# Graph Algorithms using basic python Constructs.
# Narayana Chikkam, Dec, 22, 2015.
from collections import defaultdict
from heapq import *
import itertools
import copy
from lib.unionfind import (
UnionFind
)
from lib.prioritydict import (
priorityDictionary
)
class Vertex:
def __init__(self, id):
self.id = id
self.neighbours = {}
def addNeighbour(self, id, weight):
self.neighbours[id] = weight
def __str__(self):
return str(self.id) + ': ' + str(self.neighbours.keys())
def getNeighbours(self):
return self.neighbours #.keys()
def getName(self):
return self.id
def getWeight(self, id):
return self.neighbours[id]
class Graph:
def __init__(self):
self.v = {}
self.count = 0
def addVertex(self, key):
self.count += 1
newV = Vertex(key)
self.v[key] = newV
def getVertex(self, id):
if id in self.v.keys():
return self.v[id]
return None
def __contains__(self, id):
return id in self.v.keys()
def addEdge(self, vertexOne, vertexTwo, weight=None): # vertexOne, vertexTwo, cost-of-the-edge
if vertexOne not in self.v.keys():
self.addVertex(vertexOne)
if vertexTwo not in self.v.keys():
self.addVertex(vertexTwo)
self.v[vertexOne].addNeighbour(vertexTwo, weight)
def updateEdge(self, vertexOne, vertexTwo, weight=None): # vertexOne, vertexTwo, cost-of-the-edge
self.v[vertexOne].addNeighbour(vertexTwo, weight)
def getVertices(self):
return self.v.keys()
def __str__(self):
ret = "{ "
for v in self.v.keys():
ret += str(self.v[v].__str__()) + ", "
return ret + " }"
def __iter__(self):
return iter(self.v.values())
def getNeighbours(self, vertex):
if vertex not in self.v.keys():
raise "Node %s not in graph" % vertex
return self.v[vertex].neighbours #.keys()
def getEdges(self):
edges = []
for node in self.v.keys():
neighbours = self.v[node].getNeighbours()
for w in neighbours:
edges.append((node, w, neighbours[w])) #tuple, srcVertex, dstVertex, weightBetween
return edges
def findIsolated(self):
isolated = []
for node in self.v:
deadNode = False
reachable = True
# dead node, can't reach any other node from this
if len(self.v[node].getNeighbours()) == 0:
deadNode = True
# reachable from other nodes ?
nbrs = [n.neighbours.keys() for n in self.v.values()]
# flatten the nested list
nbrs = list(itertools.chain(*nbrs))
if node not in nbrs:
reachable = False
if deadNode == True and reachable == False:
isolated.append(node)
return isolated
def getPath(self, start, end, path=[]):
path = path + [start]
if start == end:
return path
if start not in self.v:
return None
for vertex in self.v[start].getNeighbours():
if vertex not in path:
extended_path = self.getPath(vertex,
end,
path)
if extended_path:
return extended_path
return None
def getAllPaths(self, start, end, path=[]):
path = path + [start]
if start == end:
return [path]
if start not in self.v:
return []
paths = []
for vertex in self.v[start].getNeighbours():
if vertex not in path:
extended_paths = self.getAllPaths(vertex,
end,
path)
for p in extended_paths:
paths.append(p)
return paths
def inDegree(self, vertex):
"""
how many edges coming into this vertex
"""
nbrs = [n.neighbours.keys() for n in self.v.values()]
# flatten the nested list
nbrs = list(itertools.chain(*nbrs))
return nbrs.count(vertex)
def outDegree(self, vertex):
"""
how many vertices are neighbours to this vertex
"""
adj_vertices = self.v[vertex].getNeighbours()
return len(adj_vertices)
"""
The degree of a vertex is the no of edges connecting to it.
loop is counted twice
for an undirected Graph deg(v) = indegree(v) + outdegree(v)
"""
def getDegree(self, vertex):
return self.inDegree(vertex) + self.outDegree(vertex)
def verifyDegreeSumFormula(self):
"""Handshaking lemma - Vdeg(v) = 2 |E| """
degSum = 0
for v in self.v:
degSum += self.getDegree(v)
return degSum == (2* len(self.getEdges()))
def delta(self):
""" the minimum degree of the Graph V """
min = 2**64
for vertex in self.v:
vertex_degree = self.getDegree(vertex)
if vertex_degree < min:
min = vertex_degree
return min
def Delta(self):
""" the maximum degree of the Graph V """
max = -2**64
for vertex in self.v:
vertex_degree = self.getDegree(vertex)
if vertex_degree > max:
max = vertex_degree
return max
def degreeSequence(self):
"""
degree sequence is the reverse sorder of the vertices degrees
Isomorphic graphs have the same degree sequence. However,
two graphs with the same degree sequence are not necessarily
isomorphic.
More-Info:
http://en.wikipedia.org/wiki/Graph_realization_problem
"""
seq = []
for vertex in self.v:
seq.append(self.getDegree(vertex))
seq.sort(reverse=True)
return tuple(seq)
# helper to check if the given sequence is in non-increasing Order ;)
@staticmethod
def sortedInDescendingOrder(seq):
return all (x>=y for x,y in zip(seq, seq[1:]))
@staticmethod
def isGraphicSequence(seq):
"""
Assumes that the degreeSequence is a list of non negative integers
http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Gallai_theorem
"""
# Check to ensure there are an even number of odd degrees
if sum(seq)%2 != 0: return False
# Erdos-Gallai theorem
for k in range(1, len(seq)+1):
leftSum = sum(seq[:(k)])
rightSum = k * (k-1) + sum([min(x, k) for x in seq[k:]])
if leftSum > rightSum: return False
return True
@staticmethod
def isGraphicSequenceIterative(s):
# successively reduce degree sequence by removing node of maximum degree
# as in Havel-Hakimi algorithm
while s:
s.sort() # sort in increasing order
if s[0]<0:
return False # check if removed too many from some node
d=s.pop() # pop largest degree
if d==0: return True # done! rest must be zero due to ordering
# degree must be <= number of available nodes
if d>len(s): return False
# remove edges to nodes of next higher degrees
#s.reverse() # to make it easy to get at higher degree nodes.
for i in range(len(s)-1,len(s)-(d+1),-1):
s[i]-=1
# should never get here b/c either d==0, d>len(s) or d<0 before s=[]
return False
def density(self):
"""
In mathematics, a dense graph is a graph in which the number of edges
is close to the maximal number of edges. The opposite, a graph with
only a few edges, is a sparse graph. The distinction between sparse
and dense graphs is rather vague, and depends on the context.
For undirected simple graphs, the graph density is defined as:
D = (2*No-Of-Edges)/((v*(v-1))/2)
For a complete Graph, the Density D is 1
"""
""" method to calculate the density of a graph """
V = len(self.v.keys())
E = len(self.getEdges())
return 2.0 * E / (V *(V - 1))
"""
Choose an arbitrary node x of the graph G as the starting point
Determine the set A of all the nodes which can be reached from x.
If A is equal to the set of nodes of G, the graph is connected; otherwise
it is disconnected.
"""
def isConnected(self, start=None):
if start == None:
start = self.v.keys()[0]
reachables = self.dfs(start, [])
return len(reachables) == len(self.v.keys())
"""
ToDo: USE CLR Approach for this Later
"""
def dfs(self, start, path = []):
path = path + [start]
for v in self.v[start].getNeighbours().keys():
if v not in path:
path = self.dfs(v, path)
return path
"""
CLR Sytle
"""
def CLR_Dfs(self):
paths = []
for v in self.v.keys():
explored = self.dfs(v, [])
if len(explored) == len(self.v.keys()):
paths.append(explored)
return paths
def BFS(self, start):
# initialize lists
maxV = len(self.v.keys())
processed = [False] * (maxV) # which vertices have been processed
discovered = [False] * (maxV) # which vertices have been found
parent= [-1] * (maxV) # discovery relation
q = [] # queue of vertices to visit */
# enqueue(&q,start);
q.append(start)
discovered[start] = True
while (len(q) != 0):
v = q.pop(0)
processed[v] = True
nbrs = self.v[v].getNeighbours().keys()
# print nbrs
for n in nbrs:
# if processed[n] == False
if discovered[n] == False:
q.append(n)
discovered[n] = True
parent[n] = v
return (discovered, parent)
def findPath(self, start, end, parents, path):
if ((start == end) or (end == -1)):
path.append(start)
else:
self.findPath(start, parents[end], parents, path)
path.append(end)
"""
Find path between two given nodes
"""
def find_path(self, start, end, path=[]):
path = path + [start]
if start == end:
return path
if not self.v.has_key(start):
return None
for node in self.v[start].getNeighbours().keys():
if node not in path:
newpath = self.find_path(node, end, path)
if newpath: return newpath
return None
"""
Find all paths
"""
def find_all_paths(self, start, end, path=[]):
path = path + [start]
if start == end:
return [path]
if not self.v.has_key(start):
return []
paths = []
for node in self.v[start].getNeighbours().keys():
if node not in path:
newpaths = self.find_all_paths(node, end, path)
for newpath in newpaths:
paths.append(newpath)
return paths
"""
Find shorted path w.r.t no of vertices on the path
"""
def find_shortest_path(self, start, end, path=[]):
path = path + [start]
if start == end:
return path
if not self.v.has_key(start):
return None
shortest = None
for node in self.v[start].getNeighbours().keys():
if node not in path:
newpath = self.find_shortest_path(node, end, path)
if newpath:
if not shortest or len(newpath) < len(shortest):
shortest = newpath
return shortest
"""
prim's algorithm - properties: tree could be not connected during
the finding process as it finds edges with min cost - greedy strategy
Prims always stays as a tree
If you don't know all the weight on edges use
Prim's algorithm
f you only need partial solution on the graph
use Prim's algorithm
"""
def mspPrims(self):
nodes = self.v.keys()
edges = [(u, v, c) for u in self.v.keys() for v, c in self.v[u].getNeighbours().items()]
return self.prim(nodes, edges)
def prim(self, nodes, edges):
conn = defaultdict( list )
for n1,n2,c in edges: # makes graph undirected
conn[ n1 ].append( (c, n1, n2) )
conn[ n2 ].append( (c, n2, n1) )
mst = []
used = set()
used.add( nodes[0] )
usable_edges = conn[ nodes[0] ][:]
heapify( usable_edges )
while usable_edges:
cost, n1, n2 = heappop( usable_edges )
if n2 not in used:
used.add( n2 )
mst.append( ( n1, n2, cost ) )
for e in conn[ n2 ]:
if e[ 2 ] not in used:
heappush( usable_edges, e )
return mst
"""
Kruskals begins with forest and merge into a tree
"""
def mspKrushkals(self):
nodes = self.v.keys()
edges = [(c, u, v) for u in self.v.keys() for v, c in self.v[u].getNeighbours().items()]
return self.krushkal(edges)
def pprint(self):
print ("{ ", end=" ")
for u in self.v.keys():
print (u, end=" ")
print (": { ", end=" ")
for v in self.v[u].getNeighbours().keys():
print (v, ":", self.v[u].getNeighbours()[v], end=" ")
print(" }", end= " ")
print (" }\n")
def krushkal(self, edges):
"""
Return the minimum spanning tree of an undirected graph G.
G should be represented in such a way that iter(G) lists its
vertices, iter(G[u]) lists the neighbors of u, G[u][v] gives the
length of edge u,v, and G[u][v] should always equal G[v][u].
The tree is returned as a list of edges.
"""
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
subtrees = UnionFind()
tree = []
for c,u,v in sorted(edges): # take from small weight to large in order
if subtrees[u] != subtrees[v]:
tree.append((u,v, c))
subtrees.union(u,v)
return tree
def adj(self, missing=float('inf')): # makes the adj dict will all possible cells, similar to matrix
"""
G= { 0 : { 1 : 6, 2 : 4 }
1 : { 2 : 3, 5 : 7 }
2 : { 3 : 9, 4 : 1 }
3 : { 4 : 1 }
4 : { 5 : 5, 6 : 2 }
5 : { }
6 : { }
}
adj(G) >>
{ 0: {0: 0, 1: 6, 2: 4, 3: inf, 4: inf, 5: inf, 6: inf},
1: {0: inf, 1: 0, 2: 3, 3: inf, 4: inf, 5: 7, 6: inf},
2: {0: inf, 1: inf, 2: 0, 3: 9, 4: 1, 5: inf, 6: inf},
3: {0: inf, 1: inf, 2: inf, 3: 0, 4: 1, 5: inf, 6: inf},
4: {0: inf, 1: inf, 2: inf, 3: inf, 4: 0, 5: 5, 6: 2},
5: {0: inf, 1: inf, 2: inf, 3: inf, 4: inf, 5: 0, 6: inf},
6: {0: inf, 1: inf, 2: inf, 3: inf, 4: inf, 5: inf, 6: 0}
}
"""
vertices = self.v.keys()
return {v1:
{v2: 0 if v1 == v2 else self.v[v1].getNeighbours().get(v2, missing) for v2 in vertices
}
for v1 in vertices
}
def floyds(self):
"""
All pair shortest Path
Idea:
for k in (0, n):
for i in (0, n):
for j in (0, n):
g[i][j] = min(graph[i][j], graph[i][k]+graph[k][j])
Find the shortest distance between every pair of vertices in the weighted Graph G
"""
d = self.adj() # prepare the adjacency list representation for the algorithm
vertices = self.v.keys()
for v2 in vertices:
d = {v1: {v3: min(d[v1][v3], d[v1][v2] + d[v2][v3])
for v3 in vertices}
for v1 in vertices}
return d
def reachability(self):
""" Idea: graph reachability floyd-warshall
for k in (0, n):
for i in (0, n):
for j in (0, n):
g[i][j] = graph[i][j] || (graph[i][k]&&graph[k][j]))
"""
vertices = self.v.keys()
d = self.adj(float('0'))
for u in vertices:
for v in vertices:
if u ==v or d[u][v]: d[u][v] = True
else: d[u][v] = False
for v2 in vertices:
d = {v1: {v3: d[v1][v3] or (d[v1][v2] and d[v2][v3]) # path for v1->v3 or v1->v2, v2-?v3
for v3 in vertices}
for v1 in vertices}
return d
def pathRecoveryFloydWarshall(self):
d = self.adj() # missing edges will have -1.0 value
vertices = self.v.keys()
parentMap = copy.deepcopy(d)
for v1 in vertices:
for v2 in vertices:
if (v1 == v2) or d[v1][v2] == float('inf'):
parentMap[v1][v2] = -1
else:
parentMap[v1][v2] = v1
for i in vertices:
for j in vertices:
for k in vertices:
temp = d[i][k] + d[k][j]
if temp < d[i][j]:
d[i][j] = temp
parentMap[i][j] = parentMap[k][j]
return parentMap
def getFloydPath(self, parentMap, u, v, path=[]):
"""
recursive procedure to get the path from parentMap matrix
"""
path.append(v)
if u != v and v != -1:
self.getFloydPath(parentMap, u, parentMap[u][v], path)
# from active recipes - handy thoughts to think about heap for this algorithm
def dijkstra(self, start, end=None):
"""
Find shortest paths from the start vertex to all
vertices nearer than or equal to the end.
The input graph G is assumed to have the following
representation: A vertex can be any object that can
be used as an index into a dictionary. G is a
dictionary, indexed by vertices. For any vertex v,
G[v] is itself a dictionary, indexed by the neighbors
of v. For any edge v->w, G[v][w] is the length of
the edge. This is related to the representation in
<http://www.python.org/doc/essays/graphs.html>
Of course, G and G[v] need not be Python dict objects;
they can be any other object that obeys dict protocol,
for instance a wrapper in which vertices are URLs
and a call to G[v] loads the web page and finds its links.
The output is a pair (D,P) where D[v] is the distance
from start to v and P[v] is the predecessor of v along
the shortest path from s to v.
Dijkstra's algorithm is only guaranteed to work correctly
when all edge lengths are positive. This code does not
verify this property for all edges (only the edges seen
before the end vertex is reached), but will correctly
compute shortest paths even for some graphs with negative
edges, and will raise an exception if it discovers that
a negative edge has caused it to make a mistake.
Introduction to Algorithms, 1st edition), page 528:
G = { 's':{'u':10, 'x':5},
' u':{'v':1, 'x':2},
'v':{'y':4},
'x':{'u':3, 'v':9, 'y':2},
'y':{'s':7, 'v':6}
}
"""
G = self.adj()
D = {} # dictionary of final distances
P = {} # dictionary of predecessors
Q = priorityDictionary() # est.dist. of non-final vert.
Q[start] = 0
for v in Q:
D[v] = Q[v]
if v == end: break
for w in G[v]:
vwLength = D[v] + G[v][w]
if w in D:
if vwLength < D[w]:
raise (ValueError, "Dijkstra: found better path to already-final vertex")
elif w not in Q or vwLength < Q[w]:
Q[w] = vwLength
P[w] = v
return D,P
def shortestPathDijkstra(self, start, end):
"""
Find a single shortest path from the given start vertex
to the given end vertex.
The input has the same conventions as Dijkstra().
The output is a list of the vertices in order along
the shortest path.
"""
D, P = self.dijkstra(start, end)
Path = []
while 1:
Path.append(end)
if end == start: break
end = P[end]
Path.reverse()
return Path
"""
smart snippet on the dijkstra alg:
def shortestPath(graph, start, end):
queue = [(0, start, [])]
seen = set()
while True:
(cost, v, path) = heapq.heappop(queue)
if v not in seen:
path = path + [v]
seen.add(v)
if v == end:
return cost, path
for (next, c) in graph[v].iteritems():
heapq.heappush(queue, (cost + c, next, path))
"""
def strongly_connected_components(self):
"""
Tarjan's Algorithm (named for its discoverer, Robert Tarjan) is a graph theory algorithm
for finding the strongly connected components of a graph.
Based on: http://en.wikipedia.org/wiki/Tarjan's_strongly_connected_components_algorithm
"""
index_counter = [0]
stack = []
lowlinks = {}
index = {}
result = []
def strongconnect(node):
# set the depth index for this node to the smallest unused index
index[node] = index_counter[0]
lowlinks[node] = index_counter[0]
index_counter[0] += 1
stack.append(node)
# Consider successors of `node`
try:
successors = self.v[node].getNeighbours().keys()
print(node, successors)
except:
successors = []
for successor in successors:
if successor not in lowlinks:
# Successor has not yet been visited; recurse on it
strongconnect(successor)
lowlinks[node] = min(lowlinks[node],lowlinks[successor])
elif successor in stack:
# the successor is in the stack and hence in the current strongly connected component (SCC)
lowlinks[node] = min(lowlinks[node],index[successor])
# If `node` is a root node, pop the stack and generate an SCC
if lowlinks[node] == index[node]:
connected_component = []
while True:
successor = stack.pop()
connected_component.append(successor)
if successor == node: break
component = tuple(connected_component)
# storing the result
print(component)
result.append(component)
for node in self.v.keys():
if node not in lowlinks:
strongconnect(node)
return result
def computeFirstUsingSCC(self, initFirst):
index_counter = [0]
stack = []
lowlinks = {}
index = {}
result = []
first = {}
def computeFirst(node):
# set the depth index for this node to the smallest unused index
index[node] = index_counter[0]
lowlinks[node] = index_counter[0]
index_counter[0] += 1
stack.append(node)
# Consider successors of `node`
try:
successors = self.v[node].getNeighbours().keys()
except:
successors = []
for successor in successors:
if successor not in lowlinks:
# Successor has not yet been visited; recurse on it
computeFirst(successor)
lowlinks[node] = min(lowlinks[node],lowlinks[successor])
elif successor in stack:
# the successor is in the stack and hence in the current strongly connected component (SCC)
lowlinks[node] = min(lowlinks[node],index[successor])
first[node] |= set(first[successor] - set(['epsilon'])).union(set(initFirst[node])) #(*union!*)
# If `node` is a root node, pop the stack and generate an SCC
if lowlinks[node] == index[node]:
connected_component = []
while True:
successor = stack.pop()
#FIRST[w] := FIRST[v]; (*distribute!*)
first[successor] = set(first[node] - set(['epsilon'])).union(set(initFirst[successor]) )#(*distribute!*)
connected_component.append(successor)
if successor == node: break
component = tuple(connected_component)
# storing the result
result.append(component)
for v in initFirst:
first[v] = initFirst[v] #(*init!*)
#print "init First assignment: ", first
for node in self.v.keys():
if node not in lowlinks:
computeFirst(node)
return first
def computeFollowUsingSCC(self, FIRST, initFollow):
index_counter = [0]
stack = []
lowlinks = {}
index = {}
result = []
follow = {}
def computeFollow(node):
# set the depth index for this node to the smallest unused index
index[node] = index_counter[0]
lowlinks[node] = index_counter[0]
index_counter[0] += 1
stack.append(node)
# Consider successors of `node`
try:
successors = self.v[node].getNeighbours().keys()
except:
successors = []
for successor in successors:
if successor not in lowlinks:
# Successor has not yet been visited; recurse on it
computeFollow(successor)
lowlinks[node] = min(lowlinks[node],lowlinks[successor])
elif successor in stack:
# the successor is in the stack and hence in the current strongly connected component (SCC)
lowlinks[node] = min(lowlinks[node],index[successor])
follow[node] |= follow[successor] #(*union!*)
# If `node` is a root node, pop the stack and generate an SCC
if lowlinks[node] == index[node]:
connected_component = []
while True:
successor = stack.pop()
follow[successor] = follow[node]
connected_component.append(successor)
if successor == node: break
component = tuple(connected_component)
# storing the result
result.append(component)
for v in initFollow:
follow[v] = initFollow[v] #(*init!*)
for node in self.v.keys():
if node not in lowlinks:
computeFollow(node)
return follow
Diff to Previous Revision
--- revision 1 2015-12-22 10:46:28
+++ revision 2 2015-12-22 10:49:59
@@ -1,4 +1,4 @@
-# Graph Algorithms using basic pythos Constructs.
+# Graph Algorithms using basic python Constructs.
# Narayana Chikkam, Dec, 22, 2015.
from collections import defaultdict
