437 lines
12 KiB
Python
437 lines
12 KiB
Python
from sympy.core import Basic
|
|
from sympy.core.containers import Tuple
|
|
from sympy.tensor.array import Array
|
|
from sympy.core.sympify import _sympify
|
|
from sympy.utilities.iterables import flatten, iterable
|
|
from sympy.utilities.misc import as_int
|
|
|
|
from collections import defaultdict
|
|
|
|
|
|
class Prufer(Basic):
|
|
"""
|
|
The Prufer correspondence is an algorithm that describes the
|
|
bijection between labeled trees and the Prufer code. A Prufer
|
|
code of a labeled tree is unique up to isomorphism and has
|
|
a length of n - 2.
|
|
|
|
Prufer sequences were first used by Heinz Prufer to give a
|
|
proof of Cayley's formula.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://mathworld.wolfram.com/LabeledTree.html
|
|
|
|
"""
|
|
_prufer_repr = None
|
|
_tree_repr = None
|
|
_nodes = None
|
|
_rank = None
|
|
|
|
@property
|
|
def prufer_repr(self):
|
|
"""Returns Prufer sequence for the Prufer object.
|
|
|
|
This sequence is found by removing the highest numbered vertex,
|
|
recording the node it was attached to, and continuing until only
|
|
two vertices remain. The Prufer sequence is the list of recorded nodes.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr
|
|
[3, 3, 3, 4]
|
|
>>> Prufer([1, 0, 0]).prufer_repr
|
|
[1, 0, 0]
|
|
|
|
See Also
|
|
========
|
|
|
|
to_prufer
|
|
|
|
"""
|
|
if self._prufer_repr is None:
|
|
self._prufer_repr = self.to_prufer(self._tree_repr[:], self.nodes)
|
|
return self._prufer_repr
|
|
|
|
@property
|
|
def tree_repr(self):
|
|
"""Returns the tree representation of the Prufer object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr
|
|
[[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]
|
|
>>> Prufer([1, 0, 0]).tree_repr
|
|
[[1, 2], [0, 1], [0, 3], [0, 4]]
|
|
|
|
See Also
|
|
========
|
|
|
|
to_tree
|
|
|
|
"""
|
|
if self._tree_repr is None:
|
|
self._tree_repr = self.to_tree(self._prufer_repr[:])
|
|
return self._tree_repr
|
|
|
|
@property
|
|
def nodes(self):
|
|
"""Returns the number of nodes in the tree.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes
|
|
6
|
|
>>> Prufer([1, 0, 0]).nodes
|
|
5
|
|
|
|
"""
|
|
return self._nodes
|
|
|
|
@property
|
|
def rank(self):
|
|
"""Returns the rank of the Prufer sequence.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]])
|
|
>>> p.rank
|
|
778
|
|
>>> p.next(1).rank
|
|
779
|
|
>>> p.prev().rank
|
|
777
|
|
|
|
See Also
|
|
========
|
|
|
|
prufer_rank, next, prev, size
|
|
|
|
"""
|
|
if self._rank is None:
|
|
self._rank = self.prufer_rank()
|
|
return self._rank
|
|
|
|
@property
|
|
def size(self):
|
|
"""Return the number of possible trees of this Prufer object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer([0]*4).size == Prufer([6]*4).size == 1296
|
|
True
|
|
|
|
See Also
|
|
========
|
|
|
|
prufer_rank, rank, next, prev
|
|
|
|
"""
|
|
return self.prev(self.rank).prev().rank + 1
|
|
|
|
@staticmethod
|
|
def to_prufer(tree, n):
|
|
"""Return the Prufer sequence for a tree given as a list of edges where
|
|
``n`` is the number of nodes in the tree.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
|
|
>>> a.prufer_repr
|
|
[0, 0]
|
|
>>> Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4)
|
|
[0, 0]
|
|
|
|
See Also
|
|
========
|
|
prufer_repr: returns Prufer sequence of a Prufer object.
|
|
|
|
"""
|
|
d = defaultdict(int)
|
|
L = []
|
|
for edge in tree:
|
|
# Increment the value of the corresponding
|
|
# node in the degree list as we encounter an
|
|
# edge involving it.
|
|
d[edge[0]] += 1
|
|
d[edge[1]] += 1
|
|
for i in range(n - 2):
|
|
# find the smallest leaf
|
|
for x in range(n):
|
|
if d[x] == 1:
|
|
break
|
|
# find the node it was connected to
|
|
y = None
|
|
for edge in tree:
|
|
if x == edge[0]:
|
|
y = edge[1]
|
|
elif x == edge[1]:
|
|
y = edge[0]
|
|
if y is not None:
|
|
break
|
|
# record and update
|
|
L.append(y)
|
|
for j in (x, y):
|
|
d[j] -= 1
|
|
if not d[j]:
|
|
d.pop(j)
|
|
tree.remove(edge)
|
|
return L
|
|
|
|
@staticmethod
|
|
def to_tree(prufer):
|
|
"""Return the tree (as a list of edges) of the given Prufer sequence.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> a = Prufer([0, 2], 4)
|
|
>>> a.tree_repr
|
|
[[0, 1], [0, 2], [2, 3]]
|
|
>>> Prufer.to_tree([0, 2])
|
|
[[0, 1], [0, 2], [2, 3]]
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://hamberg.no/erlend/posts/2010-11-06-prufer-sequence-compact-tree-representation.html
|
|
|
|
See Also
|
|
========
|
|
tree_repr: returns tree representation of a Prufer object.
|
|
|
|
"""
|
|
tree = []
|
|
last = []
|
|
n = len(prufer) + 2
|
|
d = defaultdict(lambda: 1)
|
|
for p in prufer:
|
|
d[p] += 1
|
|
for i in prufer:
|
|
for j in range(n):
|
|
# find the smallest leaf (degree = 1)
|
|
if d[j] == 1:
|
|
break
|
|
# (i, j) is the new edge that we append to the tree
|
|
# and remove from the degree dictionary
|
|
d[i] -= 1
|
|
d[j] -= 1
|
|
tree.append(sorted([i, j]))
|
|
last = [i for i in range(n) if d[i] == 1] or [0, 1]
|
|
tree.append(last)
|
|
|
|
return tree
|
|
|
|
@staticmethod
|
|
def edges(*runs):
|
|
"""Return a list of edges and the number of nodes from the given runs
|
|
that connect nodes in an integer-labelled tree.
|
|
|
|
All node numbers will be shifted so that the minimum node is 0. It is
|
|
not a problem if edges are repeated in the runs; only unique edges are
|
|
returned. There is no assumption made about what the range of the node
|
|
labels should be, but all nodes from the smallest through the largest
|
|
must be present.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer.edges([1, 2, 3], [2, 4, 5]) # a T
|
|
([[0, 1], [1, 2], [1, 3], [3, 4]], 5)
|
|
|
|
Duplicate edges are removed:
|
|
|
|
>>> Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K
|
|
([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7)
|
|
|
|
"""
|
|
e = set()
|
|
nmin = runs[0][0]
|
|
for r in runs:
|
|
for i in range(len(r) - 1):
|
|
a, b = r[i: i + 2]
|
|
if b < a:
|
|
a, b = b, a
|
|
e.add((a, b))
|
|
rv = []
|
|
got = set()
|
|
nmin = nmax = None
|
|
for ei in e:
|
|
for i in ei:
|
|
got.add(i)
|
|
nmin = min(ei[0], nmin) if nmin is not None else ei[0]
|
|
nmax = max(ei[1], nmax) if nmax is not None else ei[1]
|
|
rv.append(list(ei))
|
|
missing = set(range(nmin, nmax + 1)) - got
|
|
if missing:
|
|
missing = [i + nmin for i in missing]
|
|
if len(missing) == 1:
|
|
msg = 'Node %s is missing.' % missing.pop()
|
|
else:
|
|
msg = 'Nodes %s are missing.' % sorted(missing)
|
|
raise ValueError(msg)
|
|
if nmin != 0:
|
|
for i, ei in enumerate(rv):
|
|
rv[i] = [n - nmin for n in ei]
|
|
nmax -= nmin
|
|
return sorted(rv), nmax + 1
|
|
|
|
def prufer_rank(self):
|
|
"""Computes the rank of a Prufer sequence.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
|
|
>>> a.prufer_rank()
|
|
0
|
|
|
|
See Also
|
|
========
|
|
|
|
rank, next, prev, size
|
|
|
|
"""
|
|
r = 0
|
|
p = 1
|
|
for i in range(self.nodes - 3, -1, -1):
|
|
r += p*self.prufer_repr[i]
|
|
p *= self.nodes
|
|
return r
|
|
|
|
@classmethod
|
|
def unrank(self, rank, n):
|
|
"""Finds the unranked Prufer sequence.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> Prufer.unrank(0, 4)
|
|
Prufer([0, 0])
|
|
|
|
"""
|
|
n, rank = as_int(n), as_int(rank)
|
|
L = defaultdict(int)
|
|
for i in range(n - 3, -1, -1):
|
|
L[i] = rank % n
|
|
rank = (rank - L[i])//n
|
|
return Prufer([L[i] for i in range(len(L))])
|
|
|
|
def __new__(cls, *args, **kw_args):
|
|
"""The constructor for the Prufer object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
|
|
A Prufer object can be constructed from a list of edges:
|
|
|
|
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
|
|
>>> a.prufer_repr
|
|
[0, 0]
|
|
|
|
If the number of nodes is given, no checking of the nodes will
|
|
be performed; it will be assumed that nodes 0 through n - 1 are
|
|
present:
|
|
|
|
>>> Prufer([[0, 1], [0, 2], [0, 3]], 4)
|
|
Prufer([[0, 1], [0, 2], [0, 3]], 4)
|
|
|
|
A Prufer object can be constructed from a Prufer sequence:
|
|
|
|
>>> b = Prufer([1, 3])
|
|
>>> b.tree_repr
|
|
[[0, 1], [1, 3], [2, 3]]
|
|
|
|
"""
|
|
arg0 = Array(args[0]) if args[0] else Tuple()
|
|
args = (arg0,) + tuple(_sympify(arg) for arg in args[1:])
|
|
ret_obj = Basic.__new__(cls, *args, **kw_args)
|
|
args = [list(args[0])]
|
|
if args[0] and iterable(args[0][0]):
|
|
if not args[0][0]:
|
|
raise ValueError(
|
|
'Prufer expects at least one edge in the tree.')
|
|
if len(args) > 1:
|
|
nnodes = args[1]
|
|
else:
|
|
nodes = set(flatten(args[0]))
|
|
nnodes = max(nodes) + 1
|
|
if nnodes != len(nodes):
|
|
missing = set(range(nnodes)) - nodes
|
|
if len(missing) == 1:
|
|
msg = 'Node %s is missing.' % missing.pop()
|
|
else:
|
|
msg = 'Nodes %s are missing.' % sorted(missing)
|
|
raise ValueError(msg)
|
|
ret_obj._tree_repr = [list(i) for i in args[0]]
|
|
ret_obj._nodes = nnodes
|
|
else:
|
|
ret_obj._prufer_repr = args[0]
|
|
ret_obj._nodes = len(ret_obj._prufer_repr) + 2
|
|
return ret_obj
|
|
|
|
def next(self, delta=1):
|
|
"""Generates the Prufer sequence that is delta beyond the current one.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
|
|
>>> b = a.next(1) # == a.next()
|
|
>>> b.tree_repr
|
|
[[0, 2], [0, 1], [1, 3]]
|
|
>>> b.rank
|
|
1
|
|
|
|
See Also
|
|
========
|
|
|
|
prufer_rank, rank, prev, size
|
|
|
|
"""
|
|
return Prufer.unrank(self.rank + delta, self.nodes)
|
|
|
|
def prev(self, delta=1):
|
|
"""Generates the Prufer sequence that is -delta before the current one.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.combinatorics.prufer import Prufer
|
|
>>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]])
|
|
>>> a.rank
|
|
36
|
|
>>> b = a.prev()
|
|
>>> b
|
|
Prufer([1, 2, 0])
|
|
>>> b.rank
|
|
35
|
|
|
|
See Also
|
|
========
|
|
|
|
prufer_rank, rank, next, size
|
|
|
|
"""
|
|
return Prufer.unrank(self.rank -delta, self.nodes)
|