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implemented function to generate upsets
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Saatvik Rao authored and Saatvik Rao committed Apr 15, 2024
1 parent 8ea5214 commit b264dc2
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4 changes: 4 additions & 0 deletions src/doc/en/reference/references/index.rst
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Expand Up @@ -5872,6 +5872,10 @@ REFERENCES:
Journal of Algorithms, Volume 26, Issue 2, Pages 275 - 290, 1998.
(https://doi.org/10.1006/jagm.1997.0891)
.. [Sq2002] Matthew B. Squire.
*Enumerating the Ideals of a Poset*.
(https://citeseer.ist.psu.edu/465417.html)
.. [SS1983] Shorey and Stewart. "On the Diophantine equation a
x^{2t} + b x^t y + c y^2 = d and pure powers in recurrence
sequences." Mathematica Scandinavica, 1983.
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50 changes: 50 additions & 0 deletions src/sage/categories/posets.py
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Expand Up @@ -304,6 +304,56 @@ def order_filter(self, elements):
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]
"""

def generate_upsets(U, Y, Poset):
r"""
Enumerates the ideals (upsets) of a given poset.
ALGORITHM:
The algorithm is based on [Sq2002]_.
Current algorithms for genertating ideals require an amortized
time of `O(n)` per ideal in the worst case, where `n` is the number
of elements in the poset. This algorithm generates ideals in an
amortized time of `O(log n)` per ideal.
The function recursively generates all possible upsets (subsets) of
a given set `Y` based on a partially ordered set (Poset), ensuring that
if `Xi < Xm < Xj`, then Xi must also be less than `Xj`. It iterates through
`Y`, partitioning it into two subsets based on the relationship between
elements and yields the resulting upsets.
INPUT:
- ``U`` -- a list of the current upset being constructed.
- ``Y`` -- a list of remaining elements to be considered for inclusion.
- ``Poset`` -- a dictionary representing the partial order.
OUTPUT:
- A subset of Y that satisfies the upset condition.
EXAMPLES::
"""
if not Y:
yield U
else:
Xm = Y[len(Y) // 2]
Y1 = []
Y2 = []
for Xi in Y:
if Poset[(Xi, Xm)] == 1:
U[Y.index(Xi)] = 0
else:
Y1.append(Xi)
yield from Posets.generate_upsets(U.copy(), Y1, Poset)
for Xj in Y:
if Poset[(Xm, Xj)] == 1:
U[Y.index(Xj)] = 1
else:
Y2.append(Xj)
yield from Posets.generate_upsets(U.copy(), Y2, Poset)

def directed_subset(self, elements, direction):
r"""
Return the order filter or the order ideal generated by a
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