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graph.py
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graph.py
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from queue import Queue
import disjoint_sets_forest as dsf
import sys
from typing import Iterable, Optional, Tuple, Set, Dict
from enum import Enum
Color = Enum('Color', ('WHITE', 'GREY', 'BLACK'))
class Vertex:
def __init__(self, key):
self.key = key
def __repr__(self):
return f"Vertex({self.key})"
def print_path(self, v):
"""
print out the vertices on a shortest path from s to
v, assuming that BFS has already computed a breadth-first tree
"""
if self == v:
print(self)
elif v.p is None:
print("No path from {} to {} exists".format(self.key, v.key))
else:
self.print_path(v.p)
print(v)
Vertices = Optional[Iterable[Vertex]]
Edge = Tuple[Vertex, Vertex]
Edges = Optional[Iterable[Edge]]
class Graph:
def __init__(self, vertices: Optional[Vertices] = None,
edges: Optional[Edges] = None, directed: bool = True):
self.directed: bool = directed
self.vertices: Set[Vertex] = set() if vertices is None else set(vertices)
self.edges: Set[Edge] = set()
self.adj: Dict[Vertex, Set[Vertex]] = dict()
self._time: int = 0
for u in self.vertices:
self.adj[u] = set()
if edges is not None:
for u, v in edges:
self._add_edge(u, v)
def __eq__(self, graph2: 'Graph'):
graph1 = self
return (graph1.directed == graph2.directed) and (graph1.vertices == graph2.vertices) and (
graph1.edges == graph2.edges)
def _add_edge(self, u: Vertex, v: Vertex):
if self.directed:
self.adj[u].add(v)
self.edges.add((u, v))
elif u != v: # undirected graph does not allow self loop
self.adj[u].add(v)
self.edges.add((u, v))
self.adj[v].add(u)
self.edges.add((v, u))
def _add_vertex(self, u: Vertex, edges: Optional[Edges] = None):
self.vertices.add(u)
if edges is not None:
for u, v in edges:
self._add_edge(u, v)
def copy(self):
return Graph(self.vertices, self.edges, self.directed)
def transpose(self):
return Graph(self.vertices, [(v, u) for u, v in self.edges], self.directed)
def bfs(self, s: Vertex):
for u in self.vertices:
u.distance = float("Inf")
u.color = Color.WHITE
u.parent = None
s.color = Color.GREY
s.distance = 0
s.parent = None
queue = Queue()
queue.put(s)
while not queue.empty():
u = queue.get()
for v in self.adj[u]:
if v.color == Color.WHITE:
v.color = Color.GREY
v.distance = u.distance + 1
v.parent = u
queue.put(v)
u.color = Color.BLACK
def dfs(self):
self._time = 0
for u in self.vertices:
u.color = Color.WHITE
u.p = None
for u in self.vertices:
if u.color == Color.WHITE:
self._dfs_visit(u)
def _dfs_visit(self, u: Vertex):
self._time += 1
u.d = self._time
u.color = Color.GREY
for v in self.adj[u]:
if v.color == Color.WHITE:
v.p = u
self._dfs_visit(v)
u.color = Color.BLACK
self._time += 1
u.f = self._time
def is_cyclic(self):
for u in self.vertices:
u.color = Color.WHITE
for u in self.vertices:
if u.color == Color.WHITE:
if self._is_cyclic_aux(u):
return True
return False
def _is_cyclic_aux(self, u):
u.color = Color.GREY
for v in self.adj[u]:
if v.color == Color.WHITE:
if self._is_cyclic_aux(v):
return True
elif v.color == Color.GREY:
return True
u.color = Color.BLACK
return False
def topological_sort(self):
"""
Perform topological sort for dag
A topological sort of a dag(directed acyclic graph) G = (V, E) is a linear ordering of all the vertices
such that if G contains an edge (u, v), then u appears before v in the ordering.
"""
assert (not self.is_cyclic()) and self.directed
self.dfs()
return sorted(self.vertices, key=lambda x: x.f, reverse=True)
def print_all_edges(self):
s = next(iter(self.vertices))
self.bfs(s)
self._print_all_edges_aux(s)
def _print_all_edges_aux(self, u):
for v in self.adj[u]:
if u == v.p:
print((u, v))
self._print_all_edges_aux(v)
print((v, u))
else:
pass
def printAllEdges(self):
self.status = dict()
s = next(iter(self.vertices))
print("key of s is {}".format(s.key))
self.printAllEdges_aux(s)
def printAllEdges_aux(self, u):
for v in self.adj[u]:
try:
status = self.status[(u, v)]
except KeyError:
self.status[(u, v)] = 1
self.status[(v, u)] = 1
print((u, v))
self.printAllEdges_aux(v)
print((v, u))
def path_num(self, s, t):
"""
A linear-time algorithm that takes as input a directed acyclic graph
G = (V, E) and two vertices s and t, and returns the number of simple
paths from s to t in G.
"""
for u in self.vertices:
u.color = 0
u.num = 0
t.color = 2
t.num = 1
return self._path_num_aux(s, t)
def _path_num_aux(self, s, t):
s.color = 1
for v in self.adj[s]:
if v.color == 2:
s.num = s.num + v.num
elif v.color == 0:
s.num = s.num + self._path_num_aux(v, t)
s.color = 2
return s.num
def strongly_connected_components(self):
global time, cc
self.dfs()
t = self.transpose()
for u in t.vertices:
u.color = 0
u.p = None
time = 0
cc = 0
for u in sorted(self.vertices, key=lambda u: u.f, reverse=True):
if u.color == 0:
cc = cc + 1
t.strongly_connected_components_dfs_visit(u)
def strongly_connected_components_dfs_visit(self, u):
global time, cc
u.cc = cc
time = time + 1
u.d = time
u.color = 1
for v in self.adj[u]:
if v.color == 0:
v.p = u
self.strongly_connected_components_dfs_visit(v)
u.color = 2
time = time + 1
u.f = time
def simplified(self):
"""
create a simplified graph that has the same strong
connected components and component graph as G and that is as small
as possible
"""
self.dfs()
t = self.transpose()
return t.simplified_dfs()
def simplified_dfs(self):
global time, cc, status
status = dict()
s = Graph(self.vertices)
for u in self.vertices:
u.color = 0
u.p = None
time = 0
cc = 0
for u in sorted(self.vertices, key=lambda u: u.f, reverse=True):
if u.color == 0:
stack = []
cc = cc + 1
self.simplified_dfs_visit(u, stack, s)
for i in range(len(stack) - 1):
s._add_edge(stack[i], stack[i + 1])
if len(stack) > 1:
s._add_edge(stack[len(stack) - 1], stack[0])
return s
def simplified_dfs_visit(self, u, stack, s):
global time, cc, status
stack.append(u)
u.cc = cc
time = time + 1
u.d = time
u.color = 1
for v in self.adj[u]:
if v.color == 0:
v.p = u
self.simplified_dfs_visit(v, stack, s)
elif v.color == 2 and v.cc < u.cc:
try:
st = status[(v.cc, u.cc)]
except KeyError:
status[(v.cc, u.cc)] = 1
s._add_edge(v, u)
u.color = 2
time = time + 1
u.f = time
def component_graph(self):
"""
compute the component graph of a directed graph
there is at most one edge between two vertices in the component graph
:return:
"""
global time, cc, cg, status, vertices_list
self.dfs()
t = self.transpose()
for u in t.vertices:
u.color = 0
u.p = None
time = 0
cc = 0
status = dict()
vertices_list = list()
cg = Graph()
for u in sorted(self.vertices, key=lambda u: u.f, reverse=True):
if u.color == 0:
cc = cc + 1
vertices_list.append(Vertex(cc))
cg._add_vertex(vertices_list[cc - 1])
t.component_graph_dfs_visit(u)
return cg
def component_graph_dfs_visit(self, u):
global time, cc, cg, vertices_list
u.cc = cc
time = time + 1
u.d = time
u.color = 1
for v in self.adj[u]:
if v.color == 0:
v.p = u
self.component_graph_dfs_visit(v)
elif v.color == 2 and v.cc < u.cc:
try:
st = status[(v.cc, u.cc)]
except KeyError:
status[(v.cc, u.cc)] = 1
cg._add_edge(
vertices_list[v.cc - 1],
vertices_list[u.cc - 1])
u.color = 2
time = time + 1
u.f = time
def semiconnected(self):
cg = self.component_graph()
vertices_list = sorted(cg.vertices, key=lambda u: u.key, reverse=False)
for i in range(len(vertices_list) - 1):
if vertices_list[i + 1] not in cg.adj[vertices_list[i]]:
return False
return True
def cut(self, x, y, w):
"""
For a given edge (x, y) contained in some minimum spanning tree,
form a minimum spanning tree that contains (x, y) using a method like Prim's algorithm,
and construct a cut (S, V - S) such that (x, y) is the light edge crossing
the cut, S = {u: u.root = x}
"""
for v in self.vertices:
v.weight = float("Inf")
v.p = None
v.root = None
x.weight = 0
y.weight = 0
x.root = x
y.root = y
q = min_priority_queue(self.vertices, 'weight')
while q.heap_size > 0:
u = q.heap_extract_min()
for v in self.adj[u]:
if v in q and w(u, v) < v.weight:
v.root = u.root
q.heap_decrease_key(v.index, w(u, v))
v.p = u
def alledges_undirected_dfs(self):
global time, l
for u in self.vertices:
u.color = 0
u.p = None
time = 0
l = []
for u in self.vertices:
if u.color == 0:
self.alledges_undirected_dfs_visit(u)
return l
def alledges_undirected_dfs_visit(self, u):
global time, l
time = time + 1
u.d = time
u.color = 1
for v in self.adj[u]:
if v.color == 0:
l.append((u, v))
v.p = u
self.alledges_undirected_dfs_visit(v)
elif v.color == 1 and u.p != v:
l.append((u, v))
u.color = 2
time = time + 1
u.f = time
def Kruskal(self, w):
A = set()
for v in self.vertices:
DfsNode(v)
# ls = self.alledges_undirected_dfs()
for u, v in sorted(self.edges, key=lambda x: w(x[0], x[1]), reverse=False):
if u.index.find_set() != v.index.find_set():
A = A.union({(u, v)})
u.index.union(v.index)
return A
def Prim(self, w, r):
"""
G.Prim(weight, root) -- Given weight function
and an arbitrary vertex root of the graph G,
compute minimum spanning tree using Prim's algorithm
"""
for v in self.vertices:
v.weight = float("Inf")
v.p = None
r.weight = 0
q = min_priority_queue(self.vertices, 'weight')
while q.heap_size > 0:
u = q.heap_extract_min()
for v in self.adj[u]:
if v in q and w(u, v) < v.weight:
v.p = u
q.heap_decrease_key(v.index, w(u, v))
# def Prim_vEB(self, w, r, bound):
# """G.Prim(weight, root) -- Given weight function
# and an arbitrary vertex root of the graph G,
# compute minimum spanning tree using Prim's algorithm"""
# for v in self.vertices:
# v.weight = bound - 1
# v.p = None
# r.weight = 0
# t = vEB_node(bound)
# for u in self.vertices:
# t.insert(u)
# while t.size > 0:
# u = t.minimum()
# t.delete(u)
# for v in self.adj[u]:
# if t.member(v) and w(u, v) < v.weight:
# v.p = u
# t.delete(v)
# v.weight = w(u, v)
# t.insert(v)
def Bellman_Ford(self, w, s):
"""
The Bellman-Ford algorithm solves the single-source
shortest-paths problem in the general case in which edge
weights may be negative.
If there is a negative-weight cycle that is reachable from
the source s, this function returns False and indicates that
no solution exists.
If there is no such cycle, this function returns True and produces
the shortest paths and their weights.
"""
self.initialize_signle_source(s)
for i in range(1, len(self.vertices)):
for u, v in self.edges:
self.relax(u, v, w)
for u, v in self.edges:
if v.d > u.d + w(u, v):
return False
return True
def initialize_signle_source(self, s):
for v in self.vertices:
v.d = float("Inf")
v.p = None
s.d = 0
def relax(self, u, v, w):
if v.d > u.d + w(u, v):
v.d = u.d + w(u, v)
v.p = u
def Bellman_Ford_modified(self, w, s):
"""
Given a weighted, directed graph G = (V, E) with
no negative-weight cycles, let m be the maximum
over all vertices v of the minimum number of edges
in a shortest path from the source s to v. This variant to
the Bellman-Ford algorithm terminates in m + 1 passes, even
if m is not known in advance.
"""
modified = True
number = 0
self.initialize_signle_source(s)
for i in range(1, len(self.vertices)):
if modified:
for u, v in self.edges:
number = self.relax_modified(u, v, w) + number
if number == 0:
modified = False
number = 0
else:
break
for u, v in self.edges:
if v.d > u.d + w(u, v):
return False
return True
def relax_modified(self, u, v, w):
if v.d > u.d + w(u, v):
v.d = u.d + w(u, v)
v.p = u
return 1
else:
return 0
def dag_shortest_paths(self, w, s):
"""
compute shortest paths from a single source
s for a directed acyclic graph with a weight function w
"""
l = self.topological_sort()
self.initialize_signle_source(s)
for u in l:
for v in self.adj[u]:
self.relax(u, v, w)
def dag_shortest_paths_modified(self, s):
"""
In the PERT chart analysis, vertices repre
sent jobs and edges represent sequencing
contraints; that is, edge (u, v) would
indicate that job u must be performed
before job v. The weight attribute of
every vertex represent the time to
perform the job. This variant to
DAG-SHORTEST-PATHS algorithm gives a
solution to find a longest path in a
directed acyclic graph with weighted
vertices in linear time.
This function return a list of vertices in
a longest path
"""
sink = Vertex("sink")
vertices = self.vertices.union({sink})
edges = self.edges.union(set([(v, sink) for v in G.vertices]))
Ga = Graph(vertices, edges)
Ga.dag_shortest_paths(lambda u, v: -u.weight, s)
u = sink
l = []
while u.p is not None:
l.append(u.p)
u = u.p
return l[::-1]
def total_path_number(self):
"""
A algorithm to count the total number of paths in
a directed acyclic graph
"""
number = 0
self._total_path_number_dfs()
for v in self.vertices:
number = number + v.num
return number
def _total_path_number_dfs(self):
global time
for u in self.vertices:
u.color = 0
u.p = None
u.num = 0
time = 0
for u in self.vertices:
if u.color == 0:
self._total_path_number_dfs_visit(u)
def _total_path_number_dfs_visit(self, u):
global time
time = time + 1
u.d = time
u.color = 1
for v in self.adj[u]:
if v.color == 0:
v.p = u
self._total_path_number_dfs_visit(v)
u.num = u.num + v.num + 1
u.color = 2
time = time + 1
u.f = time
def Dijkstra(self, w, s):
"""
Dijkstra's algorithm solves the single-source shortest-paths problem
on a weighted, directed graph G = (V, E) for the case in which all edge
weights are nonnegative.
"""
self.initialize_signle_source(s)
S = set()
Q = min_priority_queue(self.vertices, 'd')
while Q.heap_size > 1:
u = Q.heap_extract_min()
S = S.union({u})
for v in self.adj[u]:
if v.d > u.d + w(u, v):
v.d = u.d + w(u, v)
v.p = u
Q.heap_decrease_key(v.index, u.d + w(u, v))
def Dijkstra_modified(self, w, s, W):
"""
A algorithm to the the case when the values of
the weight function w is in the range {0, 1, ..., W}
for some nonnegative integer W.
"""
self.initialize_signle_source(s)
A = []
for i in range(W * len(self.vertices) + 1):
A.append(set())
A[0].add(s)
i = 0
S = set()
while True:
while i <= W * len(self.vertices) and len(A[i]) == 0:
i = i + 1
if i > W * len(self.vertices):
break
print("i = {}".format(i))
u = A[i].pop()
print(u)
print(A[i])
S.add(u)
for v in self.adj[u]:
if v.d > u.d + w(u, v):
if v.d < float("Inf"):
A[int(v.d)].remove(v)
v.d = u.d + w(u, v)
A[int(v.d)].add(v)
v.p = u
def single_edge(self):
"""
An algorithm that given an adjacency-list representation
of a multigraph G = (V, E), compute the adjacency-list
representation of the "equivalent" undirected graph
G2 = (V, E2), where E2 consists of
the edges in E with all multiple edges between two vertices
replaced by a single edge and with all self-loops removed
"""
return Graph(self.vertices, self.edges, directed=False)
def union(self, G2):
if self.directed != G2.directed:
print("The two graphs must be either both directed graphs or both undirected graphs")
return None
vertices = self.vertices | G2.vertices
edges = self.edges | G2.edges
return Graph(vertices, edges, directed=self.directed)
def square(self):
"""
The square of a directed graph G = (V, E) is the graph
G^2 = (V, E^2) such that (u, v) belongs to E^2
if and only if G contains a path with at most
two edges between u and v.
"""
sqrt = self.copy()
for u in self.vertices:
for v in self.adj[u]:
for w in self.adj[v]:
sqrt._add_edge(u, w)
return sqrt
def height(self, u):
maximum = 0
print(u.key)
for v in self.adj[u]:
if v.p == u:
maximum = max(maximum, self.height(v) + 1)
u.h = maximum
print(u.key, u.h)
return u.h
def mht(self):
s = next(iter(self.vertices))
self.bfs(s)
self.height(s)
s.mh = s.h
for u in self.adj[s]:
self._mht_aux(u)
def _mht_aux(self, u):
u.mh = max(u.h, u.p.mh + 1)
for v in self.adj[u]:
if v.p == u:
self._mht_aux(v)
class DfsNode(dsf.node):
def __init__(self, key):
self.key = key
key.index = self
self.p = self
self.rank = 0
self.child = []
class max_heap(list):
def __init__(self, data, attr):
list.__init__(self, data)
for i in range(len(data)):
self[i].index = i
self.length = len(data)
self.attr = attr
self.heap_size = self.length
self.build_max_heap()
def __contains__(self, y):
return y in self[0:self.heap_size]
def left(self, i):
return 2 * i + 1
def right(self, i):
return 2 * i + 2
def parent(self, i):
return (i - 1) // 2
def max_heapify(self, i):
l = self.left(i)
r = self.right(i)
if (l <= (self.heap_size - 1)) and (self[l].__dict__[self.attr] > self[i].__dict__[self.attr]):
largest = l
else:
largest = i
if (r <= (self.heap_size - 1)) and (self[r].__dict__[self.attr] > self[largest].__dict__[self.attr]):
largest = r
if largest != i:
self[i], self[largest] = self[largest], self[i]
self[i].index = i
self[largest].index = largest
self.max_heapify(largest)
def build_max_heap(self):
self.heap_size = self.length
for i in range(self.length // 2 - 1, -1, -1):
self.max_heapify(i)
class max_priority_queue(max_heap):
def heap_maximum(self):
return self[0]
def heap_extract_max(self):
if self.heap_size < 1:
sys.exit("heap underflow")
maximum = self[0]
self[0] = self[self.heap_size - 1]
self[0].index = 0
self.heap_size = self.heap_size - 1
self.max_heapify(0)
return maximum
def heap_increase_key(self, i, key):
if key < self[i].__dict__[self.attr]:
sys.exit("new key is smaller than current key")
self[i].__dict__[self.attr] = key
while i > 0 and self[self.parent(i)].__dict__[self.attr] < self[i].__dict__[self.attr]:
self[i], self[self.parent(i)] = self[self.parent(i)], self[i]
self[i].index = i
self[self.parent(i)].index = self.parent(i)
i = self.parent(i)
def max_heap_insert(self, element):
if self.heap_size >= self.length:
sys.exit("heap overflow")
self.heap_size = self.heap_size + 1
self[self.heap_size - 1] = element
element.index = self.heap_size - 1
key = element.__dict__[self.attr]
element.__dict__[self.attr] = float("-Inf")
self.heap_increase_key(self.heap_size - 1, key)
class min_heap(list):
def __init__(self, data, attr):
"""
data: input data for heap
attr: the attribute of input date used as compare key
"""
list.__init__(self, data)
for i in range(len(data)):
self[i].index = i
self.attr = attr
self.length = len(data)
self.heap_size = self.length
self.build_min_heap()
def __contains__(self, y):
return y in self[0:self.heap_size]
def left(self, i):
return 2 * i + 1
def right(self, i):
return 2 * i + 2
def parent(self, i):
return (i - 1) // 2
def min_heapify(self, i):
l = self.left(i)
r = self.right(i)
if (l <= (self.heap_size - 1)) and (self[l].__dict__[self.attr] < self[i].__dict__[self.attr]):
smallest = l
else:
smallest = i
if (r <= (self.heap_size - 1)) and (self[r].__dict__[self.attr] < self[smallest].__dict__[self.attr]):
smallest = r
if smallest != i:
self[i], self[smallest] = self[smallest], self[i]
self[i].index = i
self[smallest].index = smallest
self.min_heapify(smallest)
def build_min_heap(self):
self.heap_size = self.length
for i in range(self.length // 2 - 1, -1, -1):
self.min_heapify(i)
class min_priority_queue(min_heap):
def heap_minimum(self):
return self[0]
def heap_extract_min(self):
if self.heap_size < 1:
sys.exit("heap underflow")
minimum = self[0]
self[0] = self[self.heap_size - 1]
self[0].index = 0
self.heap_size = self.heap_size - 1
self.min_heapify(0)
return minimum
def heap_decrease_key(self, i, key):
if key > self[i].__dict__[self.attr]:
sys.exit("new key is larger than current key")
self[i].__dict__[self.attr] = key
while i > 0 and self[self.parent(i)].__dict__[self.attr] > self[i].__dict__[self.attr]:
self[i], self[self.parent(i)] = self[self.parent(i)], self[i]
self[i].index = i
self[self.parent(i)].index = self.parent(i)
i = self.parent(i)
def min_heap_insert(self, element):
if self.heap_size >= self.length:
sys.exit("heap overflow")
self.heap_size = self.heap_size + 1
self[self.heap_size - 1] = element
element.index = self.heap_size - 1
key = element.__dict__[self.attr]
element.__dict__[self.attr] = float("Inf")
self.heap_decrease_key(self.heap_size - 1, key)