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Improve PolynomialSequence.connected_components() #35518

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Apr 23, 2023
Merged

Improve PolynomialSequence.connected_components() #35518

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6eda76a
connected components without graph in multi_polinomial_sequence.py
dcoudert Apr 15, 2023
2b2e9f4
PR #35518: avoid call to self.variables()
dcoudert Apr 16, 2023
1f0371a
PR #35518: avoid sorting in connected_components
dcoudert Apr 16, 2023
9bc14b3
PR #35518: some care
dcoudert Apr 16, 2023
bb61d57
PR #35518: doctest to check the order of the output
dcoudert Apr 18, 2023
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92 changes: 51 additions & 41 deletions src/sage/rings/polynomial/multi_polynomial_sequence.py
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Expand Up @@ -76,38 +76,39 @@

We separate the system in independent subsystems::

sage: C = Sequence(r2).connected_components(); C
sage: C = Sequence(sorted(r2)).connected_components(); C
[[w213 + k113 + x111 + x112 + x113,
w212 + k112 + x110 + x111 + x112 + 1,
w211 + k111 + x110 + x111 + x113 + 1,
w210 + k110 + x110 + x112 + x113,
x110*w112 + x111*w111 + x112*w110 + x113*w113 + 1,
x110*w112 + x111*w110 + x111*w111 + x111*w113 + x112*w111 + x113*w110 + x113*w112 + w111,
x110*w111 + x111*w110 + x111*w112 + x112*w110 + x113*w111 + x113*w113 + w113,
x110*w111 + x110*w113 + x111*w111 + x111*w112 + x112*w110 + x112*w113 + x113*w111 + x111,
x110*w111 + x110*w112 + x111*w110 + x111*w113 + x112*w111 + x113*w113 + x113,
x110*w111 + x110*w112 + x111*w110 + x111*w111 + x112*w110 + x112*w113 + x113*w112,
x110*w110 + x110*w113 + x111*w112 + x112*w111 + x113*w110,
x110*w110 + x110*w112 + x111*w110 + x111*w112 + x111*w113 + x112*w110 + x112*w111 + x113*w112 + x112,
x110*w110 + x110*w112 + x110*w113 + x111*w110 + x111*w111 + x112*w112 + x113*w110 + x110,
x110*w110 + x110*w111 + x111*w110 + x111*w113 + x112*w112 + x113*w111,
x110*w110 + x110*w111 + x110*w113 + x111*w111 + x112*w110 + x112*w112 + x113*w110 + w110,
x110*w110 + x110*w111 + x110*w112 + x111*w112 + x112*w110 + x112*w111 + x112*w113 + x113*w111 + w112],
[w203 + k103 + x101 + x102 + x103,
w202 + k102 + x100 + x101 + x102 + 1,
w201 + k101 + x100 + x101 + x103 + 1,
w200 + k100 + x100 + x102 + x103,
x100*w102 + x101*w101 + x102*w100 + x103*w103 + 1,
x100*w102 + x101*w100 + x101*w101 + x101*w103 + x102*w101 + x103*w100 + x103*w102 + w101,
x100*w101 + x101*w100 + x101*w102 + x102*w100 + x103*w101 + x103*w103 + w103,
x100*w101 + x100*w103 + x101*w101 + x101*w102 + x102*w100 + x102*w103 + x103*w101 + x101,
x100*w101 + x100*w102 + x101*w100 + x101*w103 + x102*w101 + x103*w103 + x103, x100*w101 + x100*w102 + x101*w100 + x101*w101 + x102*w100 + x102*w103 + x103*w102,
x100*w100 + x100*w103 + x101*w102 + x102*w101 + x103*w100,
x100*w100 + x100*w102 + x101*w100 + x101*w102 + x101*w103 + x102*w100 + x102*w101 + x103*w102 + x102,
x100*w100 + x100*w102 + x100*w103 + x101*w100 + x101*w101 + x102*w102 + x103*w100 + x100,
x100*w100 + x100*w101 + x101*w100 + x101*w103 + x102*w102 + x103*w101,
x100*w100 + x100*w101 + x100*w103 + x101*w101 + x102*w100 + x102*w102 + x103*w100 + w100,
x100*w100 + x100*w101 + x100*w102 + x101*w102 + x102*w100 + x102*w101 + x102*w103 + x103*w101 + w102]]
w212 + k112 + x110 + x111 + x112 + 1,
w211 + k111 + x110 + x111 + x113 + 1,
w210 + k110 + x110 + x112 + x113,
x110*w112 + x111*w111 + x112*w110 + x113*w113 + 1,
x110*w112 + x111*w110 + x111*w111 + x111*w113 + x112*w111 + x113*w110 + x113*w112 + w111,
x110*w111 + x111*w110 + x111*w112 + x112*w110 + x113*w111 + x113*w113 + w113,
x110*w111 + x110*w113 + x111*w111 + x111*w112 + x112*w110 + x112*w113 + x113*w111 + x111,
x110*w111 + x110*w112 + x111*w110 + x111*w113 + x112*w111 + x113*w113 + x113,
x110*w111 + x110*w112 + x111*w110 + x111*w111 + x112*w110 + x112*w113 + x113*w112,
x110*w110 + x110*w113 + x111*w112 + x112*w111 + x113*w110,
x110*w110 + x110*w112 + x111*w110 + x111*w112 + x111*w113 + x112*w110 + x112*w111 + x113*w112 + x112,
x110*w110 + x110*w112 + x110*w113 + x111*w110 + x111*w111 + x112*w112 + x113*w110 + x110,
x110*w110 + x110*w111 + x111*w110 + x111*w113 + x112*w112 + x113*w111,
x110*w110 + x110*w111 + x110*w113 + x111*w111 + x112*w110 + x112*w112 + x113*w110 + w110,
x110*w110 + x110*w111 + x110*w112 + x111*w112 + x112*w110 + x112*w111 + x112*w113 + x113*w111 + w112],
[w203 + k103 + x101 + x102 + x103,
w202 + k102 + x100 + x101 + x102 + 1,
w201 + k101 + x100 + x101 + x103 + 1,
w200 + k100 + x100 + x102 + x103,
x100*w102 + x101*w101 + x102*w100 + x103*w103 + 1,
x100*w102 + x101*w100 + x101*w101 + x101*w103 + x102*w101 + x103*w100 + x103*w102 + w101,
x100*w101 + x101*w100 + x101*w102 + x102*w100 + x103*w101 + x103*w103 + w103,
x100*w101 + x100*w103 + x101*w101 + x101*w102 + x102*w100 + x102*w103 + x103*w101 + x101,
x100*w101 + x100*w102 + x101*w100 + x101*w103 + x102*w101 + x103*w103 + x103,
x100*w101 + x100*w102 + x101*w100 + x101*w101 + x102*w100 + x102*w103 + x103*w102,
x100*w100 + x100*w103 + x101*w102 + x102*w101 + x103*w100,
x100*w100 + x100*w102 + x101*w100 + x101*w102 + x101*w103 + x102*w100 + x102*w101 + x103*w102 + x102,
x100*w100 + x100*w102 + x100*w103 + x101*w100 + x101*w101 + x102*w102 + x103*w100 + x100,
x100*w100 + x100*w101 + x101*w100 + x101*w103 + x102*w102 + x103*w101,
x100*w100 + x100*w101 + x100*w103 + x101*w101 + x102*w100 + x102*w102 + x103*w100 + w100,
x100*w100 + x100*w101 + x100*w102 + x101*w102 + x102*w100 + x102*w101 + x102*w103 + x103*w101 + w102]]
sage: C[0].groebner_basis()
Polynomial Sequence with 30 Polynomials in 16 Variables

Expand Down Expand Up @@ -943,17 +944,26 @@ def connected_components(self):
Polynomial Sequence with 128 Polynomials in 128 Variables,
Polynomial Sequence with 128 Polynomials in 128 Variables]
"""
g = self.connection_graph()
C = sorted(g.connected_components())
# precompute the list of variables in each polynomial
vss = [f.variables() for f in self]

# Use a union-find data structure to encode relationships between
# variables, i.e., that they belong to a same polynomial
from sage.sets.disjoint_set import DisjointSet
DS = DisjointSet(set().union(*vss))
for u, *vs in vss:
for v in vs:
DS.union(u, v)

Ps = {} # map root element -> polynomials in this component
for f, vs in zip(self, vss):
r = DS.find(vs[0])
if r in Ps:
Ps[r].append(f)
else:
Ps[r] = [f]

P = [[] for _ in range(len(C))]
for f in self:
for i,c in enumerate(C):
if len(set(f.variables()).difference(c)) == 0:
P[i].append(f)
break
P = sorted([PolynomialSequence(sorted(p)) for p in P])
return P
return [PolynomialSequence(self.ring(), p) for p in Ps.values()]

def _groebner_strategy(self):
"""
Expand Down