dx_dx, L() * G_1_dx),
subs_marked_vertex_2
) * G_1_dx,
merge
)
+ Bij(
Bij(
(G_1_dx + L() * G_1_dx_dx) * LSubs(G_2_dx_dx, L() * G_1_dx),
subs_marked_vertex
) * G_1_dx_dx,
merge
),
})
grammar.set_builder(['G_1_dx'], Merger())
return grammar
def two_connected_graph_grammar():
"""Constructs the grammar for two connected planar graphs.
Returns
-------
DecompositionGrammar
The grammar for sampling from G_2_dx and G_2_dx_dx.
"""
Z = pybo.ZeroAtomSampler
L = pybo.LAtomSampler
Rule = pybo.AliasSampler
D = Rule('D')
D_dx = Rule('D_dx')
D_dx_dx = Rule('D_dx_dx')
F = Rule('F')
F_dx = Rule('F_dx')
F_dx_dx = Rule('F_dx_dx')
G_2_dy = Rule('G_2_dy')
G_2_dx_dy = Rule('G_2_dx_dy')
G_2_dx_dx_dy = Rule('G_2_dx_dx_dy')
G_2_arrow = Rule('G_2_arrow')
G_2_arrow_dx = Rule('G_2_arrow_dx')
G_2_arrow_dx_dx = Rule('G_2_arrow_dx_dx')
Trans = pybo.TransformationSampler
Bij = pybo.BijectionSampler
DxFromDy = pybo.LDerFromUDerSampler
grammar = pybo.DecompositionGrammar()
grammar.rules = network_grammar().rules
EarlyRejectionControl.grammar = grammar
grammar.add_rules({
# two connected
'G_2_arrow':
Trans(Z() + D, to_G_2_arrow,
eval_transform=divide_by_1_plus_y), # see 5.5
'F':
Bij(L()**2 * G_2_arrow, to_G_2),
'G_2_dy':
Trans(F, to_u_derived_class, eval_transform=divide_by_2),
'G_2_dx':
Bij(
DxFromDy(
G_2_dy, alpha_l_u=2.0
), # see p. 26 TODO check this, error in paper? (no error, 2 is caused by the link graph)
mark_l_atom),
# l-derived two connected
'G_2_arrow_dx':
Trans(D_dx, to_G_2_arrow_dx, eval_transform=divide_by_1_plus_y),
'F_dx':
Bij(L()**2 * G_2_arrow_dx + 2 * L() * G_2_arrow, to_G_2_dx),
'G_2_dx_dy':
Trans(F_dx, to_u_derived_class, eval_transform=divide_by_2),
'G_2_dx_dx':
Bij(
DxFromDy(G_2_dx_dy, alpha_l_u=1.0), # see 5.5
mark_2_l_atoms),
# bi-l-derived two connected
'G_2_arrow_dx_dx':
Trans(D_dx_dx, to_G_2_arrow_dx_dx, eval_transform=divide_by_1_plus_y),
'F_dx_dx':
Bij(L()**2 * G_2_arrow_dx_dx + 4 * L() * G_2_arrow_dx + 2 * G_2_arrow,
to_G_2_dx_dx),
'G_2_dx_dx_dy':
Trans(F_dx_dx, to_u_derived_class, eval_transform=divide_by_2),
'G_2_dx_dx_dx':
Bij(DxFromDy(G_2_dx_dx_dy, alpha_l_u=1.0), mark_3_l_atoms),
})
grammar.set_builder(['G_2_arrow'], ZeroAtomGraphBuilder())
return grammar
def network_grammar():
"""Constructs the grammar for networks.
Returns
-------
DecompositionGrammar
The grammar for sampling from D and D_dx.
"""
L = pybo.LAtomSampler
U = pybo.UAtomSampler
Rule = pybo.AliasSampler
G_3_arrow = Rule('G_3_arrow')
G_3_arrow_dx = Rule('G_3_arrow_dx')
G_3_arrow_dy = Rule('G_3_arrow_dy')
D = Rule('D')
D_dx = Rule('D_dx')
S = Rule('S')
S_dx = Rule('S_dx')
P = Rule('P')
P_dx = Rule('P_dx')
H = Rule('H')
H_dx = Rule('H_dx')
D_dx_dx = Rule('D_dx_dx')
S_dx_dx = Rule('S_dx_dx')
P_dx_dx = Rule('P_dx_dx')
H_dx_dx = Rule('H_dx_dx')
G_3_arrow_dx_dx = Rule('G_3_arrow_dx_dx')
G_3_arrow_dx_dy = Rule('G_3_arrow_dx_dy')
G_3_arrow_dy_dy = Rule('G_3_arrow_dy_dy')
Bij = pybo.BijectionSampler
Set = pybo.SetSampler
USubs = pybo.USubsSampler
grammar = pybo.DecompositionGrammar()
grammar.rules = three_connected_graph_grammar().rules
grammar.add_rules({
# networks
'D': U() + S + P + H,
'S': (U() + P + H) * D * L(),
'P': U() * Set(1, S + H) + Set(2, S + H),
'H': Bij(USubs(G_3_arrow, D), g_3_arrow_to_network),
# l-derived networks
'D_dx': S_dx + P_dx + H_dx,
'S_dx':
(P_dx + H_dx) * D * L()
+ (U() + P + H) * D_dx * L()
+ (U() + P + H) * D,
'P_dx':
U() * (S_dx + H_dx) * Set(0, S + H)
+ (S_dx + H_dx) * Set(1, S + H),
'H_dx':
Bij(
USubs(G_3_arrow_dx, D) + D_dx * USubs(G_3_arrow_dy, D),
g_3_arrow_to_network
),
# bi-l-derived networks
'D_dx_dx':
S_dx_dx + P_dx_dx + H_dx_dx,
'S_dx_dx':
(P_dx_dx + H_dx_dx) * D * L()
+ 2 * (P_dx + H_dx) * D_dx * L()
+ (U() + P + H) * D_dx_dx * L()
+ 2 * (P_dx + H_dx) * D
+ 2 * (U() + P + H) * D_dx,
'P_dx_dx':
U() * (S_dx_dx + H_dx_dx) * Set(0, S + H)
+ U() * (S_dx + H_dx) ** 2 * Set(0, S + H)
+ (S_dx_dx + H_dx_dx) * Set(1, S + H)
+ (S_dx + H_dx) ** 2 * Set(0, S + H),
'H_dx_dx':
Bij(
USubs(G_3_arrow_dx_dx, D)
+ 2 * D_dx * USubs(G_3_arrow_dx_dy, D)
+ D_dx_dx * USubs(G_3_arrow_dy, D)
+ D_dx ** 2 * USubs(G_3_arrow_dy_dy, D),
g_3_arrow_to_network
),
})
# Set up builders.
grammar.set_builder(['D', 'S', 'P', 'H',
'D_dx', 'S_dx', 'P_dx', 'H_dx',
'D_dx_dx', 'S_dx_dx', 'P_dx_dx', 'H_dx_dx'], NetworkBuilder())
grammar.set_builder(['P', 'P_dx', 'P_dx_dx'], PNetworkBuilder())
grammar.set_builder(['S', 'S_dx', 'S_dx_dx'], SNetworkBuilder())
EarlyRejectionControl.grammar = grammar
return grammar
def binary_tree_grammar():
"""
Builds the bicolored binary tree grammar. Must still be initialized with init().
Returns
-------
DecompositionGrammar
The grammar for sampling K and K_dx.
"""
# Some shorthands to keep the grammar readable.
L = pybo.LAtomSampler
U = pybo.UAtomSampler
Rule = pybo.AliasSampler
K_dy = Rule('K_dy')
R_b_as = Rule('R_b_as')
R_w_as = Rule('R_w_as')
R_b_head = Rule('R_b_head')
R_b_head_help = Rule('R_b_head_help')
R_w_head = Rule('R_w_head')
R_b = Rule('R_b')
R_w = Rule('R_w')
K_dy_dx = Rule('K_dy_dx')
R_b_dx = Rule('R_b_dx')
R_w_dx = Rule('R_w_dx')
R_b_as_dx = Rule('R_b_as_dx')
R_w_as_dx = Rule('R_w_as_dx')
R_b_head_dx = Rule('R_b_head_dx')
R_w_head_dx = Rule('R_w_head_dx')
Bij = pybo.BijectionSampler
Trans = pybo.TransformationSampler
Hook = pybo.HookSampler
DxFromDy = pybo.LDerFromUDerSampler
grammar = pybo.DecompositionGrammar()
# Add the decomposition rules.
grammar.rules = {
# Underived and derived binary trees.
'K':
Hook(Trans(K_dy, underive),
before=EarlyRejectionControl.activate_rejection,
after=EarlyRejectionControl.deactivate_rejection), # See 4.1.6.
'K_dx':
DxFromDy(
K_dy,
alpha_l_u=2 / 3 # See 5.3.1.
),
'K_dy':
pybo.RestartableSampler(
Hook(
Bij(R_b_as + R_w_as, to_K_dy),
EarlyRejectionControl.
reset # Reset leaf-counter before sampling from this rule.
), ),
'R_b_as':
R_w * L() * U() + U() * L() * R_w + R_w * L() * R_w,
'R_w_as':
R_b_head * U() + U() * R_b_head + R_b**2,
'R_b_head':
R_w_head * L() * R_b_head_help + R_b_head_help * L() * R_w_head +
R_w_head * L() * R_w_head,
'R_b_head_help':
U() * U(),
'R_w_head':
U() + R_b * U() + U() * R_b + R_b**2,
'R_b': (U() + R_w) * L() * (U() + R_w),
'R_w': (U() + R_b)**2,
# Bi-derived binary trees.
'K_dx_dx':
DxFromDy(
K_dy_dx,
alpha_l_u=2 / 3 # See 5.3.1, 2/3 is also valid for K_dx.
),
'K_dy_dx':
Bij(R_b_as_dx + R_w_as_dx, to_K_dy_dx),
'R_b_as_dx':
U() * L() * R_w_dx + R_w_dx * L() * U() + R_w * L() * R_w_dx +
R_w_dx * L() * R_w + U() * R_w + R_w * U() + R_w**2,
'R_w_as_dx':
R_b_head_dx * U() + U() * R_b_head_dx + R_b * R_b_dx + R_b_dx * R_b,
'R_b_head_dx':
R_b_head_help * L() * R_w_head_dx + R_w_head_dx * L() * R_b_head_help +
R_w_head * L() * R_w_head_dx + R_w_head_dx * L() * R_w_head +
R_w_head * R_b_head_help + R_b_head_help * R_w_head + R_w_head**2,
'R_w_head_dx':
R_b_dx * (R_b + U()) + (R_b + U()) * R_b_dx, # is the same as R_w_dx!
'R_b_dx': (U() + R_w)**2 + R_w_dx * L() * (U() + R_w) +
(U() + R_w) * L() * R_w_dx,
'R_w_dx':
R_b_dx * (U() + R_b) + (U() + R_b) * R_b_dx,
}
# Set builders.
grammar.set_builder([
'R_w', 'R_w_head', 'R_w_as', 'R_b_head_help', 'R_w_dx', 'R_w_head_dx',
'R_w_as_dx'
], WhiteRootedBinaryTreeBuilder())
grammar.set_builder(
['R_b', 'R_b_head', 'R_b_as', 'R_b_dx', 'R_b_head_dx', 'R_b_as_dx'],
BlackRootedBinaryTreeBuilder())
# Set the grammar in the early rejection control variables.
EarlyRejectionControl.grammar = grammar
# We do not init the grammar here.
return grammar
def three_connected_graph_grammar():
"""Builds the three-connected planar graph grammar.
Returns
-------
DecompositionGrammar
The grammar for sampling from G_3_arrow_dx and G_3_arrow_dy
"""
# Some shorthands to keep the grammar readable.
Rule = pybo.AliasSampler
J_a = Rule('J_a')
J_a_dx = Rule('J_a_dx')
J_a_dx_dx = Rule('J_a_dx_dx')
G_3_arrow = Rule('G_3_arrow')
G_3_arrow_dx = Rule('G_3_arrow_dx')
G_3_arrow_dx_dx = Rule('G_3_arrow_dx_dx')
G_3_arrow_dx_dy = Rule('G_3_arrow_dx_dy')
M_3_arrow = Rule('M_3_arrow')
M_3_arrow_dx = Rule('M_3_arrow_dx')
M_3_arrow_dx_dx = Rule('M_3_arrow_dx_dx')
Bij = pybo.BijectionSampler
Rej = pybo.RejectionSampler
Trans = pybo.TransformationSampler
DyFromDx = pybo.UDerFromLDerSampler
grammar = pybo.DecompositionGrammar()
# Depends on irreducible dissection so we add those rules.
grammar.rules = irreducible_dissection_grammar().rules
EarlyRejectionControl.grammar = grammar
grammar.add_rules({
# Non-derived 3-connected rooted planar maps/graphs.
'M_3_arrow':
Bij(J_a, primal_map),
'G_3_arrow':
Trans(M_3_arrow, eval_transform=divide_by_2), # See 4.1.9.
# Derived 3-connected rooted planar maps/graphs.
'M_3_arrow_dx':
Bij(J_a_dx, primal_map),
'G_3_arrow_dx':
Trans(M_3_arrow_dx, to_l_derived_class, eval_transform=divide_by_2),
'G_3_arrow_dy':
Bij(
DyFromDx(G_3_arrow_dx, alpha_u_l=3), # See 5.3.3.
mark_u_atom),
# Bi-derived 3-connected rooted planar maps/graphs.
'M_3_arrow_dx_dx':
Bij(J_a_dx_dx, primal_map),
'G_3_arrow_dx_dx':
Trans(M_3_arrow_dx_dx,
to_bi_l_derived_class,
eval_transform=divide_by_2),
'G_3_arrow_dx_dy':
Bij(DyFromDx(G_3_arrow_dx_dx, alpha_u_l=3), mark_u_atom),
'G_3_arrow_dy_dy':
Bij(
DyFromDx(Bij(G_3_arrow_dx_dy,
lambda gamma: gamma.invert_derivation_order()),
alpha_u_l=3), mark_2_u_atoms),
# Usual 3-connected planar graphs for testing/debugging purposes.
'G_3':
Bij(Rej(G_3_arrow, lambda g: pybo.bern(1 / g.number_of_edges)),
lambda g: HalfEdgeGraph(g.half_edge)),
})
return grammar