Create Kruskal's_Algorithm.py

Kruskal’s algorithm to find the minimum cost spanning tree uses the greedy approach. The Greedy Choice is to pick the smallest weight edge that does not cause a cycle in the MST constructed so far.

Time Complexity: O(ElogE) or O(ElogV), Sorting of edges takes O(ELogE) time. After sorting, we iterate through all edges and apply the find-union algorithm. The find and union operations can take at most O(LogV) time. So overall complexity is O(ELogE + ELogV) time. The value of E can be at most O(V2), so O(LogV) is O(LogE) the same. Therefore, the overall time complexity is O(ElogE) or O(ElogV)
Auxiliary Space: O(V + E), where V is the number of vertices and E is the number of edges in the graph

algorithms/Python/graphs/Kruskal's_Algorithm.py

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+# Python program for Kruskal's algorithm to find
+# Minimum Spanning Tree of a given connected,
+# undirected and weighted graph
+
+from collections import defaultdict
+
+# Class to represent a graph
+
+
+class Graph:
+
+	def __init__(self, vertices):
+		self.V = vertices # No. of vertices
+		self.graph = [] # default dictionary
+		# to store graph
+
+	# function to add an edge to graph
+	def addEdge(self, u, v, w):
+		self.graph.append([u, v, w])
+
+	# A utility function to find set of an element i
+	# (uses path compression technique)
+	def find(self, parent, i):
+		if parent[i] == i:
+			return i
+		return self.find(parent, parent[i])
+
+	# A function that does union of two sets of x and y
+	# (uses union by rank)
+	def union(self, parent, rank, x, y):
+		xroot = self.find(parent, x)
+		yroot = self.find(parent, y)
+
+		# Attach smaller rank tree under root of
+		# high rank tree (Union by Rank)
+		if rank[xroot] < rank[yroot]:
+			parent[xroot] = yroot
+		elif rank[xroot] > rank[yroot]:
+			parent[yroot] = xroot
+
+		# If ranks are same, then make one as root
+		# and increment its rank by one
+		else:
+			parent[yroot] = xroot
+			rank[xroot] += 1
+
+	# The main function to construct MST using Kruskal's
+		# algorithm
+	def KruskalMST(self):
+
+		result = [] # This will store the resultant MST
+
+		# An index variable, used for sorted edges
+		i = 0
+
+		# An index variable, used for result[]
+		e = 0
+
+		# Step 1: Sort all the edges in
+		# non-decreasing order of their
+		# weight. If we are not allowed to change the
+		# given graph, we can create a copy of graph
+		self.graph = sorted(self.graph,
+							key=lambda item: item[2])
+
+		parent = []
+		rank = []
+
+		# Create V subsets with single elements
+		for node in range(self.V):
+			parent.append(node)
+			rank.append(0)
+
+		# Number of edges to be taken is equal to V-1
+		while e < self.V - 1:
+
+			# Step 2: Pick the smallest edge and increment
+			# the index for next iteration
+			u, v, w = self.graph[i]
+			i = i + 1
+			x = self.find(parent, u)
+			y = self.find(parent, v)
+
+			# If including this edge doesn't
+			# cause cycle, then include it in result
+			# and increment the index of result
+			# for next edge
+			if x != y:
+				e = e + 1
+				result.append([u, v, w])
+				self.union(parent, rank, x, y)
+			# Else discard the edge
+
+		minimumCost = 0
+		print("Edges in the constructed MST")
+		for u, v, weight in result:
+			minimumCost += weight
+			print("%d -- %d == %d" % (u, v, weight))
+		print("Minimum Spanning Tree", minimumCost)
+
+
+# Driver's code
+if __name__ == '__main__':
+	g = Graph(4)
+	g.addEdge(0, 1, 10)
+	g.addEdge(0, 2, 6)
+	g.addEdge(0, 3, 5)
+	g.addEdge(1, 3, 15)
+	g.addEdge(2, 3, 4)
+
+	# Function call
+	g.KruskalMST()