add longest path based on topological sorting

details in comments.

longest_path.py

@@ -0,0 +1,93 @@
+# Python program to find the longest distance between
+# a source vertex,s, and all other vertices in a weighted
+# directed acylic graph (DAG). The algorithm used Kahn's
+# topological sorting algoritm for DAGs to sort the vertices
+# in linear order. The longest path problem is not as easy
+# as the shortest path because it doesn't have the optimal
+# substructure property. The longest path problem, in fact,
+# is NP-hard for a general graph. But it has a linear solution
+# for a DAG. Time complexity is the same as for topological
+# sorting, since that is the bulk of the algorith. But space
+# complexity is O(V + E) because vertices and edges need to be
+# stored in arrays.
+
+
+def topologicalSortUtil(v):
+	global Stack, visited, adj
+	visited[v] = True
+
+	# Recur for all the vertices adjacent to this vertex
+	# list<AdjListNode>::iterator i
+	for i in adj[v]:
+		if (not visited[i[0]]):
+			topologicalSortUtil(i[0])
+
+	# Push current vertex to stack which stores topological
+	# sort
+	Stack.append(v)
+
+# The function to find longest distances from a given vertex.
+# It uses recursive topologicalSortUtil() to get topological
+# sorting.
+def longestPath(s):
+	global Stack, visited, adj, V
+	dist = [-10**9 for i in range(V)]
+
+	# Call the recursive helper function to store Topological
+	# Sort starting from all vertices one by one
+	for i in range(V):
+		if (visited[i] == False):
+			topologicalSortUtil(i)
+	# print(Stack)
+
+	# Initialize distances to all vertices as infinite and
+	# distance to source as 0
+	dist[s] = 0
+	# Stack.append(1)
+
+	# Process vertices in topological order
+	while (len(Stack) > 0):
+	
+		# Get the next vertex from topological order
+		u = Stack[-1]
+		del Stack[-1]
+		#print(u)
+
+		# Update distances of all adjacent vertices
+		# list<AdjListNode>::iterator i
+		if (dist[u] != 10**9):
+			for i in adj[u]:
+				# print(u, i)
+				if (dist[i[0]] < dist[u] + i[1]):
+					dist[i[0]] = dist[u] + i[1]
+
+	# Print calculated longest distances
+	# print(dist)
+	for i in range(V):
+		print("INF ",end="") if (dist[i] == -10**9) else print(dist[i],end=" ")
+
+# Driver code
+if __name__ == '__main__':
+	V, Stack, visited = 6, [], [False for i in range(7)]
+	adj = [[] for i in range(7)]
+	
+	# Create a graph given in the above diagram.
+	# Here vertex numbers are 0, 1, 2, 3, 4, 5 with
+	# following mappings:
+	# 0=r, 1=s, 2=t, 3=x, 4=y, 5=z
+	adj[0].append([1, 5])
+	adj[0].append([2, 3])
+	adj[1].append([3, 6])
+	adj[1].append([2, 2])
+	adj[2].append([4, 4])
+	adj[2].append([5, 2])
+	adj[2].append([3, 7])
+	adj[3].append([5, 1])
+	adj[3].append([4, -1])
+	adj[4].append([5, -2])
+
+	s = 1
+	print("The following are the longest distances from source vertex ",s)
+	longestPath(s)
+
+