# Chapter 7: Graph Algorithms

## 7.3 Breadth-First Search

Breadth-First Search (BFS) is a graph traversal algorithm that visits nodes in a level-wise manner, starting from the root node and moving outwards. This means that it visits all the nodes closest to the root first before moving on to the next level. BFS is an efficient algorithm for finding the shortest path between two nodes in a graph or tree.

It does this by exploring all possible paths from the starting node to the ending node, ensuring that it finds the shortest path. Moreover, BFS is not only used for finding shortest paths but also for binary tree traversals. It is particularly useful for searching for the closest neighbors of a particular node in a graph or tree, as it visits the nodes in a level-wise order, ensuring that the closest neighbors are visited first.

Therefore, BFS is an essential algorithm in computer science and is widely used in various applications, such as network routing, data mining, and image processing.

Here's how the BFS algorithm works:

- Select a starting node (the 'source' node).
- Explore all the vertices adjacent to the initial node.
- Move to the next level and explore all vertices there.
- Repeat the process until all vertices have been explored.

Let's make this more concrete with a simple pseudocode example:

`function BFS(graph, root) {`

create an empty queue Q

create an array visited of size = number of nodes in the graph, and set all to False

enqueue root into Q

set visited[root] = True

while Q is not empty {

current_node = dequeue from Q

print current_node

for each node i adjacent to current_node {

if visited[i] is False {

enqueue i into Q

set visited[i] = True

}

}

}

}

The BFS function operates by maintaining a queue of nodes to visit. It starts by visiting the root, then all of its neighbors, then the neighbors' neighbors, and so on.

Here's a Python example of BFS with a simple unweighted graph represented by an adjacency list:

`from collections import deque`

def BFS(graph, root):

visited = [False] * (len(graph))

queue = deque()

queue.append(root)

visited[root] = True

while queue:

vertex = queue.popleft()

print(vertex, end=" ")

for neighbour in graph[vertex]:

if visited[neighbour] == False:

queue.append(neighbour)

visited[neighbour] = True

# defining graph

graph = {

0: [1, 2],

1: [2],

2: [0, 3],

3: [3]

}

BFS(graph, 2) # Output: 2 0 3 1

In this code, we start at node 2 and then proceed to visit every node that's adjacent to it, and then the nodes adjacent to those, and so on until all the nodes have been visited.

Breadth-First Search (BFS) is a well-known algorithm in graph theory. It is a searching algorithm that traverses the graph in a breadth-first manner, starting from the source vertex and visiting the vertices in the order of their distance from the source vertex. This algorithm has a wide range of applications in various real-world problems, such as social networking, web-crawling, network broadcasting, and many more.

BFS is a highly efficient algorithm for finding the shortest path in an unweighted graph, as it always visits the nodes in the shortest possible path from the source node. It is also useful for finding all the connected components in a graph, performing level order traversal on a binary tree, and solving puzzles such as the Rubik's cube.

This algorithm starts by visiting the source vertex and enqueuing its adjacent vertices in a queue. Then it dequeues the vertex at the front of the queue and enqueues its unvisited adjacent vertices. This process continues until all the vertices are visited.

Breadth-First Search is a powerful algorithm that every computer science student should master. It has many practical applications in real-world problems and is a go-to strategy for many graph-related tasks. So, start practicing BFS today and unlock its full potential!

### 7.3.1 BFS Time Complexity

Breadth-first search, also known as BFS, is an algorithm used to traverse graphs, trees, and other data structures. It is particularly useful for finding the shortest path between two nodes in an unweighted graph, as well as for detecting cycles in a graph.

BFS explores all the vertices and edges in the worst-case scenario, which makes its time complexity represented as O(V + E), where V is the number of vertices and E is the number of edges. However, this complexity can be improved using various optimization techniques such as using an adjacency list instead of an adjacency matrix.

In terms of space complexity, BFS requires O(V) space since the queue needs to store all the vertices of the graph in the worst case. This can be a disadvantage for large graphs, but again, there are strategies to optimize this space usage.

Overall, BFS is a versatile and efficient algorithm for exploring graphs, and its time and space complexities can be improved with some optimization techniques.

We should also discuss that BFS, while primarily known for its traversing capabilities, also has other significant applications:

1. **Shortest Path and Scheduling Algorithms**

One of the most widely used algorithms in graph theory is the Breadth-First Search (BFS) algorithm, which is commonly used to find the shortest path in an unweighted graph. This algorithm is particularly useful in scheduling processes, where it can be used to optimize the order of tasks to be performed and minimize the time required to complete them.

By analyzing the relationships between tasks and their dependencies, BFS can help identify the most efficient path to achieve the desired outcome. The versatility of BFS has led to its implementation in a wide range of scheduling algorithms, making it a fundamental tool in various fields such as computer science, logistics, and project management.

2. **Cycle Detection in Undirected Graph**

BFS can check if the graph contains a cycle or not. It can also find if there is a path between two nodes. Cycle detection in an undirected graph is an important problem in graph theory. A graph is said to contain a cycle if there is a path that starts and ends at the same vertex. And one of the common ways to detect the cycle is by using Breadth First Search (BFS) algorithm.

BFS algorithm is useful not only in detecting whether a cycle exists in a graph but also it can find if there is a path between two nodes. This algorithm starts from the source vertex and traverses all the vertices of the graph in a breadth-first manner. By doing that, it can mark the vertices that have already been visited.

In this way, BFS algorithm can determine whether a cycle exists in the graph or not. Moreover, this algorithm can also find the shortest path between two nodes. That is why it is widely used in various applications such as network routing protocols, social network analysis, and so on.

3. **Crawlers in Search Engines**

Search engines rely on web crawlers to build up an index of web pages, and BFS (Breadth First Search) is one such algorithm that can help to navigate the vast expanse of the web. The algorithm starts from a source page and follows all the links from the source to other pages and so forth, hence traversing a vast breadth of the web.

By using BFS, search engines can retrieve information from a large number of pages and build up their index, which can then be used to provide relevant search results to users. Moreover, web crawlers are essential for search engines to keep their index up-to-date, as they can detect changes to web pages and update the index accordingly. Thus, without crawlers, search engines would not be able to provide accurate and up-to-date search results to users.

4. **Broadcasting in Network:**

In networks, a broadcasted packet follows the BFS algorithm to reach all nodes. When it comes to broadcasting in a network, a packet is sent out to multiple nodes at once. This process is known as a broadcast and it can be very useful when you want to send information to many different devices.

In order to ensure that the packet reaches all nodes, the BFS (Breadth-First Search) algorithm is typically used. This algorithm works by exploring all the neighboring nodes first, before moving on to the next level of nodes. By using this method, the packet is able to reach all nodes in the network and ensure that the information is delivered to everyone who needs it.

5. **In GPS Navigation**

BFS can find all locations within a given distance from a source location. In GPS navigation, BFS (breadth-first search) is a powerful algorithm that can be used to find all locations within a certain distance from a given source location. This is especially useful when you need to find nearby points of interest, such as gas stations, restaurants, or tourist attractions.

By using BFS, you can quickly and efficiently search through a large number of locations to identify those that are within the desired distance from your starting point. Additionally, BFS can be customized to take into account different factors, such as traffic conditions or road closures, to provide you with the most accurate and up-to-date information possible. Overall, BFS is an essential tool for anyone who wants to navigate unfamiliar terrain with confidence and ease.

## 7.3 Breadth-First Search

Breadth-First Search (BFS) is a graph traversal algorithm that visits nodes in a level-wise manner, starting from the root node and moving outwards. This means that it visits all the nodes closest to the root first before moving on to the next level. BFS is an efficient algorithm for finding the shortest path between two nodes in a graph or tree.

It does this by exploring all possible paths from the starting node to the ending node, ensuring that it finds the shortest path. Moreover, BFS is not only used for finding shortest paths but also for binary tree traversals. It is particularly useful for searching for the closest neighbors of a particular node in a graph or tree, as it visits the nodes in a level-wise order, ensuring that the closest neighbors are visited first.

Therefore, BFS is an essential algorithm in computer science and is widely used in various applications, such as network routing, data mining, and image processing.

Here's how the BFS algorithm works:

- Select a starting node (the 'source' node).
- Explore all the vertices adjacent to the initial node.
- Move to the next level and explore all vertices there.
- Repeat the process until all vertices have been explored.

Let's make this more concrete with a simple pseudocode example:

`function BFS(graph, root) {`

create an empty queue Q

create an array visited of size = number of nodes in the graph, and set all to False

enqueue root into Q

set visited[root] = True

while Q is not empty {

current_node = dequeue from Q

print current_node

for each node i adjacent to current_node {

if visited[i] is False {

enqueue i into Q

set visited[i] = True

}

}

}

}

The BFS function operates by maintaining a queue of nodes to visit. It starts by visiting the root, then all of its neighbors, then the neighbors' neighbors, and so on.

Here's a Python example of BFS with a simple unweighted graph represented by an adjacency list:

`from collections import deque`

def BFS(graph, root):

visited = [False] * (len(graph))

queue = deque()

queue.append(root)

visited[root] = True

while queue:

vertex = queue.popleft()

print(vertex, end=" ")

for neighbour in graph[vertex]:

if visited[neighbour] == False:

queue.append(neighbour)

visited[neighbour] = True

# defining graph

graph = {

0: [1, 2],

1: [2],

2: [0, 3],

3: [3]

}

BFS(graph, 2) # Output: 2 0 3 1

In this code, we start at node 2 and then proceed to visit every node that's adjacent to it, and then the nodes adjacent to those, and so on until all the nodes have been visited.

Breadth-First Search (BFS) is a well-known algorithm in graph theory. It is a searching algorithm that traverses the graph in a breadth-first manner, starting from the source vertex and visiting the vertices in the order of their distance from the source vertex. This algorithm has a wide range of applications in various real-world problems, such as social networking, web-crawling, network broadcasting, and many more.

BFS is a highly efficient algorithm for finding the shortest path in an unweighted graph, as it always visits the nodes in the shortest possible path from the source node. It is also useful for finding all the connected components in a graph, performing level order traversal on a binary tree, and solving puzzles such as the Rubik's cube.

This algorithm starts by visiting the source vertex and enqueuing its adjacent vertices in a queue. Then it dequeues the vertex at the front of the queue and enqueues its unvisited adjacent vertices. This process continues until all the vertices are visited.

Breadth-First Search is a powerful algorithm that every computer science student should master. It has many practical applications in real-world problems and is a go-to strategy for many graph-related tasks. So, start practicing BFS today and unlock its full potential!

### 7.3.1 BFS Time Complexity

Breadth-first search, also known as BFS, is an algorithm used to traverse graphs, trees, and other data structures. It is particularly useful for finding the shortest path between two nodes in an unweighted graph, as well as for detecting cycles in a graph.

BFS explores all the vertices and edges in the worst-case scenario, which makes its time complexity represented as O(V + E), where V is the number of vertices and E is the number of edges. However, this complexity can be improved using various optimization techniques such as using an adjacency list instead of an adjacency matrix.

In terms of space complexity, BFS requires O(V) space since the queue needs to store all the vertices of the graph in the worst case. This can be a disadvantage for large graphs, but again, there are strategies to optimize this space usage.

Overall, BFS is a versatile and efficient algorithm for exploring graphs, and its time and space complexities can be improved with some optimization techniques.

We should also discuss that BFS, while primarily known for its traversing capabilities, also has other significant applications:

1. **Shortest Path and Scheduling Algorithms**

One of the most widely used algorithms in graph theory is the Breadth-First Search (BFS) algorithm, which is commonly used to find the shortest path in an unweighted graph. This algorithm is particularly useful in scheduling processes, where it can be used to optimize the order of tasks to be performed and minimize the time required to complete them.

By analyzing the relationships between tasks and their dependencies, BFS can help identify the most efficient path to achieve the desired outcome. The versatility of BFS has led to its implementation in a wide range of scheduling algorithms, making it a fundamental tool in various fields such as computer science, logistics, and project management.

2. **Cycle Detection in Undirected Graph**

BFS can check if the graph contains a cycle or not. It can also find if there is a path between two nodes. Cycle detection in an undirected graph is an important problem in graph theory. A graph is said to contain a cycle if there is a path that starts and ends at the same vertex. And one of the common ways to detect the cycle is by using Breadth First Search (BFS) algorithm.

BFS algorithm is useful not only in detecting whether a cycle exists in a graph but also it can find if there is a path between two nodes. This algorithm starts from the source vertex and traverses all the vertices of the graph in a breadth-first manner. By doing that, it can mark the vertices that have already been visited.

In this way, BFS algorithm can determine whether a cycle exists in the graph or not. Moreover, this algorithm can also find the shortest path between two nodes. That is why it is widely used in various applications such as network routing protocols, social network analysis, and so on.

3. **Crawlers in Search Engines**

Search engines rely on web crawlers to build up an index of web pages, and BFS (Breadth First Search) is one such algorithm that can help to navigate the vast expanse of the web. The algorithm starts from a source page and follows all the links from the source to other pages and so forth, hence traversing a vast breadth of the web.

By using BFS, search engines can retrieve information from a large number of pages and build up their index, which can then be used to provide relevant search results to users. Moreover, web crawlers are essential for search engines to keep their index up-to-date, as they can detect changes to web pages and update the index accordingly. Thus, without crawlers, search engines would not be able to provide accurate and up-to-date search results to users.

4. **Broadcasting in Network:**

In networks, a broadcasted packet follows the BFS algorithm to reach all nodes. When it comes to broadcasting in a network, a packet is sent out to multiple nodes at once. This process is known as a broadcast and it can be very useful when you want to send information to many different devices.

In order to ensure that the packet reaches all nodes, the BFS (Breadth-First Search) algorithm is typically used. This algorithm works by exploring all the neighboring nodes first, before moving on to the next level of nodes. By using this method, the packet is able to reach all nodes in the network and ensure that the information is delivered to everyone who needs it.

5. **In GPS Navigation**

BFS can find all locations within a given distance from a source location. In GPS navigation, BFS (breadth-first search) is a powerful algorithm that can be used to find all locations within a certain distance from a given source location. This is especially useful when you need to find nearby points of interest, such as gas stations, restaurants, or tourist attractions.

By using BFS, you can quickly and efficiently search through a large number of locations to identify those that are within the desired distance from your starting point. Additionally, BFS can be customized to take into account different factors, such as traffic conditions or road closures, to provide you with the most accurate and up-to-date information possible. Overall, BFS is an essential tool for anyone who wants to navigate unfamiliar terrain with confidence and ease.

## 7.3 Breadth-First Search

Breadth-First Search (BFS) is a graph traversal algorithm that visits nodes in a level-wise manner, starting from the root node and moving outwards. This means that it visits all the nodes closest to the root first before moving on to the next level. BFS is an efficient algorithm for finding the shortest path between two nodes in a graph or tree.

It does this by exploring all possible paths from the starting node to the ending node, ensuring that it finds the shortest path. Moreover, BFS is not only used for finding shortest paths but also for binary tree traversals. It is particularly useful for searching for the closest neighbors of a particular node in a graph or tree, as it visits the nodes in a level-wise order, ensuring that the closest neighbors are visited first.

Therefore, BFS is an essential algorithm in computer science and is widely used in various applications, such as network routing, data mining, and image processing.

Here's how the BFS algorithm works:

- Select a starting node (the 'source' node).
- Explore all the vertices adjacent to the initial node.
- Move to the next level and explore all vertices there.
- Repeat the process until all vertices have been explored.

Let's make this more concrete with a simple pseudocode example:

`function BFS(graph, root) {`

create an empty queue Q

create an array visited of size = number of nodes in the graph, and set all to False

enqueue root into Q

set visited[root] = True

while Q is not empty {

current_node = dequeue from Q

print current_node

for each node i adjacent to current_node {

if visited[i] is False {

enqueue i into Q

set visited[i] = True

}

}

}

}

The BFS function operates by maintaining a queue of nodes to visit. It starts by visiting the root, then all of its neighbors, then the neighbors' neighbors, and so on.

Here's a Python example of BFS with a simple unweighted graph represented by an adjacency list:

`from collections import deque`

def BFS(graph, root):

visited = [False] * (len(graph))

queue = deque()

queue.append(root)

visited[root] = True

while queue:

vertex = queue.popleft()

print(vertex, end=" ")

for neighbour in graph[vertex]:

if visited[neighbour] == False:

queue.append(neighbour)

visited[neighbour] = True

# defining graph

graph = {

0: [1, 2],

1: [2],

2: [0, 3],

3: [3]

}

BFS(graph, 2) # Output: 2 0 3 1

In this code, we start at node 2 and then proceed to visit every node that's adjacent to it, and then the nodes adjacent to those, and so on until all the nodes have been visited.

Breadth-First Search (BFS) is a well-known algorithm in graph theory. It is a searching algorithm that traverses the graph in a breadth-first manner, starting from the source vertex and visiting the vertices in the order of their distance from the source vertex. This algorithm has a wide range of applications in various real-world problems, such as social networking, web-crawling, network broadcasting, and many more.

BFS is a highly efficient algorithm for finding the shortest path in an unweighted graph, as it always visits the nodes in the shortest possible path from the source node. It is also useful for finding all the connected components in a graph, performing level order traversal on a binary tree, and solving puzzles such as the Rubik's cube.

This algorithm starts by visiting the source vertex and enqueuing its adjacent vertices in a queue. Then it dequeues the vertex at the front of the queue and enqueues its unvisited adjacent vertices. This process continues until all the vertices are visited.

Breadth-First Search is a powerful algorithm that every computer science student should master. It has many practical applications in real-world problems and is a go-to strategy for many graph-related tasks. So, start practicing BFS today and unlock its full potential!

### 7.3.1 BFS Time Complexity

Breadth-first search, also known as BFS, is an algorithm used to traverse graphs, trees, and other data structures. It is particularly useful for finding the shortest path between two nodes in an unweighted graph, as well as for detecting cycles in a graph.

BFS explores all the vertices and edges in the worst-case scenario, which makes its time complexity represented as O(V + E), where V is the number of vertices and E is the number of edges. However, this complexity can be improved using various optimization techniques such as using an adjacency list instead of an adjacency matrix.

In terms of space complexity, BFS requires O(V) space since the queue needs to store all the vertices of the graph in the worst case. This can be a disadvantage for large graphs, but again, there are strategies to optimize this space usage.

Overall, BFS is a versatile and efficient algorithm for exploring graphs, and its time and space complexities can be improved with some optimization techniques.

We should also discuss that BFS, while primarily known for its traversing capabilities, also has other significant applications:

1. **Shortest Path and Scheduling Algorithms**

One of the most widely used algorithms in graph theory is the Breadth-First Search (BFS) algorithm, which is commonly used to find the shortest path in an unweighted graph. This algorithm is particularly useful in scheduling processes, where it can be used to optimize the order of tasks to be performed and minimize the time required to complete them.

By analyzing the relationships between tasks and their dependencies, BFS can help identify the most efficient path to achieve the desired outcome. The versatility of BFS has led to its implementation in a wide range of scheduling algorithms, making it a fundamental tool in various fields such as computer science, logistics, and project management.

2. **Cycle Detection in Undirected Graph**

BFS can check if the graph contains a cycle or not. It can also find if there is a path between two nodes. Cycle detection in an undirected graph is an important problem in graph theory. A graph is said to contain a cycle if there is a path that starts and ends at the same vertex. And one of the common ways to detect the cycle is by using Breadth First Search (BFS) algorithm.

BFS algorithm is useful not only in detecting whether a cycle exists in a graph but also it can find if there is a path between two nodes. This algorithm starts from the source vertex and traverses all the vertices of the graph in a breadth-first manner. By doing that, it can mark the vertices that have already been visited.

In this way, BFS algorithm can determine whether a cycle exists in the graph or not. Moreover, this algorithm can also find the shortest path between two nodes. That is why it is widely used in various applications such as network routing protocols, social network analysis, and so on.

3. **Crawlers in Search Engines**

Search engines rely on web crawlers to build up an index of web pages, and BFS (Breadth First Search) is one such algorithm that can help to navigate the vast expanse of the web. The algorithm starts from a source page and follows all the links from the source to other pages and so forth, hence traversing a vast breadth of the web.

By using BFS, search engines can retrieve information from a large number of pages and build up their index, which can then be used to provide relevant search results to users. Moreover, web crawlers are essential for search engines to keep their index up-to-date, as they can detect changes to web pages and update the index accordingly. Thus, without crawlers, search engines would not be able to provide accurate and up-to-date search results to users.

4. **Broadcasting in Network:**

In networks, a broadcasted packet follows the BFS algorithm to reach all nodes. When it comes to broadcasting in a network, a packet is sent out to multiple nodes at once. This process is known as a broadcast and it can be very useful when you want to send information to many different devices.

In order to ensure that the packet reaches all nodes, the BFS (Breadth-First Search) algorithm is typically used. This algorithm works by exploring all the neighboring nodes first, before moving on to the next level of nodes. By using this method, the packet is able to reach all nodes in the network and ensure that the information is delivered to everyone who needs it.

5. **In GPS Navigation**

BFS can find all locations within a given distance from a source location. In GPS navigation, BFS (breadth-first search) is a powerful algorithm that can be used to find all locations within a certain distance from a given source location. This is especially useful when you need to find nearby points of interest, such as gas stations, restaurants, or tourist attractions.

By using BFS, you can quickly and efficiently search through a large number of locations to identify those that are within the desired distance from your starting point. Additionally, BFS can be customized to take into account different factors, such as traffic conditions or road closures, to provide you with the most accurate and up-to-date information possible. Overall, BFS is an essential tool for anyone who wants to navigate unfamiliar terrain with confidence and ease.

## 7.3 Breadth-First Search

Here's how the BFS algorithm works:

- Select a starting node (the 'source' node).
- Explore all the vertices adjacent to the initial node.
- Move to the next level and explore all vertices there.
- Repeat the process until all vertices have been explored.

Let's make this more concrete with a simple pseudocode example:

`function BFS(graph, root) {`

create an empty queue Q

create an array visited of size = number of nodes in the graph, and set all to False

enqueue root into Q

set visited[root] = True

while Q is not empty {

current_node = dequeue from Q

print current_node

for each node i adjacent to current_node {

if visited[i] is False {

enqueue i into Q

set visited[i] = True

}

}

}

}

Here's a Python example of BFS with a simple unweighted graph represented by an adjacency list:

`from collections import deque`

def BFS(graph, root):

visited = [False] * (len(graph))

queue = deque()

queue.append(root)

visited[root] = True

while queue:

vertex = queue.popleft()

print(vertex, end=" ")

for neighbour in graph[vertex]:

if visited[neighbour] == False:

queue.append(neighbour)

visited[neighbour] = True

# defining graph

graph = {

0: [1, 2],

1: [2],

2: [0, 3],

3: [3]

}

BFS(graph, 2) # Output: 2 0 3 1

### 7.3.1 BFS Time Complexity

1. **Shortest Path and Scheduling Algorithms**

2. **Cycle Detection in Undirected Graph**

3. **Crawlers in Search Engines**

4. **Broadcasting in Network:**

5. **In GPS Navigation**