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Algorithms: Shortest Paths in Graphs - the Bellman Ford Algorithm with C Program source code




Bellman –Ford

The bellman-Ford algorithm solves the single source shortest path problems even in the cases in which edge weights are negative. This algorithm returns a Boolean value indicating whether or not there is a negative weight cycle that is reachable from the source. If there is such a cycle, the algorithm indicates that no solution exists and it there is no such cycle, it produces the shortest path and their weights.

Complexity 

The Bellman-Ford algorithm runs in time O(VE), since the initialization takes Θ(V) time, each of the |V| - 1 passes over the edges takes O(E) time and calculating the distance takes O(E) times.

Complete Tutorial with Example :



Bellman Ford Algorithm - C Program source code


#include<stdio.h>
#include<assert.h>
/* maxVertices represents maximum number of vertices that can be present in the graph. */
#define maxVertices   100
#define INF 123456789
/* Input Format: Graph is directed and weighted. First two integers must be number of vertces and edges
   which must be followed by pairs of vertices which has an edge between them.
 */

void BellmanFord(int graph[][maxVertices],int cost[][maxVertices],int size[],int source,int vertices)
{
       
int distance[vertices];
       
int iter,jter,from,to;
       
for(iter=0;iter<vertices;iter++)
       
{
                distance
[iter] = INF;
       
}
        distance
[source] = 0;
       
/* We have to repeatedly update the distance |V|-1 times where |V| represents
           number of vertices */

       
for(iter=0;iter<vertices-1;iter++)
       
{
               
for(from=0;from<vertices;from++)
               
{
                       
for(jter=0;jter<size[from];jter++)
                       
{
                                to
= graph[from][jter];
                               
if(distance[from] + cost[from][jter] < distance[to])
                               
{
                                        distance
[to] = distance[from] + cost[from][jter];
                               
}
                       
}
               
}
       
}
       
for(iter=0;iter<vertices;iter++)
       
{
                printf
("The shortest distance to %d is %d\n",iter,distance[iter]);
       
}


}
int main()
{
       
int graph[maxVertices][maxVertices],size[maxVertices]={0},visited[maxVertices]={0};
       
int cost[maxVertices][maxVertices];
       
int vertices,edges,iter,jter;
       
/* vertices represent number of vertices and edges represent number of edges in the graph. */
        scanf
("%d%d",&vertices,&edges);
       
int vertex1,vertex2,weight;
       
/* Here graph[i][j] represent the weight of edge joining i and j */
       
for(iter=0;iter<edges;iter++)
       
{
                scanf
("%d%d%d",&vertex1,&vertex2,&weight);
                assert
(vertex1>=0 && vertex1<vertices);
                assert
(vertex2>=0 && vertex2<vertices);
                graph
[vertex1][size[vertex1]] = vertex2;
                cost
[vertex1][size[vertex1]] = weight;
                size
[vertex1]++;
       
}
       
int source;
        scanf
("%d",&source);
       
BellmanFord(graph,cost,size,source,vertices);
       
return 0;
}
Rough notes about the Algorithm and how it is implemented in the code above:

Input Format: Graph is directed and weighted. 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.
cost[maxVertices][maxVertices] represents the cost of going from one vertex to another.
visited[maxVertices]={0} represents the vertex that have been visited.
Initialize the graph and input the source vertex.
BellmanFord function is called to get the shortest path.

BellmanFord function: This function takes the graph obtained (graph[ ][ maxVertices]), cost
(cost[ ][maxVertices]) of going from one vertex to other, size (size[maxVertices]) of vertices,
source vertex and the total number of vertices (vertices) as arguments.

Initialize an array distance[vertices] to store the shortest distance travelled to reach vertex i. 
Initially distance[source] = 0 i.e. distance to reach the source vertex is zero and distance to reach every other vertex is initialed to infinity (distance[i] = INF, where INF is a very large integer value).
We have to repeatedly update the distance |V|-1 times where |V| represents number of vertices as we visit the vertices.
for iter=0 to vertices-2
for from=0 to vertices-1
for jter=0 to size[from]
to = graph[from][jter]
if(distance[from] + cost[from][jter] < distance[to])
distance[to] = distance[from] + cost[from][jter]
jter + 1
from + 1
iter + 1
Lastly for every vertex print the shortest distance traveled to reach that vertex.


Related Tutorials (basic Graph Algorithms) :

 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.

Some Important Data Structures and Algorithms, at a glance:

Arrays : Popular Sorting and Searching Algorithms

 

  

Bubble Sort  

Insertion Sort 

Selection Sort Shell Sort

Merge Sort  

Quick Sort 

 
Heap Sort
 
Binary Search Algorithm

Basic Data Structures  and Operations on them


  

Stacks 

Queues  

 
 Single Linked List 

Double Linked List

Circular Linked List 















Basic Data Structures and Algorithms



Sorting- at a glance

 Bubble Sort One of the most elementary sorting algorithms to implement - and also very inefficient. Runs in quadratic time. A good starting point to understand sorting in general, before moving on to more advanced techniques and algorithms. A general idea of how the algorithm works and a the code for a C program.

Insertion Sort - Another quadratic time sorting algorithm - an example of dynamic programming. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort.

Selection Sort - Another quadratic time sorting algorithm - an example of a greedy algorithm. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs selection sort.

Shell Sort- An inefficient but interesting algorithm, the complexity of which is not exactly known.

Merge Sort An example of a Divide and Conquer algorithm. Works in O(n log n) time. The memory complexity for this is a bit of a disadvantage.

Quick Sort In the average case, this works in O(n log n) time. No additional memory overhead - so this is better than merge sort in this regard. A partition element is selected, the array is restructured such that all elements greater or less than the partition are on opposite sides of the partition. These two parts of the array are then sorted recursively.

Heap Sort- Efficient sorting algorithm which runs in O(n log n) time. Uses the Heap data structure.

Binary Search Algorithm- Commonly used algorithm used to find the position of an element in a sorted array. Runs in O(log n) time.

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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