# -*- encoding: utf-8 -*-
# Copyright (C) 2015-2019 by
# Jeffrey Finkelstein <jeffrey.finkelstein@gmail.com>
# NetworkX developers
# All rights reserved.
# BSD license.
#
# Authors: Jeffrey Finkelstein <jeffrey.finkelstein@gmail.com>
"""Functions for generating trees."""
from collections import defaultdict
import networkx as nx
from networkx.utils import generate_unique_node
from networkx.utils import py_random_state
__all__ = ['prefix_tree', 'random_tree']
#: The nil node, the only leaf node in a prefix tree.
#:
#: Each predecessor of the nil node corresponds to the end of a path
#: used to generate the prefix tree.
NIL = 'NIL'
[docs]def prefix_tree(paths):
"""Creates a directed prefix tree from the given list of iterables.
Parameters
----------
paths: iterable of lists
An iterable over "paths", which are themselves lists of
nodes. Common prefixes among these paths are converted into
common initial segments in the generated tree.
Most commonly, this may be an iterable over lists of integers,
or an iterable over Python strings.
Returns
-------
T: DiGraph
A directed graph representing an arborescence consisting of the
prefix tree generated by `paths`. Nodes are directed "downward",
from parent to child. A special "synthetic" root node is added
to be the parent of the first node in each path. A special
"synthetic" leaf node, the "nil" node, is added to be the child
of all nodes representing the last element in a path. (The
addition of this nil node technically makes this not an
arborescence but a directed acyclic graph; removing the nil node
makes it an arborescence.)
Each node has an attribute 'source' whose value is the original
element of the path to which this node corresponds. The 'source'
of the root node is None, and the 'source' of the nil node is
:data:`.NIL`.
The root node is the only node of in-degree zero in the graph,
and the nil node is the only node of out-degree zero. For
convenience, the nil node can be accessed via the :data:`.NIL`
attribute; for example::
>>> from networkx.generators.trees import NIL
>>> paths = ['ab', 'abs', 'ad']
>>> T, root = nx.prefix_tree(paths)
>>> T.predecessors(NIL) # doctest: +SKIP
root : string
The randomly generated uuid of the root node.
Notes
-----
The prefix tree is also known as a *trie*.
Examples
--------
Create a prefix tree from a list of strings with some common
prefixes::
>>> strings = ['ab', 'abs', 'ad']
>>> T, root = nx.prefix_tree(strings)
Continuing the above example, to recover the original paths that
generated the prefix tree, traverse up the tree from the
:data:`.NIL` node to the root::
>>> from networkx.generators.trees import NIL
>>>
>>> strings = ['ab', 'abs', 'ad']
>>> T, root = nx.prefix_tree(strings)
>>> recovered = []
>>> for v in T.predecessors(NIL):
... s = ''
... while v != root:
... # Prepend the character `v` to the accumulator `s`.
... s = str(T.node[v]['source']) + s
... # Each non-nil, non-root node has exactly one parent.
... v = next(T.predecessors(v))
... recovered.append(s)
>>> sorted(recovered)
['ab', 'abs', 'ad']
"""
def _helper(paths, root, B):
"""Recursively create a trie from the given list of paths.
`paths` is a list of paths, each of which is itself a list of
nodes, relative to the given `root` (but not including it). This
list of paths will be interpreted as a tree-like structure, in
which two paths that share a prefix represent two branches of
the tree with the same initial segment.
`root` is the parent of the node at index 0 in each path.
`B` is the "accumulator", the :class:`networkx.DiGraph`
representing the branching to which the new nodes and edges will
be added.
"""
# Create a mapping from each head node to the list of tail paths
# remaining beneath that node.
children = defaultdict(list)
for path in paths:
# If the path is the empty list, that represents the empty
# string, so we add an edge to the NIL node.
if not path:
B.add_edge(root, NIL)
continue
# TODO In Python 3, this should be `child, *rest = path`.
child, rest = path[0], path[1:]
# `child` may exist as the head of more than one path in `paths`.
children[child].append(rest)
# Add a node for each child found above and add edges from the
# root to each child. In this loop, `head` is the child and
# `tails` is the list of remaining paths under that child.
for head, tails in children.items():
# We need to relabel each child with a unique name. To do
# this we simply change each key in the dictionary to be a
# (key, uuid) pair.
new_head = generate_unique_node()
# Ensure the new child knows the name of the old child so
# that the user can recover the mapping to the original
# nodes.
B.add_node(new_head, source=head)
B.add_edge(root, new_head)
_helper(tails, new_head, B)
# Initialize the prefix tree with a root node and a nil node.
T = nx.DiGraph()
root = generate_unique_node()
T.add_node(root, source=None)
T.add_node(NIL, source=NIL)
# Populate the tree.
_helper(paths, root, T)
return T, root
# From the Wikipedia article on Prüfer sequences:
#
# > Generating uniformly distributed random Prüfer sequences and
# > converting them into the corresponding trees is a straightforward
# > method of generating uniformly distributed random labelled trees.
#
[docs]@py_random_state(1)
def random_tree(n, seed=None):
"""Returns a uniformly random tree on `n` nodes.
Parameters
----------
n : int
A positive integer representing the number of nodes in the tree.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
NetworkX graph
A tree, given as an undirected graph, whose nodes are numbers in
the set {0, …, *n* - 1}.
Raises
------
NetworkXPointlessConcept
If `n` is zero (because the null graph is not a tree).
Notes
-----
The current implementation of this function generates a uniformly
random Prüfer sequence then converts that to a tree via the
:func:`~networkx.from_prufer_sequence` function. Since there is a
bijection between Prüfer sequences of length *n* - 2 and trees on
*n* nodes, the tree is chosen uniformly at random from the set of
all trees on *n* nodes.
"""
if n == 0:
raise nx.NetworkXPointlessConcept('the null graph is not a tree')
# Cannot create a Prüfer sequence unless `n` is at least two.
if n == 1:
return nx.empty_graph(1)
sequence = [seed.choice(range(n)) for i in range(n - 2)]
return nx.from_prufer_sequence(sequence)