Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to BFS
Unlock Audio Lesson
Today, we're going to learn about the Breadth First Search algorithm, often abbreviated as BFS. Who can tell me what a graph is?
A graph consists of vertices and edges connecting them.
That's correct! Now, BFS helps us explore those graphs. Can anyone describe how BFS begins its exploration?
It starts with a source vertex and then explores all directly connected vertices before moving further.
Exactly! BFS goes level by level. Remember the acronym 'BFS' for Breadth First Search—it's a great way to recall its systematic approach!
Graph Representation
Unlock Audio Lesson
We can represent graphs using adjacency matrices or lists. What do you know about these representations?
An adjacency matrix is a two-dimensional array showing edges.
Good! But if a graph is sparse, the adjacency list is often better because it only lists connected vertices. Can you see how this impacts BFS efficiency?
Yes, if we only check the neighbors, it makes the process quicker with fewer edges.
Correct! Always remember, ‘Simplicity’ in representation leads to efficiency!
How BFS Works
Unlock Audio Lesson
Let’s discuss how BFS actually explores the graph. What data structure do we use?
A queue!
That's right! Can someone explain the queue's role during exploration?
It stores the vertices that need to be explored next after marking them as visited.
Excellent! Always remember: Queue = Next in Line. Now, let's recap: What's the significance of marking vertices?
Marking helps prevent revisiting vertices that have already been explored.
Exactly! That's crucial for preventing cycles. Well done!
Path Reconstruction
Unlock Audio Lesson
Besides finding connected vertices, how else can BFS be useful?
It can help find the path taken to reach each vertex!
Great observation! If we remember the parent of each vertex, we can reconstruct the entire path back to the source. Can anyone share how we can keep track of parents during BFS?
By storing the vertex from which we first visit each vertex.
That's exactly correct! And remember the term 'parent node' as a helpful mnemonic.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
BFS utilizes a queue data structure to manage the exploration of vertices. Starting from a source vertex, it discovers all adjacent vertices before moving on to the next level, marking each vertex as visited. This method efficiently identifies the shortest path in an unweighted graph and can be adapted to track the path taken.
Detailed
Breadth First Search (BFS)
Breadth First Search (BFS) is an algorithm used to explore vertices in a graph systematically. It starts at a source vertex and traverses the graph by exploring all direct neighbors first before moving on to vertices two edges away, and so on.
Key Concepts:
- Graph Representation: Graphs can be represented using either an adjacency matrix or adjacency lists. An adjacency matrix indicates the presence or absence of edges (1 or 0) for every pair of vertices, which can yield an inefficient representation in sparse graphs. An adjacency list, on the other hand, records only the vertices that are directly connected, making it more efficient for sparse graphs.
- Queue Data Structure: BFS uses a queue to keep track of the vertices to be visited. As vertices are explored, they are marked as visited and added to the queue.
- Exploration Process: Each vertex is marked as visited upon its first encounter, after which all unvisited connected vertices are added to the queue for future exploration. The process continues until there are no more vertices to visit, at which point all reachable vertices from the source have been explored.
- Path Reconstruction: BFS can also track parent vertices to reconstruct paths back to the source vertex and can store the depth or level of each vertex relative to the source.
- Complexity Analysis: BFS runs in O(n + m) time complexity when using an adjacency list, where n is the number of vertices and m is the number of edges, making it efficient for connected graphs.
BFS is particularly useful for finding the shortest path in unweighted graphs due to its level-order exploration.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Graphs and Edges
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, let us look at a formal algorithm to explore a graph. So, recall that a graph consists of a set of vertices and a set of edges, the edges are the connections between the vertices, they may be directed or undirected.
So, if we look at it undirected graph, the problem we are looking at is to find out, whether the source vertex is connected to a target vertex. And we said that this amounts to finding a path from v_0 the source vertex to v_k the target vertex, where each pair of vertices on the path is connected by an edge in the graph.
Detailed Explanation
This chunk introduces the fundamental concepts of graph theory essential for understanding the BFS algorithm. A graph is made up of vertices (or nodes) which are connected through edges. The key problem being solved here is whether there is a pathway from a given source vertex to a target vertex. When we’re looking for a path, it’s not just about having one connection; all vertices in the path need to be connected by edges.
Examples & Analogies
Think of a graph like a road map where the vertices represent towns and the edges represent roads connecting those towns. When we ask if you can travel from one town to another, we are essentially asking if there’s a route (path) connecting those two towns (vertices) via the roads (edges).
Graph Representation: Adjacency Matrix vs. Adjacency List
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, we argued that we could easily do this for small graphs visually, but if you want to write an algorithm, we need a way to represent the graph. ... If you want to find out whether a given pair i j is connected, we have to look at the list for i and see, if j appears in it.
Detailed Explanation
This chunk discusses how to represent graphs in a way that algorithms can process. An adjacency matrix is a 2D array where if there's an edge from vertex i to vertex j, the entry is 1, otherwise, it’s 0. This method, while straightforward, can be inefficient for sparsely connected graphs. An adjacency list, on the other hand, records only the neighbors of each vertex, making it more efficient for graphs where the number of edges is significantly less than the square of the number of vertices.
Examples & Analogies
Imagine a social network where you have a list of friends (adjacency list) rather than a huge chart showing all possible friendships (adjacency matrix). If you want to know who a person is friends with, you just check their list of friends. This is much easier than checking through a large matrix.
Breadth First Search Strategy
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, our strategy for finding a path connecting a source vertex in a target vertex, which is follows, we would start at the source vertex and keep exploring the graph systematically. ... So, today we are going to look at breadth first search.
Detailed Explanation
The BFS strategy involves exploring the graph level by level. Starting from the source vertex, it explores all direct neighbors first, then moves to neighbors of those neighbors, and so on. This systematic approach ensures that you find the shortest path in terms of the number of edges because you’re exploring all vertices at a given level before moving deeper into the graph.
Examples & Analogies
Consider a close-knit community where you want to find out how many people you can reach. You would first ask your immediate friends (level 1) about their friends (level 2) before asking your friends' friends about their friends (level 3). This way, you minimize the number of connections needed to reach the maximum number of people.
Data Structures for BFS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, the way to do this is to use some data structures to keep track of these quantities... So, we want to explore in the order in which they will visited, a natural data structure to keep the list of unexplored, but visited vertices is a queue.
Detailed Explanation
In BFS, we use two main data structures: an array to track whether a vertex has been visited and a queue for exploring vertices in the order they were discovered. When a vertex is visited for the first time, it is marked as visited, and added to the queue to be explored later. This ensures that we explore all connections level by level, as required by the algorithm.
Examples & Analogies
Imagine you’re waiting in line at a bank. The people at the front who entered the queue first get served first. Similarly, vertices that are added to the queue during BFS are explored in the same order they were discovered, ensuring a level-by-level exploration.
Pseudocode for Breadth First Search
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, else some high level Pseudo code for breadth first search... If at some stage, there are no vertices in the queue, it means that every vertex, we have visited have been explored and so there is no new information left and we can stop the exploration.
Detailed Explanation
This chunk formalizes the BFS process into pseudocode. The algorithm begins by marking the source vertex as visited and adding it to the queue. As the algorithm iterates, it dequeues a vertex, checks its neighbors, and if they’re unvisited, it marks them as visited and adds them to the queue. When the queue is empty, it indicates all reachable vertices have been explored.
Examples & Analogies
It’s like searching for a friend in a large crowd by making sure to ask everyone in your immediate vicinity first, then moving outwards to those nearby until everyone you can reach has been asked. When no one is left to ask, you've completed your search.
Example of BFS Execution
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, before giving the actual code for breadth first search, let us workout the example that we have and see, how this data structures are updated... If we had add more vertices, for example, if we had add a component in this graph; that is nothing to stop us from having many components.
Detailed Explanation
In this example, the BFS algorithm is executed on a graph with ten vertices where the source is vertex 1. As each vertex is explored, the queue and visited array are updated, demonstrating how the algorithm works practically. If additional disconnected vertices are added, they would remain unvisited, showcasing BFS’s ability to explore only connected components of the graph.
Examples & Analogies
Imagine launching a search party in a park with different sections. If you start from one section, you can only gather information from that part until you decide to explore another section. Sections with no entry paths remain unvisited, just like disconnected vertices in a graph.
Complexity Analysis of BFS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, how do we analyze the complexity of an algorithm... So, therefore, the graph is connected first of all every vertex will get into the queue.
Detailed Explanation
The complexity of BFS is analyzed by counting how many times each vertex and edge is processed during the execution of the algorithm. For a graph represented using an adjacency list, the time complexity is O(n + m), where n is the number of vertices and m is the number of edges, making BFS efficient especially for sparse graphs.
Examples & Analogies
Think of an efficient plan to map all routes in a transportation network. If you only look at relevant connections (roads), you save time and resources instead of analyzing every possible route even if there’s no road. The focus on relevant paths makes the search process faster.
Tracking Path Information with BFS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, we can do something more in breadth first search... the path information and also the distance information at the same time, when you are computing breadth first search.
Detailed Explanation
While BFS explores vertices, it can also track from which vertex each vertex was reached, thus allowing the reconstruction of paths. By maintaining a 'parent' data structure, we can trace back from any vertex to the starting vertex, providing both the path taken and the distance in terms of edges.
Examples & Analogies
If you're drawing a map of how you traveled through a city, you can note down each turn you took. Later, if someone asks how you got from point A to point B, you can retrace your steps using your notes to guide them along the exact path you took.
Levels in BFS for Shortest Path
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, breadth first search if we add this level predicate, it actually gives us the shortest path to each node in terms of number of edges.
Detailed Explanation
In addition to finding paths, BFS can determine how far each vertex is from the starting point in terms of levels. Each vertex's level corresponds to how many edges away it is from the start vertex. This is particularly useful for unweighted graphs, as BFS provides the shortest path in terms of edge count.
Examples & Analogies
You can think of leveling as a height measurement. If you start on the ground (level 0) and move up a staircase, every step counts as moving to the next level. Similarly, each edge traversed in BFS elevates you to the next level, helping you measure how far you've come from the start.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Graph Representation: Graphs can be represented using either an adjacency matrix or adjacency lists. An adjacency matrix indicates the presence or absence of edges (1 or 0) for every pair of vertices, which can yield an inefficient representation in sparse graphs. An adjacency list, on the other hand, records only the vertices that are directly connected, making it more efficient for sparse graphs.
-
Queue Data Structure: BFS uses a queue to keep track of the vertices to be visited. As vertices are explored, they are marked as visited and added to the queue.
-
Exploration Process: Each vertex is marked as visited upon its first encounter, after which all unvisited connected vertices are added to the queue for future exploration. The process continues until there are no more vertices to visit, at which point all reachable vertices from the source have been explored.
-
Path Reconstruction: BFS can also track parent vertices to reconstruct paths back to the source vertex and can store the depth or level of each vertex relative to the source.
-
Complexity Analysis: BFS runs in O(n + m) time complexity when using an adjacency list, where n is the number of vertices and m is the number of edges, making it efficient for connected graphs.
-
BFS is particularly useful for finding the shortest path in unweighted graphs due to its level-order exploration.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
In a social network, BFS can be used to find all friends of a user within two degrees of separation.
-
In city navigation systems, BFS helps in mapping routes to find the shortest path between two locations.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Explore the graph without a gap, level by level, that's the map!
📖 Fascinating Stories
-
Imagine you're a detective trying to find everyone at a party—first you talk to those closest to you, then expand your search to friends of friends. That's how BFS works!
🧠 Other Memory Gems
-
Remember 'BFS' by thinking: Best Friend Search—find all nearby friends first!
🎯 Super Acronyms
BFS
- Breadth First Search - Begin from the source and Find nearest neighbors Sequentially.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Graph
Definition:
A collection of vertices connected by edges.
-
Term: Vertex
Definition:
A node in a graph.
-
Term: Edge
Definition:
A connection between two vertices in a graph.
-
Term: Adjacency Matrix
Definition:
A 2D array used to represent graphs, indicating the presence or absence of edges.
-
Term: Adjacency List
Definition:
A data structure that lists all the vertices connected to each vertex.
-
Term: Queue
Definition:
A data structure that follows the First In, First Out (FIFO) principle, used in BFS to manage exploration.
-
Term: Visited
Definition:
A status indicating that a vertex has been explored.
-
Term: Parent Node
Definition:
The vertex from which another vertex was first discovered during exploration.
-
Term: Level
Definition:
The depth of a vertex from the source vertex in BFS.