In python. # primsfunctions.py from Weighted_Graph import # kruskals functions.p
ID: 3715550 • Letter: I
Question
In python.
# primsfunctions.py from Weighted_Graph import # kruskals functions.py from Weighted Graph import n This file should store all functions that This file should store all functions that you write which will be needed in the steps of Kruskal's algorithm" you write which will be needed in the steps of Prim's algorithm def c(G, e): def c(G, e): .. # Prims.py import prims functions.py # Kruskalspy import kruskals functions.py This file should implement Prim's This file should implement Kruskal's algorithmm algorithm def Kruskals(textfile):.. Grading Rubric: # MST.py from Prims import Prims from Kruskals import Kruskals Solve correctly the MST for a given graph: 90 its n This file should be your main program Prims or Kruskals (bonus for both) 1. Correct Optimal Cost 60 pts 2. Correct File Structure 20 pts 3. Creative user input and program design/ error checks: 10 pts (textfile, algorithm 'Prims'): if algorithm-'Prims' Write a detailed argument, or proof, for the fact that Prims (or Kruskals) always terminates with a optimal spanning tree: 10 pts return Prims(textfile) else: return Kruskals(textfile)Explanation / Answer
#prims implementation
import sys # Library for INT_MAX
class Graph():
def __init__(self, vertices):
self.V = vertices
self.graph = [[0 for column in range(vertices)]
for row in range(vertices)]
# A utility function to print the constructed MST stored in parent[]
def printMST(self, parent):
print "Edge Weight"
for i in range(1,self.V):
print parent[i],"-",i," ",self.graph[i][ parent[i] ]
# A utility function to find the vertex with minimum distance value, from
# the set of vertices not yet included in shortest path tree
def minKey(self, key, mstSet):
# Initilaize min value
min = sys.maxint
for v in range(self.V):
if key[v] < min and mstSet[v] == False:
min = key[v]
min_index = v
return min_index
# Function to construct and print MST for a graph represented using
# adjacency matrix representation
def primMST(self):
#Key values used to pick minimum weight edge in cut
key = [sys.maxint] * self.V
parent = [None] * self.V # Array to store constructed MST
key[0] = 0 # Make key 0 so that this vertex is picked as first vertex
mstSet = [False] * self.V
parent[0] = -1 # First node is always the root of
for cout in range(self.V):
# Pick the minimum distance vertex from the set of vertices not
# yet processed. u is always equal to src in first iteration
u = self.minKey(key, mstSet)
# Put the minimum distance vertex in the shortest path tree
mstSet[u] = True
# Update dist value of the adjacent vertices of the picked vertex
# only if the current distance is greater than new distance and
# the vertex in not in the shotest path tree
for v in range(self.V):
# graph[u][v] is non zero only for adjacent vertices of m
# mstSet[v] is false for vertices not yet included in MST
# Update the key only if graph[u][v] is smaller than key[v]
if self.graph[u][v] > 0 and mstSet[v] == False and
key[v] > self.graph[u][v]:
key[v] = self.graph[u][v]
parent[v] = u
self.printMST(parent)
g = Graph(5)
g.graph = [ [0, 2, 0, 6, 0],
[2, 0, 3, 8, 5],
[0, 3, 0, 0, 7],
[6, 8, 0, 0, 9],
[0, 5, 7, 9, 0],
]
g.primMST();
#kruskals implementation
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
i = 0 # An index variable, used for sorted edges
e = 0 # An index variable, used for result[]
# 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 does't cause cycle,
# 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
# print the contents of result[] to display the built MST
print "Following are the edges in the constructed MST"
for u,v,weight in result:
#print str(u) + " -- " + str(v) + " == " + str(weight)
print ("%d -- %d == %d" % (u,v,weight))
# Driver code
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)
g.KruskalMST()
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