"""Functions for generating trees."""fromcollectionsimportdefaultdictimportnetworkxasnxfromnetworkx.utilsimportpy_random_state__all__=["prefix_tree","random_tree","prefix_tree_recursive"]
[docs]defprefix_tree(paths):"""Creates a directed prefix tree from a list of paths. Usually the paths are described as strings or lists of integers. A "prefix tree" represents the prefix structure of the strings. Each node represents a prefix of some string. The root represents the empty prefix with children for the single letter prefixes which in turn have children for each double letter prefix starting with the single letter corresponding to the parent node, and so on. More generally the prefixes do not need to be strings. A prefix refers to the start of a sequence. The root has children for each one element prefix and they have children for each two element prefix that starts with the one element sequence of the parent, and so on. Note that this implementation uses integer nodes with an attribute. Each node has an attribute "source" whose value is the original element of the path to which this node corresponds. For example, suppose `paths` consists of one path: "can". Then the nodes `[1, 2, 3]` which represent this path have "source" values "c", "a" and "n". All the descendants of a node have a common prefix in the sequence/path associated with that node. From the returned tree, the prefix for each node can be constructed by traversing the tree up to the root and accumulating the "source" values along the way. The root node is always `0` and has "source" attribute `None`. The root is the only node with in-degree zero. The nil node is always `-1` and has "source" attribute `"NIL"`. The nil node is the only node with out-degree zero. Parameters ---------- paths: iterable of paths An iterable of paths which are themselves sequences. Matching prefixes among these sequences are identified with nodes of the prefix tree. One leaf of the tree is associated with each path. (Identical paths are associated with the same leaf of the tree.) Returns ------- tree: 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 `-1`, 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.) Notes ----- The prefix tree is also known as a *trie*. Examples -------- Create a prefix tree from a list of strings with common prefixes:: >>> paths = ["ab", "abs", "ad"] >>> T = nx.prefix_tree(paths) >>> list(T.edges) [(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)] The leaf nodes can be obtained as predecessors of the nil node:: >>> root, NIL = 0, -1 >>> list(T.predecessors(NIL)) [2, 3, 4] To recover the original paths that generated the prefix tree, traverse up the tree from the node `-1` to the node `0`:: >>> recovered = [] >>> for v in T.predecessors(NIL): ... prefix = "" ... while v != root: ... prefix = str(T.nodes[v]["source"]) + prefix ... v = next(T.predecessors(v)) # only one predecessor ... recovered.append(prefix) >>> sorted(recovered) ['ab', 'abs', 'ad'] """defget_children(parent,paths):children=defaultdict(list)# Populate dictionary with key(s) as the child/children of the root and# value(s) as the remaining paths of the corresponding child/childrenforpathinpaths:# If path is empty, we add an edge to the NIL node.ifnotpath:tree.add_edge(parent,NIL)continuechild,*rest=path# `child` may exist as the head of more than one path in `paths`.children[child].append(rest)returnchildren# Initialize the prefix tree with a root node and a nil node.tree=nx.DiGraph()root=0tree.add_node(root,source=None)NIL=-1tree.add_node(NIL,source="NIL")children=get_children(root,paths)stack=[(root,iter(children.items()))]whilestack:parent,remaining_children=stack[-1]try:child,remaining_paths=next(remaining_children)# Pop item off stack if there are no remaining childrenexceptStopIteration:stack.pop()continue# We relabel each child with an unused name.new_name=len(tree)-1# The "source" node attribute stores the original node name.tree.add_node(new_name,source=child)tree.add_edge(parent,new_name)children=get_children(new_name,remaining_paths)stack.append((new_name,iter(children.items())))returntree
defprefix_tree_recursive(paths):"""Recursively creates a directed prefix tree from a list of paths. The original recursive version of prefix_tree for comparison. It is the same algorithm but the recursion is unrolled onto a stack. Usually the paths are described as strings or lists of integers. A "prefix tree" represents the prefix structure of the strings. Each node represents a prefix of some string. The root represents the empty prefix with children for the single letter prefixes which in turn have children for each double letter prefix starting with the single letter corresponding to the parent node, and so on. More generally the prefixes do not need to be strings. A prefix refers to the start of a sequence. The root has children for each one element prefix and they have children for each two element prefix that starts with the one element sequence of the parent, and so on. Note that this implementation uses integer nodes with an attribute. Each node has an attribute "source" whose value is the original element of the path to which this node corresponds. For example, suppose `paths` consists of one path: "can". Then the nodes `[1, 2, 3]` which represent this path have "source" values "c", "a" and "n". All the descendants of a node have a common prefix in the sequence/path associated with that node. From the returned tree, ehe prefix for each node can be constructed by traversing the tree up to the root and accumulating the "source" values along the way. The root node is always `0` and has "source" attribute `None`. The root is the only node with in-degree zero. The nil node is always `-1` and has "source" attribute `"NIL"`. The nil node is the only node with out-degree zero. Parameters ---------- paths: iterable of paths An iterable of paths which are themselves sequences. Matching prefixes among these sequences are identified with nodes of the prefix tree. One leaf of the tree is associated with each path. (Identical paths are associated with the same leaf of the tree.) Returns ------- tree: 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 `-1`, 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.) Notes ----- The prefix tree is also known as a *trie*. Examples -------- Create a prefix tree from a list of strings with common prefixes:: >>> paths = ["ab", "abs", "ad"] >>> T = nx.prefix_tree(paths) >>> list(T.edges) [(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)] The leaf nodes can be obtained as predecessors of the nil node. >>> root, NIL = 0, -1 >>> list(T.predecessors(NIL)) [2, 3, 4] To recover the original paths that generated the prefix tree, traverse up the tree from the node `-1` to the node `0`:: >>> recovered = [] >>> for v in T.predecessors(NIL): ... prefix = "" ... while v != root: ... prefix = str(T.nodes[v]["source"]) + prefix ... v = next(T.predecessors(v)) # only one predecessor ... recovered.append(prefix) >>> sorted(recovered) ['ab', 'abs', 'ad'] """def_helper(paths,root,tree):"""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. `tree` is the "accumulator", the :class:`networkx.DiGraph` representing the branching to which the new nodes and edges will be added. """# For each path, remove the first node and make it a child of root.# Any remaining paths then get processed recursively.children=defaultdict(list)forpathinpaths:# If path is empty, we add an edge to the NIL node.ifnotpath:tree.add_edge(root,NIL)continuechild,*rest=path# `child` may exist as the head of more than one path in `paths`.children[child].append(rest)# Add a node for each child, connect root, recurse to remaining pathsforchild,remaining_pathsinchildren.items():# We relabel each child with an unused name.new_name=len(tree)-1# The "source" node attribute stores the original node name.tree.add_node(new_name,source=child)tree.add_edge(root,new_name)_helper(remaining_paths,new_name,tree)# Initialize the prefix tree with a root node and a nil node.tree=nx.DiGraph()root=0tree.add_node(root,source=None)NIL=-1tree.add_node(NIL,source="NIL")# Populate the tree._helper(paths,root,tree)returntree# 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)defrandom_tree(n,seed=None,create_using=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>`. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. 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. Examples -------- >>> tree = nx.random_tree(n=10, seed=0) >>> print(nx.forest_str(tree, sources=[0])) ╙── 0 ├── 3 └── 4 ├── 6 │ ├── 1 │ ├── 2 │ └── 7 │ └── 8 │ └── 5 └── 9 >>> tree = nx.random_tree(n=10, seed=0, create_using=nx.DiGraph) >>> print(nx.forest_str(tree)) ╙── 0 ├─╼ 3 └─╼ 4 ├─╼ 6 │ ├─╼ 1 │ ├─╼ 2 │ └─╼ 7 │ └─╼ 8 │ └─╼ 5 └─╼ 9 """ifn==0:raisenx.NetworkXPointlessConcept("the null graph is not a tree")# Cannot create a Prüfer sequence unless `n` is at least two.ifn==1:utree=nx.empty_graph(1,create_using)else:sequence=[seed.choice(range(n))foriinrange(n-2)]utree=nx.from_prufer_sequence(sequence)ifcreate_usingisNone:tree=utreeelse:tree=nx.empty_graph(0,create_using)iftree.is_directed():# Use a arbitrary root node and dfs to define edge directionsedges=nx.dfs_edges(utree,source=0)else:edges=utree.edges# Populate the specified graph typetree.add_nodes_from(utree.nodes)tree.add_edges_from(edges)returntree