Breadth-First SearchBreadth-first search is one of the simplest algorithms for searching a graph. Given a graph and a distinguished source vertex, breadth-first search explores the edges of the graph to find every vertex reachable from source. It computes the distance (fewest number of edges) from source to all reachable vertices and produces a “breadth-first tree” with source vertex as root, which contains all such reachable vertices. It works on both directed and undirected graphs. This algorithm uses a first-in, first-out Queue Q to manage the vertices. Complete Tutorial with Examples :
Breadth First Search - C Program Source Code#include<stdio.h>
Rough Notes about the Algorithm : Input Format: Graph is directed and unweighted. First two integers must be number of vertices and edges which must be followed by pairs of vertices which has an edge between them. maxVertices represents maximum number of vertices that can be present in the graph. vertices represent number of vertices and edges represent number of edges in the graph. graph[i][j] represent the weight of edge joining i and j. size[maxVertices] is initialed to{0}, represents the size of every vertex i.e. the number of edges corresponding to the vertex. visited[maxVertices]={0} represents the vertex that have been visited. Initialize the graph. For presentVertex = 0 to vertices if visited[presentVertex] is 0, i.e. if the vertex has not been visited then call Bfs function. presentVertex represents the vertex that is being tackled. Bfs function is called to get the shortest path. Bfs function: This function takes the graph obtained (graph[ ][ maxVertices]), pointer to the array size and visited, and the presentValue as arguments. visited[presentVertex] = 1 as the vertex has now been visited. Iterate through all the vertices connected to the presentVertex and perform bfs on those vertices if they are not visited before. Create a queue Q using createQueue function and enqueue the presentVertex using Enqueue function. Until the Q is not empty i.e. Q->size ≠ 0 • Store the front element of Q using Front function in presentVertex • Print the vertex that is being visited now, which is presentVertex • Remove the element from the front of Q using Dequeue function • For iter=0 to size[presentVertex] – 1 If (!visited[graph[presentVertex][iter]]) visited[graph[presentVertex][iter]] = 1 Enqueue(Q,graph[presentVertex][iter]) Iter + 1 This for loop visits every vertex that is adjacent to the presentVertex and has not been visited yet. These vertices are then inserted in the Q using Enqueue function and their visited status is updated to 1. The Queue has five properties - capacity stands for the maximum number of elements Queue can hold, Size stands for the current size of the Queue, elements is the array of elements, front is the index of first element (the index at which we remove the element) and rear is the index of last element (the index at which we insert the element). Functions on Queue 1. createQueue function takes argument the maximum number of elements the Queue can hold, creates a Queue according to it and returns a pointer to the Queue. It initializes Q- >size to 0, Q->capacity to maxElements, Q->front to 0 and Q->rear to -1. 2. enqueue function - This function takes the pointer to the top of the queue Q and the item (element) to be inserted as arguments. Check for the emptiness of queue a. If Q->size is equal to Q->capacity, we cannot push an element into Q as there is no space for it. b. Else, enqueue an element at the end of Q, increase its size by one. Increase the value of Q->rear to Q->rear + 1. As we fill the queue in circular fashion, if Q->rear is equal to Q->capacity make Q->rear = 0. Now, Insert the element in its rear side Q->elements[Q->rear] = element 3. dequeue function - This function takes the pointer to the top of the stack S as an argument. a. If Q->size is equal to zero, then it is empty. So, we cannot dequeue. b. Else, remove an element which is equivalent to incrementing index of front by one. Decrease the size by 1. As we fill elements in circular fashion, if Q->front is equal to Q->capacity make Q->front=0. 4. front function – This function takes the pointer to the top of the queue Q as an argument and returns the front element of the queue Q. It first checks if the queue is empty (Q->size is equal to zero). If it’s not it returns the element which is at the front of the queue. Q->elements[Q->front] Related Tutorials (basic Graph Algorithms) :
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| Basic Data Structures and Algorithms Sorting- at a glance
Basic Data Structures and Algorithms Stacks Last In First Out data structures ( LIFO ). Like a stack of cards from which you pick up the one on the top ( which is the last one to be placed on top of the stack ). Documentation of the various operations and the stages a stack passes through when elements are inserted or deleted. C program to help you get an idea of how a stack is implemented in code. Queues First in First Out data structure (FIFO). Like people waiting to buy tickets in a queue - the first one to stand in the queue, gets the ticket first and gets to leave the queue first. Documentation of the various operations and the stages a queue passes through as elements are inserted or deleted. C Program source code to help you get an idea of how a queue is implemented in code. Single Linked List A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind. Double Linked List- A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind in front of it, as well as another of its own kind, which happens to be behind it in the sequence. Circular Linked List Linked list with no head and tail - elements point to each other in a circular fashion. Binary Search Trees A basic form of tree data structures. Inserting and deleting elements in them. Different kind of binary tree traversal algorithms. Heaps - A tree like data structure where every element is lesser (or greater) than the one above it. Heap formation, sorting using heaps in O(n log n) time. Height Balanced Trees - Ensuring that trees remain balanced to optimize complexity of operations which are performed on them. Graphs Depth First Search - Traversing through a graph using Depth First Search in which unvisited neighbors of the current vertex are pushed into a stack and visited in that order. Breadth First Search - Traversing through a graph using Breadth First Search in which unvisited neighbors of the current vertex are pushed into a queue and then visited in that order. Minimum Spanning Trees: Kruskal Algorithm- Finding the Minimum Spanning Tree using the Kruskal Algorithm which is a greedy technique. Introducing the concept of Union Find. Minumum Spanning Trees: Prim's Algorithm- Finding the Minimum Spanning Tree using the Prim's Algorithm. Dijkstra Algorithm for Shortest Paths- Popular algorithm for finding shortest paths : Dijkstra Algorithm. Floyd Warshall Algorithm for Shortest Paths- All the all shortest path algorithm: Floyd Warshall Algorithm Bellman Ford Algorithm - Another common shortest path algorithm : Bellman Ford Algorithm. Dynamic Programming A technique used to solve optimization problems, based on identifying and solving sub-parts of a problem first. Integer Knapsack problemAn elementary problem, often used to introduce the concept of dynamic programming. Matrix Chain Multiplication Given a long chain of matrices of various sizes, how do you parenthesize them for the purpose of multiplication - how do you chose which ones to start multiplying first? Longest Common Subsequence Given two strings, find the longest common sub sequence between them. Elementary cases : Fractional Knapsack Problem, Task Scheduling - Elementary problems in Greedy algorithms - Fractional Knapsack, Task Scheduling. Along with C Program source code. Data Compression using Huffman TreesCompression using Huffman Trees. A greedy technique for encoding information. |
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