from __future__ import print_function, division
from sympy.core import Basic
from sympy.core.compatibility import iterable, as_int, range
from sympy.utilities.iterables import flatten
from collections import defaultdict
[docs]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] http://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
[docs] @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
[docs] @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
==========
- 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
[docs] @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.' % list(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
[docs] 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
[docs] @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]]
"""
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.' % list(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
[docs] 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)
[docs] 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)
