from __future__ import annotations __copyright__ = """ Copyright (C) 2009-2013 Andreas Kloeckner Copyright (C) 2020 Matt Wala Copyright (C) 2020 James Stevens Copyright (C) 2024 Addison Alvey-Blanco """ __license__ = """ Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. """ __doc__ = """ Graph Algorithms ================ .. note:: These functions are mostly geared towards directed graphs (digraphs). .. autofunction:: reverse_graph .. autofunction:: a_star .. autofunction:: compute_sccs .. autoexception:: CycleError .. autofunction:: compute_topological_order .. autofunction:: compute_transitive_closure .. autofunction:: contains_cycle .. autofunction:: compute_induced_subgraph .. autofunction:: as_graphviz_dot .. autofunction:: validate_graph .. autofunction:: is_connected .. autofunction:: undirected_graph_from_edges .. autofunction:: get_reachable_nodes Type Variables Used ------------------- .. class:: _SupportsLT A :class:`~typing.Protocol` for `__lt__` support. .. class:: NodeT Type of a graph node, can be any hashable type. .. class:: GraphT A :class:`collections.abc.Mapping` representing a directed graph. The mapping contains one key representing each node in the graph, and this key maps to a :class:`collections.abc.Collection` of its successor nodes. Note that most functions expect that every graph node is included as a key in the graph. """ from collections.abc import ( Callable, Collection, Hashable, Iterable, Iterator, Mapping, MutableSet, ) from dataclasses import dataclass from typing import ( Any, Generic, Protocol, TypeAlias, TypeVar, ) NodeT = TypeVar("NodeT", bound=Hashable) GraphT: TypeAlias = Mapping[NodeT, Collection[NodeT]] # {{{ reverse_graph def reverse_graph(graph: GraphT[NodeT]) -> GraphT[NodeT]: """ Reverses a graph *graph*. :returns: A :class:`dict` representing *graph* with edges reversed. """ result: dict[NodeT, set[NodeT]] = {} for node_key, successor_nodes in graph.items(): # Make sure every node is in the result even if it has no successors result.setdefault(node_key, set()) for successor in successor_nodes: result.setdefault(successor, set()).add(node_key) return {k: frozenset(v) for k, v in result.items()} # }}} # {{{ a_star @dataclass(frozen=True) class _AStarNode(Generic[NodeT]): state: NodeT parent: _AStarNode[NodeT] | None path_cost: float | int def a_star( initial_state: NodeT, goal_state: NodeT, neighbor_map: GraphT[NodeT], estimate_remaining_cost: Callable[[NodeT], float] | None = None, get_step_cost: Callable[[Any, NodeT], float] = lambda x, y: 1 ) -> list[NodeT]: """ With the default cost and heuristic, this amounts to Dijkstra's algorithm. """ from heapq import heappop, heappush if estimate_remaining_cost is None: def estimate_remaining_cost(x: NodeT) -> float: if x != goal_state: return 1 return 0 inf = float("inf") init_remcost = estimate_remaining_cost(initial_state) assert init_remcost != inf queue = [(init_remcost, _AStarNode(initial_state, parent=None, path_cost=0))] visited_states = set() while queue: _, top = heappop(queue) visited_states.add(top.state) if top.state == goal_state: result = [] it: _AStarNode[NodeT] | None = top while it is not None: result.append(it.state) it = it.parent return result[::-1] for state in neighbor_map[top.state]: if state in visited_states: continue remaining_cost = estimate_remaining_cost(state) if remaining_cost == inf: continue step_cost = get_step_cost(top, state) estimated_path_cost = top.path_cost+step_cost+remaining_cost heappush(queue, (estimated_path_cost, _AStarNode(state, top, path_cost=top.path_cost + step_cost))) raise RuntimeError("no solution") # }}} # {{{ compute SCCs with Tarjan's algorithm def compute_sccs(graph: GraphT[NodeT]) -> list[list[NodeT]]: to_search = set(graph.keys()) visit_order: dict[NodeT, int] = {} scc_root = {} sccs = [] while to_search: top = next(iter(to_search)) call_stack: list[tuple[NodeT, Iterator[NodeT], NodeT | None]] = ( [(top, iter(graph[top]), None)]) visit_stack = [] visiting = set() scc: list[NodeT] = [] while call_stack: top, children, last_popped_child = call_stack.pop() if top not in visiting: # Unvisited: mark as visited, initialize SCC root. count = len(visit_order) visit_stack.append(top) visit_order[top] = count scc_root[top] = count visiting.add(top) to_search.discard(top) # Returned from a recursion, update SCC. if last_popped_child is not None: scc_root[top] = min( scc_root[top], scc_root[last_popped_child]) for child in children: if child not in visit_order: # Recurse. call_stack.append((top, children, child)) call_stack.append((child, iter(graph[child]), None)) break if child in visiting: scc_root[top] = min( scc_root[top], visit_order[child]) else: if scc_root[top] == visit_order[top]: scc = [] while visit_stack[-1] != top: scc.append(visit_stack.pop()) scc.append(visit_stack.pop()) for item in scc: visiting.remove(item) sccs.append(scc) return sccs # }}} # {{{ compute topological order class CycleError(Exception): """ Raised when a topological ordering cannot be computed due to a cycle. :attr node: Node in a directed graph that is part of a cycle. """ def __init__(self, node: NodeT) -> None: self.node = node class _SupportsLT(Protocol): def __lt__(self, other: Any) -> bool: ... @dataclass(frozen=True) class _HeapEntry(Generic[NodeT]): """ Helper class to compare associated keys while comparing the elements in heap operations. Only needs to define :func:`pytools.graph.__lt__` according to . """ node: NodeT key: _SupportsLT def __lt__(self, other: _HeapEntry[NodeT]) -> bool: return self.key < other.key def compute_topological_order( graph: GraphT[NodeT], key: Callable[[NodeT], _SupportsLT] | None = None, ) -> list[NodeT]: """Compute a topological order of nodes in a directed graph. :arg key: A custom key function may be supplied to determine the order in break-even cases. Expects a function of one argument that is used to extract a comparison key from each node of the *graph*. :returns: A :class:`list` representing a valid topological ordering of the nodes in the directed graph. .. note:: * Requires the keys of the mapping *graph* to be hashable. * Implements `Kahn's algorithm `__. .. versionadded:: 2020.2 """ # all nodes have the same keys when not provided keyfunc = key if key is not None else (lambda x: 0) from heapq import heapify, heappop, heappush order = [] # {{{ compute nodes_to_num_predecessors nodes_to_num_predecessors = dict.fromkeys(graph, 0) for node in graph: for child in graph[node]: nodes_to_num_predecessors[child] = ( nodes_to_num_predecessors.get(child, 0) + 1) # }}} total_num_nodes = len(nodes_to_num_predecessors) # heap: list of instances of HeapEntry(n) where 'n' is a node in # 'graph' with no predecessor. Nodes with no predecessors are the # schedulable candidates. heap = [_HeapEntry(n, keyfunc(n)) for n, num_preds in nodes_to_num_predecessors.items() if num_preds == 0] heapify(heap) while heap: # pick the node with least key node_to_be_scheduled = heappop(heap).node order.append(node_to_be_scheduled) # discard 'node_to_be_scheduled' from the predecessors of its # successors since it's been scheduled for child in graph.get(node_to_be_scheduled, ()): nodes_to_num_predecessors[child] -= 1 if nodes_to_num_predecessors[child] == 0: heappush(heap, _HeapEntry(child, keyfunc(child))) if len(order) != total_num_nodes: # any node which has a predecessor left is a part of a cycle raise CycleError(next(iter(n for n, num_preds in nodes_to_num_predecessors.items() if num_preds != 0))) return order # }}} # {{{ compute transitive closure def compute_transitive_closure( graph: Mapping[NodeT, MutableSet[NodeT]]) -> GraphT[NodeT]: """Compute the transitive closure of a directed graph using Warshall's algorithm. :arg graph: A :class:`collections.abc.Mapping` representing a directed graph. The mapping contains one key representing each node in the graph, and this key maps to a :class:`collections.abc.MutableSet` of nodes that are connected to the node by outgoing edges. This graph may contain cycles. This object must be picklable. Every graph node must be included as a key in the graph. :returns: The transitive closure of the graph, represented using the same data type. .. versionadded:: 2020.2 """ # Warshall's algorithm from copy import deepcopy closure = deepcopy(graph) # (assumes all graph nodes are included in keys) for k in graph: for n1 in graph: for n2 in graph: if k in closure[n1] and n2 in closure[k]: closure[n1].add(n2) return closure # }}} # {{{ check for cycle def contains_cycle(graph: GraphT[NodeT]) -> bool: """Determine whether a graph contains a cycle. :returns: A :class:`bool` indicating whether the graph contains a cycle. .. versionadded:: 2020.2 """ try: compute_topological_order(graph) return False except CycleError: return True # }}} # {{{ compute induced subgraph def compute_induced_subgraph(graph: Mapping[NodeT, set[NodeT]], subgraph_nodes: set[NodeT]) -> GraphT[NodeT]: """Compute the induced subgraph formed by a subset of the vertices in a graph. :arg graph: A :class:`collections.abc.Mapping` representing a directed graph. The mapping contains one key representing each node in the graph, and this key maps to a :class:`collections.abc.Set` of nodes that are connected to the node by outgoing edges. :arg subgraph_nodes: A :class:`collections.abc.Set` containing a subset of the graph nodes in the graph. :returns: A :class:`dict` representing the induced subgraph formed by the subset of the vertices included in `subgraph_nodes`. .. versionadded:: 2020.2 """ new_graph = {} for node, children in graph.items(): if node in subgraph_nodes: new_graph[node] = children & subgraph_nodes return new_graph # }}} # {{{ as_graphviz_dot def as_graphviz_dot(graph: GraphT[NodeT], node_labels: Callable[[NodeT], str] | None = None, edge_labels: Callable[[NodeT, NodeT], str] | None = None, ) -> str: """ Create a visualization of the graph *graph* in the `dot `__ language. :arg node_labels: An optional function that returns node labels for each node. :arg edge_labels: An optional function that returns edge labels for each pair of nodes. :returns: A string in the `dot `__ language. """ from pytools import UniqueNameGenerator id_gen = UniqueNameGenerator(forced_prefix="mynode") from pytools.graphviz import dot_escape if node_labels is None: def node_labels(x: NodeT) -> str: return str(x) if edge_labels is None: def edge_labels(x: NodeT, y: NodeT) -> str: return "" node_to_id = {} for node, targets in graph.items(): if node not in node_to_id: node_to_id[node] = id_gen() for t in targets: if t not in node_to_id: node_to_id[t] = id_gen() # Add nodes content = "\n".join( [f'{node_to_id[node]} [label="{dot_escape(node_labels(node))}"];' for node in node_to_id]) content += "\n" # Add edges content += "\n".join( [f"{node_to_id[node]} -> {node_to_id[t]} " f'[label="{dot_escape(edge_labels(node, t))}"];' for (node, targets) in graph.items() for t in targets]) return f"digraph mygraph {{\n{content}\n}}\n" # }}} # {{{ validate graph def validate_graph(graph: GraphT[NodeT]) -> None: """ Validates that all successor nodes of each node in *graph* are keys in *graph* itself. Raises a :class:`ValueError` if not. """ seen_nodes: set[NodeT] = set() for children in graph.values(): seen_nodes.update(children) if not seen_nodes <= graph.keys(): raise ValueError( f"invalid graph, missing keys: {seen_nodes-graph.keys()}") # }}} # {{{ is_connected def is_connected(graph: GraphT[NodeT]) -> bool: """ Returns whether all nodes in *graph* are connected, ignoring the edge direction. :returns: A :class:`bool` indicating whether the graph is connected. """ if not graph: # https://cs.stackexchange.com/questions/52815/is-a-graph-of-zero-nodes-vertices-connected return True visited = set() undirected_graph = {node: set(children) for node, children in graph.items()} for node, children in graph.items(): for child in children: undirected_graph[child].add(node) def dfs(node: NodeT) -> None: visited.add(node) for child in undirected_graph[node]: if child not in visited: dfs(child) dfs(next(iter(graph.keys()))) return visited == graph.keys() # }}} def undirected_graph_from_edges( edges: Iterable[tuple[NodeT, NodeT]], ) -> GraphT[NodeT]: """ Constructs an undirected graph using *edges*. :arg edges: An :class:`Iterable` of pairs of related :class:`NodeT` s. :returns: A :class:`GraphT` that is the undirected graph. """ undirected_graph: dict[NodeT, set[NodeT]] = {} for lhs, rhs in edges: if lhs == rhs: raise TypeError("Found loop in edges," f" LHS, RHS = {lhs}") undirected_graph.setdefault(lhs, set()).add(rhs) undirected_graph.setdefault(rhs, set()).add(lhs) return undirected_graph def get_reachable_nodes( undirected_graph: GraphT[NodeT], source_node: NodeT, exclude_nodes: Collection[NodeT] | None = None) -> frozenset[NodeT]: """ Returns a :class:`frozenset` of all nodes in *undirected_graph* that are reachable from *source_node*. If any node from *exclude_nodes* lies on a path between *source_node* and some other node :math:`u` in *undirected_graph* and there are no other viable paths, then :math:`u` is considered not reachable from *source_node*. In the case where *source_node* is in *exclude_nodes*, then no node is reachable from *source_node*, so an empty :class:`frozenset` is returned. """ if exclude_nodes is not None and source_node in exclude_nodes: return frozenset() nodes_visited: set[NodeT] = set() reachable_nodes: set[NodeT] = set() nodes_to_visit = {source_node} if exclude_nodes is None: exclude_nodes = set() while nodes_to_visit: current_node = nodes_to_visit.pop() nodes_visited.add(current_node) reachable_nodes.add(current_node) neighbors = undirected_graph[current_node] nodes_to_visit.update({ node for node in neighbors if node not in nodes_visited and node not in exclude_nodes }) return frozenset(reachable_nodes) # vim: foldmethod=marker