(Problem R-14.16, page 678 of the text) Let G be a graph whose vertices are the...
(Problem R-14.16, page 678 of the text) Let G be a graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below:
Vertex adjacent vertices
1 (2,3,4)
2 (1,3,4)
3 (1,2,4)
4 (1,2,3,6)
5 (6,7,8)
6 (4,5,7)
7 (5,6,8)
8 (5,7)
Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the table above.
a) Draw G.
b) Give the sequence of vertices of G visited using a DFS traversal starting at vertex 1
c) Give the sequence of vertices visited using a BFS traversal starting at vertex 1.
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(Problem R-14.16, page 678 of the text) Let G be a graph whose
vertices are the...
Problem 3 (15 points). Let G (V,E) be the following directed graph. a. 1. Draw the...
Problem 3 (15 points). Let G (V,E) be the following directed graph. a. 1. Draw the reverse graph G of G. 2. Run DFS on G to obtain a post number for each vertex. Assume that in the adjacency list representation of G, vertices are stored alphabetically, and in the list for each vertex, its adjacent vertices are also sorted alphabetically. In other words, the DFS algorithm needs to examine all vertices alphabetically, and when it traverses the adjacent vertices...
from collections import defaultdict # This class represents a directed graph using # adjacency list...
from collections import defaultdict # This class represents a directed graph using # adjacency list representation class Graph: # Constructor def __init__(self): # default dictionary to store graph self.graph = defaultdict(list) # function to add an edge to graph def addEdge(self,u,v): self.graph[u].append(v) # Function to print a BFS of graph def BFS(self, s): # Mark all the vertices as not visited visited = [False] * (len(self.graph)) # Create a queue for BFS queue...
Let G (V, E) be a directed graph with n vertices and m edges. It is known that in dfsTrace of G t...
Let G (V, E) be a directed graph with n vertices and m edges. It is known that in dfsTrace of G the function dfs is called n times, once for each vertex It is also seen that dfs contains a loop whose body gets executed while visiting v once for each vertex w adjacent to v; that is the body gets executed once for each edge (v, w). In the worst case there are n adjacent vertices. What do...
#include <iostream> #include <queue> using namespace std; class Graph { public: Graph(int n); ~Graph(); void addEdge(int...
#include <iostream>
#include <queue>
using namespace std;
class Graph {
public:
Graph(int n);
~Graph();
void addEdge(int src, int tar);
void BFTraversal();
void DFTraversal();
void printVertices();
void printEdges();
private:
int vertexCount;
int edgeCount;
bool** adjMat;
void BFS(int n, bool marked[]);
void DFS(int n, bool marked[]);
};
Graph::Graph(int n=0) {
vertexCount = n;
edgeCount = 0;
if(n == 0)
adjMat = 0;
else {
adjMat = new bool* [n];
for(int i=0; i < n; i++)
adjMat[i] = new bool [n];
for(int i=0;...
# Problem 4 problem4_breadth_first_traversal = [0,] problem4_depth_first_traversal = [0,] def bfs(g,start): start.set...
# Problem 4
problem4_breadth_first_traversal = [0,]
problem4_depth_first_traversal = [0,]
def bfs(g,start):
start.setDistance(0)
start.setPred(None)
vertQueue = Queue()
vertQueue.enqueue(start)
while (vertQueue.size() > 0):
currentVert = vertQueue.dequeue()
for nbr in currentVert.getConnections():
if (nbr.getColor() == 'white'):
nbr.setColor('gray')
nbr.setDistance(currentVert.getDistance() + 1)
nbr.setPred(currentVert)
vertQueue.enqueue(nbr)
currentVert.setColor('black')
class DFSGraph(Graph):
def __init__(self):
super().__init__()
self.time = 0
def dfs(self):
for aVertex in self:
aVertex.setColor('white')
aVertex.setPred(-1)
for aVertex in self:
if aVertex.getColor() == 'white':
self.dfsvisit(aVertex)
def dfsvisit(self,startVertex):
startVertex.setColor('gray')
self.time += 1
startVertex.setDiscovery(self.time)
for nextVertex in startVertex.getConnections():
if nextVertex.getColor() == 'white':
nextVertex.setPred(startVertex)...
/* Graph read from file, and represnted as adjacency list. To implement DFS and BFS on...
/* Graph read from file, and represnted as adjacency list.
To implement DFS and BFS on the graph
*/
#include <iostream>
#include <sstream>
#include <fstream>
#include <vector>
#include <utility>
#include <unordered_map>
#include <set>
#include <queue>
using namespace std;
// Each vertex has an integer id.
typedef vector<vector<pair<int,int>>> adjlist; //
Pair: (head vertex, edge weight)
adjlist makeGraph(ifstream& ifs);
void printGraph(const adjlist& alist);
vector<int> BFS(const adjlist& alist, int source); //
Return vertices in BFS order
vector<int> DFS(const adjlist& alist, int source); //...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a sequence of distinct vertices V-(vi,v2, ,%)-V connected by a sequence of edges {(vi, t), (Ug, ta), , (4-1,Un)) : We say that (V, E) is a path from tovn. Sketch a graph with 10 vertices and a path consisting of 5 vertices and four edges. Formulate a binary integer program that could be used to find the path of least total weight from one...
Problem 2.13 - page 31. Let G be an n-vertex graph such that for any non-adjacent...
Problem 2.13 - page 31. Let G be an n-vertex graph such that for any non-adjacent vertices U, V EV(G), d(u) + d(u) > n. Prove that G is Hamiltonian
Bonus 1 A walk in a graph G is a sequence of vertices V1, V2, ...,...
Bonus 1 A walk in a graph G is a sequence of vertices V1, V2, ..., Uk such that {Vi, Vi+1} is an edge of G. Informally, a walk is a sequence of vertices where each step is taken along an edge. Note that a walk may visit the same vertex more than once. A closed walk is a walk where the first and last vertex are equal, i.e. v1 = Uk. The length of a walk is the number...
Below is the Graph file that needs to be modified(using Python3) : #!/usr/bin/python3 # Simple Vertex...
Below is the Graph file that
needs to be modified(using Python3) :
#!/usr/bin/python3
# Simple Vertex class
class Vertex:
""" Lightweight vertex structure for a graph.
Vertices can have the following labels:
UNEXPLORED
VISITED
Assuming the element of a vertex is string type
"""
__slots__ = '_element', '_label'
def __init__(self, element, label="UNEXPLORED"):
""" Constructor. """
self._element = element
self._label = label
def element(self):
""" Return element associated with this vertex. """
return self._element
def getLabel(self):
""" Get label assigned to...
Problem 3 (15 points). Let G (V,E) be the following directed graph. a. 1. Draw the reverse graph G of G. 2. Run DFS on G to obtain a post number for each vertex. Assume that in the adjacency list representation of G, vertices are stored alphabetically, and in the list for each vertex, its adjacent vertices are also sorted alphabetically. In other words, the DFS algorithm needs to examine all vertices alphabetically, and when it traverses the adjacent vertices...
Let G (V, E) be a directed graph with n vertices and m edges. It is known that in dfsTrace of G the function dfs is called n times, once for each vertex It is also seen that dfs contains a loop whose body gets executed while visiting v once for each vertex w adjacent to v; that is the body gets executed once for each edge (v, w). In the worst case there are n adjacent vertices. What do...
#include <iostream>
#include <queue>
using namespace std;
class Graph {
public:
Graph(int n);
~Graph();
void addEdge(int src, int tar);
void BFTraversal();
void DFTraversal();
void printVertices();
void printEdges();
private:
int vertexCount;
int edgeCount;
bool** adjMat;
void BFS(int n, bool marked[]);
void DFS(int n, bool marked[]);
};
Graph::Graph(int n=0) {
vertexCount = n;
edgeCount = 0;
if(n == 0)
adjMat = 0;
else {
adjMat = new bool* [n];
for(int i=0; i < n; i++)
adjMat[i] = new bool [n];
for(int i=0;...
# Problem 4
problem4_breadth_first_traversal = [0,]
problem4_depth_first_traversal = [0,]
def bfs(g,start):
start.setDistance(0)
start.setPred(None)
vertQueue = Queue()
vertQueue.enqueue(start)
while (vertQueue.size() > 0):
currentVert = vertQueue.dequeue()
for nbr in currentVert.getConnections():
if (nbr.getColor() == 'white'):
nbr.setColor('gray')
nbr.setDistance(currentVert.getDistance() + 1)
nbr.setPred(currentVert)
vertQueue.enqueue(nbr)
currentVert.setColor('black')
class DFSGraph(Graph):
def __init__(self):
super().__init__()
self.time = 0
def dfs(self):
for aVertex in self:
aVertex.setColor('white')
aVertex.setPred(-1)
for aVertex in self:
if aVertex.getColor() == 'white':
self.dfsvisit(aVertex)
def dfsvisit(self,startVertex):
startVertex.setColor('gray')
self.time += 1
startVertex.setDiscovery(self.time)
for nextVertex in startVertex.getConnections():
if nextVertex.getColor() == 'white':
nextVertex.setPred(startVertex)...
/* Graph read from file, and represnted as adjacency list.
To implement DFS and BFS on the graph
*/
#include <iostream>
#include <sstream>
#include <fstream>
#include <vector>
#include <utility>
#include <unordered_map>
#include <set>
#include <queue>
using namespace std;
// Each vertex has an integer id.
typedef vector<vector<pair<int,int>>> adjlist; //
Pair: (head vertex, edge weight)
adjlist makeGraph(ifstream& ifs);
void printGraph(const adjlist& alist);
vector<int> BFS(const adjlist& alist, int source); //
Return vertices in BFS order
vector<int> DFS(const adjlist& alist, int source); //...
Problem #1 Let a "path" on a weighted graph G = (V,E,W) be defined as a sequence of distinct vertices V-(vi,v2, ,%)-V connected by a sequence of edges {(vi, t), (Ug, ta), , (4-1,Un)) : We say that (V, E) is a path from tovn. Sketch a graph with 10 vertices and a path consisting of 5 vertices and four edges. Formulate a binary integer program that could be used to find the path of least total weight from one...
Problem 2.13 - page 31. Let G be an n-vertex graph such that for any non-adjacent vertices U, V EV(G), d(u) + d(u) > n. Prove that G is Hamiltonian
Bonus 1 A walk in a graph G is a sequence of vertices V1, V2, ..., Uk such that {Vi, Vi+1} is an edge of G. Informally, a walk is a sequence of vertices where each step is taken along an edge. Note that a walk may visit the same vertex more than once. A closed walk is a walk where the first and last vertex are equal, i.e. v1 = Uk. The length of a walk is the number...
Below is the Graph file that
needs to be modified(using Python3) :
#!/usr/bin/python3
# Simple Vertex class
class Vertex:
""" Lightweight vertex structure for a graph.
Vertices can have the following labels:
UNEXPLORED
VISITED
Assuming the element of a vertex is string type
"""
__slots__ = '_element', '_label'
def __init__(self, element, label="UNEXPLORED"):
""" Constructor. """
self._element = element
self._label = label
def element(self):
""" Return element associated with this vertex. """
return self._element
def getLabel(self):
""" Get label assigned to...
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