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Code Algorithm

# Understanding the Bellman-Ford Algorithm and Showing the Shortest Path

The Bellman-Ford algorithm is a widely used graph traversal algorithm that is used to find the shortest path from a source node to all other nodes in a weighted graph. It is particularly useful when dealing with graphs that may contain negative weight edges. In this article, we will explore the Bellman-Ford algorithm and provide a step-by-step explanation of the code provided.

## What is the Bellman-Ford Algorithm?

The Bellman-Ford algorithm is an algorithm that solves the single-source shortest path problem in a weighted directed graph. It can handle graphs with negative weight edges and detects negative cycles. The algorithm maintains a distance array, where each element represents the minimum distance from the source node to that particular node.

### Code Solution

``````//C Program
//Show path in Bellman Ford algorithm
#include <stdio.h>
#include <stdlib.h>
#include <limits.h> //for INT_MAX

#define INF INT_MAX
struct AjlistNode
{
int vId;//Vertices id
int weight;
struct AjlistNode*next;
};

struct Graph
{
int data; //node key value
struct AjlistNode*next;
};

int size; //number of nodes

//set node key value
void set_data(struct Graph*node)
{
if(node!=NULL && size>0)
{
int index=0;
for(index;index<size;index++)
{
//set vertic node data
node[index].data=index;//set node key
//Initial no AjlistNode
//set NULL Value
node[index].next=NULL;
}
}else
{
printf("Vertic Node is Empty");
}
}
void connect_edge(struct Graph*node, int V ,int E,int weight)
{

struct AjlistNode *newEdge=(struct AjlistNode*)malloc(
sizeof(struct AjlistNode)
);
if(newEdge!=NULL)
{

newEdge->next=NULL;
newEdge->vId=E;
newEdge->weight=weight;

struct AjlistNode *temp=node[V].next;

if(temp==NULL)
{
node[V].next=newEdge;
}else
{
while(temp->next!=NULL)
{
temp=temp->next;
}
temp->next=newEdge;
}
}else
{
printf("\n Memory overflow");

}
}
//Add Edge from Two given Nodes
void add_edge(struct Graph*node, int V ,int E,int weight)
{
//add edge form V to E
//V and E is Node location
if(V<size && E <size)
{
connect_edge(node,V,E,weight);
}else
{
//not valid Vertices
printf("Invalid Node Vertices %d  %d", V,E);
}
}
void print_graph(struct Graph*node)

{
if(node!=NULL)
{
struct AjlistNode *temp=NULL;
for(int index=0;index<size;index++)
{
printf("\n Adjacency list of vertex %d  :",index);
temp=node[index].next;
while(temp!=NULL)
{
//temp->vId is graph node vertices
//in this case temp->vId is same as
//node[temp->vId].data

printf("  %d",node[temp->vId].data);
temp=temp->next;
}
}
}else
{
printf("Empty Graph");
}
}
void view_path(int path[], int location)
{

if (path[location] == - 1)
{

return;
}

view_path(path, path[location]);

printf("%d ", location);
}
void bellman_ford(struct Graph*root,int source)
{

if(root!=NULL)
{

int distance[size],path[size];

for (int i = 0; i < size; ++i)
{
//Initial distance is infinity
distance[i] = INF;

path[i]=-1;

}

distance[source] = 0;

struct AjlistNode *temp=NULL;

printf("\n Source node : %d\n",source );

for (int i = 1; i < size; ++i)
{

for (int j = 0; j < size; ++j)
{
temp=root[j].next;
//compare node (J) outgoing edges
while(temp!=NULL)
{
if(distance[j] != INF && distance[j] + temp->weight < distance[temp->vId]  )
{
distance[temp->vId] = distance[j] + temp->weight;
path[temp->vId]=j;
}
temp=temp->next;
}
}
}

for (int i = 1; i < size; ++i)
{

for (int j = 0; j < size; ++j)
{
temp=root[j].next;

//compare node (J) outgoing edges
while(temp!=NULL)
{
if(distance[j] != INF && distance[j] + temp->weight < distance[temp->vId]  )
{
printf("\n Negative Cycle exist node (%d %d)\n",temp->vId,j);
return;

}
temp=temp->next;
}
}
}
printf("\nVertex \tDistance Path\n");

for (int i = 0; i < size; ++i)
{
if(distance[i]==INF)
{
printf(" %d \t  INF\n", i);
}
else
{
printf("\n %d \t %d\t %d ", i, distance[i],source);

}

view_path(path,i);
}
}
else
{
printf("Empty Graph");
}
}

int main()

{

size=5;
struct Graph*node=NULL;

node=(struct Graph*)malloc(sizeof(struct Graph)*size);

if(node==NULL)
{
printf("\n Memory overflow");
}else
{
//First set node keys
set_data(node);

//Connected two node with Edges

print_graph(node);

bellman_ford(node,3);

}
return 0;
}```
```

#### Output

`````` Adjacency list of vertex 0  :  1  2  4
Adjacency list of vertex 1  :  4
Adjacency list of vertex 2  :  1
Adjacency list of vertex 3  :  0  2  4
Adjacency list of vertex 4  :  2
Source node : 3

Vertex 	Distance Path

0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````// C++ program
// Show path in Bellman Ford algorithm
#include<iostream>
#include <limits.h>
using namespace std;
class AjlistNode {
public:

//Vertices node key
int id;
AjlistNode *next;
int weight;
AjlistNode(int id, int weight) {
//Set value of node key
this->id = id;
this->next = NULL;
this->weight = weight;
}
};
class Vertices {
public:
int data;
AjlistNode *next;
Vertices()
{
this->data = 0;
this->next = NULL;
}
Vertices(int data) {
this->data = data;
this->next = NULL;
}
};
class MyGraph {
public:

//number of Vertices
int size;
Vertices *node;
MyGraph(int size) {
this->size = size;
this->node = new Vertices[size];
//set initial values of graph node
this->set_data();
}
//Set initial node value
void set_data() {
if (this->node == NULL) {
cout << "\nEmpty Graph";
} else {
int index = 0;
while (index < this->size) {
this->node[index] = index;
index++;
}
}
}
//Connect two node
void add_edge(int start, int last, int weight) {
AjlistNode *newEdge = new AjlistNode(last, weight);
if (this->node[start].next == NULL) {
//Include first adjacency list node of location start
this->node[start].next = newEdge;
} else {
AjlistNode *temp = this->node[start].next;
//Add new node at the end of edge
while (temp->next != NULL) {
temp = temp->next;
}
temp->next = newEdge;
}
}
//Display graph elements
void print_graph() {
if (this->size > 0 &&
this->node != NULL) {
int index = 0;
while (index < this->size) {
cout << "\nAdjacency list of vertex " << index << " : ";
AjlistNode *temp = this->node[index].next;
while (temp != NULL) {
cout << this->node[temp->id].data << " ";
temp = temp->next;
}
index++;
}
}
}
void view_path(int path[], int location) {
if (path[location] == -1) {
return;
}
this->view_path(path, path[location]);
cout << " " << location;
}
void bellman_ford(int source) {
if (source < 0 &&
source >= this->size) {
return;
}
if (this->node != NULL) {
int *distance = new int[size];
int *path = new int[size];
for (int i = 0; i < this->size; ++i) {
//Initial distance is Integer.MAX_VALUE is infinity
distance[i] = INT_MAX;
path[i] = -1;
}
distance[source] = 0;
AjlistNode *temp = NULL;
cout << "\n Source Node " << source << " \n";
for (int i = 1; i < this->size; ++i) {
for (int j = 0; j < this->size; ++j) {
temp = this->node[j].next;
//compare node (J) outgoing edges
while (temp != NULL) {
if (distance[j] != INT_MAX &&
distance[j] + temp->weight < distance[temp->id]) {
distance[temp->id] = distance[j] + temp->weight;
path[temp->id] = j;
}
temp = temp->next;
}
}
}
for (int i = 1; i < this->size; ++i) {
for (int j = 0; j < this->size; ++j) {
temp = this->node[j].next;
//compare node (J) outgoing edges
while (temp != NULL) {
if (distance[j] != INT_MAX &&
distance[j] + temp->weight < distance[temp->id]) {
cout << "\n Negative Cycle exist node (" << temp->id << " " << j << ")\n";
return;
}
temp = temp->next;
}
}
}
cout << "\nVertex Distance Path\n";
for (int i = 0; i < this->size; ++i) {
if (distance[i] == INT_MAX) {
cout << " " << i << " \t \t Integer MAX VALUE\n";
} else {
cout << " " << i << " \t " << distance[i] << "\t " << source;
}
this->view_path(path, i);
cout << "\n";
}
} else {
cout << "Empty Graph";
}
}
};
int main() {
//Number of nodes
int totalNode = 5;
MyGraph g =  MyGraph(totalNode);
//Connected two node with Edges
//Third parameter is weight
g.print_graph();
//Start location
int start = 3;
g.bellman_ford(start);
return 0;
}```
```

#### Output

``````Adjacency list of vertex 0 : 1 2 4
Adjacency list of vertex 1 : 4
Adjacency list of vertex 2 : 1
Adjacency list of vertex 3 : 0 2 4
Adjacency list of vertex 4 : 2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````// Java program
// Show path in Bellman Ford algorithm

class AjlistNode {
//Vertices node key
public int id;
public AjlistNode next;
public int weight;
public AjlistNode(int id, int weight) {
//Set value of node key
this.id = id;
this.next = null;
this.weight = weight;
}
}
class Vertices {

public int data;
public AjlistNode next;

public Vertices(int data) {
this.data = data;
this.next = null;
}
}

public class MyGraph {

//number of Vertices
private int size;

private Vertices[] node;

public MyGraph(int size) {

this.size = size;

this.node = new Vertices[size];
//set initial values of graph node
this.set_data();

}

//Set initial node value
public void set_data() {
if (node == null) {
System.out.println("\nEmpty Graph");
} else {

int index = 0;

while (index < size) {

node[index] = new Vertices(index);

index++;
}
}
}

//Connect two node
public void add_edge(int start, int last, int weight) {

AjlistNode newEdge = new AjlistNode(last, weight);

if (node[start].next == null) {
//Include first adjacency list node of location start
node[start].next = newEdge;
} else {
AjlistNode temp = node[start].next;
//Add new node at the end of edge
while (temp.next != null) {
temp = temp.next;
}
temp.next = newEdge;
}
}
//Display graph elements
public void print_graph() {

if (size > 0 && node != null) {
int index = 0;
while (index < size) {
System.out.print("\nAdjacency list of vertex " + index + " : ");
AjlistNode temp = node[index].next;
while (temp != null) {
System.out.print(node[temp.id].data + "  ");
temp = temp.next;
}
index++;
}
}
}
public void view_path(int []path, int location)
{

if (path[location] == - 1)
{

return;
}

view_path(path, path[location]);

System.out.print("  "+ location);
}
public void bellman_ford(int source) {
if (source < 0 && source >= this.size) {
return;
}
if (node != null) {

int[] distance = new int[size];

int []path = new int[size];

for (int i = 0; i < size; ++i) {
//Initial distance is Integer.MAX_VALUE is infinity
distance[i] = Integer.MAX_VALUE;
path[i]=-1;
}

distance[source] = 0;

AjlistNode temp = null;
System.out.print("\n Source Node "+ source +" \n");

for (int i = 1; i < size; ++i) {

for (int j = 0; j < size; ++j) {
temp = node[j].next;
//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != Integer.MAX_VALUE && distance[j] + temp.weight < distance[temp.id]) {
distance[temp.id] = distance[j] + temp.weight;
path[temp.id]=j;
}
temp = temp.next;
}
}
}

for (int i = 1; i < size; ++i) {

for (int j = 0; j < size; ++j) {
temp = node[j].next;

//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != Integer.MAX_VALUE && distance[j] + temp.weight < distance[temp.id]) {
System.out.print("\n Negative Cycle exist node (" + temp.id + " " + j + ")\n");
return;

}
temp = temp.next;
}
}
}

System.out.print("\nVertex   Distance    Path\n");

for (int i = 0; i < size; ++i) {
if (distance[i] == Integer.MAX_VALUE) {
System.out.print(" " + i + " \t \t Integer MAX VALUE\n");
} else {
System.out.print(" " + i + " \t  " + distance[i] +"\t   "+source);
}
view_path(path,i);
System.out.print("\n");

}

} else {
System.out.print("Empty Graph");
}
}

public static void main(String[] args) {
//Number of nodes
int totalNode = 5;

MyGraph g = new MyGraph(totalNode);
//Connected two node with Edges
//Third parameter is weight
g.print_graph();

//Start location
int start = 3;

g.bellman_ford(start);

}
} ```
```

#### Output

``````Adjacency list of vertex 0 : 1 2 4
Adjacency list of vertex 1 : 4
Adjacency list of vertex 2 : 1
Adjacency list of vertex 3 : 0 2 4
Adjacency list of vertex 4 : 2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````// C# program
// Show path in Bellman Ford algorithm
using System;
public class AjlistNode {
//Vertices node key
public int id;
public AjlistNode next;
public int weight;
public AjlistNode(int id, int weight) {
//Set value of node key
this.id = id;
this.next = null;
this.weight = weight;
}
}
public class Vertices {
public int data;
public AjlistNode next;
public Vertices(int data) {
this.data = data;
this.next = null;
}
}
public class MyGraph {
//number of Vertices
private int size;
private Vertices[] node;
public MyGraph(int size) {
this.size = size;
this.node = new Vertices[size];
this.set_data();
}
//Set initial node value
public void set_data() {
if (node == null) {
Console.WriteLine("\nEmpty Graph");
} else {
int index = 0;
while (index < size) {
node[index] = new Vertices(index);
index++;
}
}
}
//Connect two node
public void add_edge(int start, int last, int weight) {
AjlistNode newEdge = new AjlistNode(last, weight);
if (node[start].next == null) {
//Include first adjacency list node of location start
node[start].next = newEdge;
} else {
AjlistNode temp = node[start].next;
//Add new node at the end of edge
while (temp.next != null) {
temp = temp.next;
}
temp.next = newEdge;
}
}
//Display graph elements
public void print_graph() {
if (size > 0 &&
node != null) {
int index = 0;
while (index < size) {
Console.Write("\nAdjacency list of vertex " + index + " : ");
AjlistNode temp = node[index].next;
while (temp != null) {
Console.Write(node[temp.id].data + " ");
temp = temp.next;
}
index++;
}
}
}
public void view_path(int[] path, int location) {
if (path[location] == -1) {
return;
}
view_path(path, path[location]);
Console.Write(" " + location);
}
public void bellman_ford(int source) {
if (source < 0 &&
source >= this.size) {
return;
}
if (node != null) {
int[] distance = new int[size];
int[] path = new int[size];
for (int i = 0; i < size; ++i) {
//Initial distance is Integer.MAX_VALUE is infinity
distance[i] = int.MaxValue;
path[i] = -1;
}
distance[source] = 0;
AjlistNode temp = null;
Console.Write("\n Source Node " + source + " \n");
for (int i = 1; i < size; ++i) {
for (int j = 0; j < size; ++j) {
temp = node[j].next;
//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != int.MaxValue &&
distance[j] + temp.weight < distance[temp.id]) {
distance[temp.id] = distance[j] + temp.weight;
path[temp.id] = j;
}
temp = temp.next;
}
}
}
for (int i = 1; i < size; ++i) {
for (int j = 0; j < size; ++j) {
temp = node[j].next;
//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != int.MaxValue &&
distance[j] + temp.weight < distance[temp.id]) {
Console.Write("\n Negative Cycle exist node (" + temp.id + " " + j + ")\n");
return;
}
temp = temp.next;
}
}
}
Console.Write("\nVertex Distance Path\n");
for (int i = 0; i < size; ++i) {
if (distance[i] == int.MaxValue) {
Console.Write(" " + i + " \t \t Integer MAX VALUE\n");
} else {
Console.Write(" " + i + " \t " + distance[i] + "\t " + source);
}
view_path(path, i);
Console.Write("\n");
}
} else {
Console.Write("Empty Graph");
}
}
public static void Main(String[] args) {
//Number of nodes
int totalNode = 5;
MyGraph g = new MyGraph(totalNode);
g.print_graph();
//Start location
int start = 3;
g.bellman_ford(start);
}
}```
```

#### Output

``````Adjacency list of vertex 0 : 1 2 4
Adjacency list of vertex 1 : 4
Adjacency list of vertex 2 : 1
Adjacency list of vertex 3 : 0 2 4
Adjacency list of vertex 4 : 2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````<?php
// Php program
// Show path in Bellman Ford algorithm
class AjlistNode {
//Vertices node key

public \$id;
public \$next;
public \$weight;

function __construct(\$id, \$weight) {
//Set value of node key
\$this->id = \$id;
\$this->next = null;
\$this->weight = \$weight;
}
}
class Vertices {
public \$data;
public \$next;

function __construct(\$data) {
\$this->data = \$data;
\$this->next = null;
}
}
class MyGraph {
//number of Vertices

private \$size;
private \$node;

function __construct(\$size) {
\$this->size = \$size;
\$this->node = array_fill(0, \$size, null);
//set initial values of graph node
\$this->set_data();
}
//Set initial node value

public 	function set_data() {
if (\$this->node == null) {
echo("\nEmpty Graph");
} else {
\$index = 0;
while (\$index < \$this->size) {
\$this->node[\$index] = new Vertices(\$index);
\$index++;
}
}
}
//Connect two node

public 	function add_edge(\$start, \$last, \$weight) {
\$newEdge = new AjlistNode(\$last, \$weight);
if (\$this->node[\$start]->next == null) {
//Include first adjacency list node of location start
\$this->node[\$start]->next = \$newEdge;
} else {
\$temp = \$this->node[\$start]->next;
//Add new node at the end of edge
while (\$temp->next != null) {
\$temp = \$temp->next;
}
\$temp->next = \$newEdge;
}
}
//Display graph elements

public 	function print_graph() {
if (\$this->size > 0 &&
\$this->node != null) {
\$index = 0;
while (\$index < \$this->size) {
echo("\nAdjacency list of vertex ". \$index ." : ");
\$temp = \$this->node[\$index]->next;
while (\$temp != null) {
echo(\$this->node[\$temp->id]->data ." ");
\$temp = \$temp->next;
}
\$index++;
}
}
}
public 	function view_path( & \$path, \$location) {
if (\$path[\$location] == -1) {
return;
}
\$this->view_path(\$path, \$path[\$location]);
echo(" ". \$location);
}
public 	function bellman_ford(\$source) {
if (\$source < 0 &&
\$source >= \$this->size) {
return;
}
if (\$this->node != null) {
\$distance = array_fill(0, \$this->size, 0);
\$path = array_fill(0, \$this->size, 0);
for (\$i = 0; \$i < \$this->size; ++\$i) {
//Initial distance is Integer.MAX_VALUE is infinity
\$distance[\$i] = PHP_INT_MAX;
\$path[\$i] = -1;
}
\$distance[\$source] = 0;
\$temp = null;
echo("\n Source Node ". \$source ." \n");
for (\$i = 1; \$i < \$this->size; ++\$i) {
for (\$j = 0; \$j < \$this->size; ++\$j) {
\$temp = \$this->node[\$j]->next;
//compare node (J) outgoing edges
while (\$temp != null) {
if (\$distance[\$j] != PHP_INT_MAX &&
\$distance[\$j] + \$temp->weight < \$distance[\$temp->id]) {
\$distance[\$temp->id] = \$distance[\$j] + \$temp->weight;
\$path[\$temp->id] = \$j;
}
\$temp = \$temp->next;
}
}
}
for (\$i = 1; \$i < \$this->size; ++\$i) {
for (\$j = 0; \$j < \$this->size; ++\$j) {
\$temp = \$this->node[\$j]->next;
//compare node (J) outgoing edges
while (\$temp != null) {
if (\$distance[\$j] != PHP_INT_MAX &&
\$distance[\$j] + \$temp->weight < \$distance[\$temp->id]) {
echo("\n Negative Cycle exist node (". \$temp->id ." ". \$j .")\n");
return;
}
\$temp = \$temp->next;
}
}
}
echo("\nVertex Distance Path\n");
for (\$i = 0; \$i < \$this->size; ++\$i) {
if (\$distance[\$i] == PHP_INT_MAX) {
echo(" ". \$i ." \t \t Integer MAX VALUE\n");
} else {
echo(" ". \$i ." \t ". \$distance[\$i] ."\t ". \$source);
}
\$this->view_path(\$path, \$i);
echo("\n");
}
} else {
echo("Empty Graph");
}
}
}

function main() {
//Number of nodes
\$totalNode = 5;
\$g = new MyGraph(\$totalNode);
//Connected two node with Edges
//Third parameter is weight
\$g->print_graph();
//Start location
\$start = 3;
\$g->bellman_ford(\$start);

}
main();```
```

#### Output

``````Adjacency list of vertex 0 : 1 2 4
Adjacency list of vertex 1 : 4
Adjacency list of vertex 2 : 1
Adjacency list of vertex 3 : 0 2 4
Adjacency list of vertex 4 : 2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````// Node Js program
// Show path in Bellman Ford algorithm
class AjlistNode {
//Vertices node key
constructor(id, weight) {
//Set value of node key
this.id = id;
this.next = null;
this.weight = weight;
}
}
class Vertices {
constructor(data) {
this.data = data;
this.next = null;
}
}
class MyGraph {
//number of Vertices
constructor(size) {
this.size = size;
this.node = Array(size).fill(null);
//set initial values of graph node
this.set_data();
}

//Set initial node value
set_data() {
if (this.node == null) {
process.stdout.write("\nEmpty Graph");
} else {
var index = 0;
while (index < this.size) {
this.node[index] = new Vertices(index);
index++;
}
}
}

//Connect two node
var newEdge = new AjlistNode(last, weight);
if (this.node[start].next == null) {
//Include first adjacency list node of location start
this.node[start].next = newEdge;
} else {
var temp = this.node[start].next;
//Add new node at the end of edge
while (temp.next != null) {
temp = temp.next;
}

temp.next = newEdge;
}
}

//Display graph elements
print_graph() {
if (this.size > 0 &&
this.node != null) {
var index = 0;
while (index < this.size) {
process.stdout.write("\nAdjacency list of vertex " + index + " : ");
var temp = this.node[index].next;
while (temp != null) {
process.stdout.write(this.node[temp.id].data + " ");
temp = temp.next;
}
index++;
}
}
}
view_path(path, location) {
if (path[location] == -1) {
return;
}
this.view_path(path, path[location]);
process.stdout.write(" " + location);
}
bellman_ford(source) {
if (source < 0 &&
source >= this.size) {
return;
}

if (this.node != null) {
var distance = Array(this.size).fill(0);
var path = Array(this.size).fill(0);
for (var i = 0; i < this.size; ++i) {
//Initial distance is Integer.MAX_VALUE is infinity
distance[i] = Number.MAX_VALUE;
path[i] = -1;
}
distance[source] = 0;
var temp = null;
process.stdout.write("\n Source Node " + source + " \n");
for (var i = 1; i < this.size; ++i) {
for (var j = 0; j < this.size; ++j) {
temp = this.node[j].next;
//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != Number.MAX_VALUE &&
distance[j] + temp.weight < distance[temp.id]) {
distance[temp.id] = distance[j] + temp.weight;
path[temp.id] = j;
}
temp = temp.next;
}
}
}

for (var i = 1; i < this.size; ++i) {
for (var j = 0; j < this.size; ++j) {
temp = this.node[j].next;
//compare node (J) outgoing edges
while (temp != null) {
if (distance[j] != Number.MAX_VALUE &&
distance[j] + temp.weight < distance[temp.id]) {
process.stdout.write("\n Negative Cycle exist node (" + temp.id + " " + j + ")\n");
return;
}
temp = temp.next;
}
}
}

process.stdout.write("\nVertex Distance Path\n");
for (var i = 0; i < this.size; ++i) {
if (distance[i] == Number.MAX_VALUE) {
process.stdout.write(" " + i + " \t \t Integer MAX VALUE\n");
} else {
process.stdout.write(" " + i + " \t " + distance[i] + "\t " + source);
}
this.view_path(path, i);
process.stdout.write("\n");
}
} else {
process.stdout.write("Empty Graph");
}
}
}

function main(args) {
//Number of nodes
var totalNode = 5;
var g = new MyGraph(totalNode);
//Connected two node with Edges
//Third parameter is weight
g.print_graph();
//Start location
var start = 3;
g.bellman_ford(start);
}

main();```
```

#### Output

``````Adjacency list of vertex 0 : 1 2 4
Adjacency list of vertex 1 : 4
Adjacency list of vertex 2 : 1
Adjacency list of vertex 3 : 0 2 4
Adjacency list of vertex 4 : 2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````#  Python 3 program
#  Show path in Bellman Ford algorithm
import sys
class AjlistNode :
# Vertices node key
def __init__(self, id, weight) :
# Set value of node key
self.id = id
self.next = None
self.weight = weight

class Vertices :

def __init__(self, data) :
self.data = data
self.next = None

class MyGraph :
# number of Vertices
def __init__(self, size) :
self.size = size
self.node = [None] * size
self.set_data()

# Set initial node value
def set_data(self) :
if (self.node == None) :
print("\nEmpty Graph", end = "")
else :
index = 0
while (index < self.size) :
self.node[index] = Vertices(index)
index += 1

# Connect two node
def add_edge(self, start, last, weight) :
newEdge = AjlistNode(last, weight)
if (self.node[start].next == None) :
# Include first adjacency list node of location start
self.node[start].next = newEdge
else :
temp = self.node[start].next
# Add new node at the end of edge
while (temp.next != None) :
temp = temp.next

temp.next = newEdge

# Display graph elements
def print_graph(self) :
if (self.size > 0 and self.node != None) :
index = 0
while (index < self.size) :
print("\nAdjacency list of vertex ", index ," : ", end = "")
temp = self.node[index].next
while (temp != None) :
print(self.node[temp.id].data ," ", end = "")
temp = temp.next

index += 1

def view_path(self, path, location) :
if (path[location] == -1) :
return

self.view_path(path, path[location])
print(" ", location, end = "")

def bellman_ford(self, source) :
if (source < 0 and source >= self.size) :
return

if (self.node != None) :
distance = [0] * self.size
path = [0] * self.size
i = 0
while (i < self.size) :
# Initial distance is Integer.MAX_VALUE is infinity
distance[i] = sys.maxsize
path[i] = -1
i += 1

distance[source] = 0
temp = None
print("\n Source Node ", source ," \n", end = "")
i = 1
while (i < self.size) :
j = 0
while (j < self.size) :
temp = self.node[j].next
# compare node (J) outgoing edges
while (temp != None) :
if (distance[j] != sys.maxsize and distance[j] + temp.weight < distance[temp.id]) :
distance[temp.id] = distance[j] + temp.weight
path[temp.id] = j

temp = temp.next

j += 1

i += 1

i = 1
while (i < self.size) :
j = 0
while (j < self.size) :
temp = self.node[j].next
# compare node (J) outgoing edges
while (temp != None) :
if (distance[j] != sys.maxsize and distance[j] + temp.weight < distance[temp.id]) :
print("\n Negative Cycle exist node (", temp.id ," ", j ,")\n", end = "")
return

temp = temp.next

j += 1

i += 1

print("\nVertex Distance Path\n", end = "")
i = 0
while (i < self.size) :
if (distance[i] == sys.maxsize) :
print(" ", i ," \t \t Integer MAX VALUE\n", end = "")
else :
print(" ", i ," \t ", distance[i] ,"\t ", source, end = "")

self.view_path(path, i)
print("\n", end = "")
i += 1

else :
print("Empty Graph", end = "")

def main() :
# Number of nodes
totalNode = 5
g = MyGraph(totalNode)
# Connected two node with Edges
# Third parameter is weight
g.print_graph()
# Start location
start = 3
g.bellman_ford(start)

if __name__ == "__main__":
main()```
```

#### Output

``````Adjacency list of vertex  0  : 1  2  4
Adjacency list of vertex  1  : 4
Adjacency list of vertex  2  : 1
Adjacency list of vertex  3  : 0  2  4
Adjacency list of vertex  4  : 2
Source Node  3

Vertex Distance Path
0  	  4 	  3  0
1  	  -4 	  3  0  2  1
2  	  1 	  3  0  2
3  	  0 	  3
4  	  1 	  3  0  2  1  4``````
``````#  Ruby program
#  Show path in Bellman Ford algorithm
class AjlistNode
# Define the accessor and reader of class AjlistNode
attr_accessor :id, :next, :weight
# Vertices node key
def initialize(id, weight)
# Set value of node key
self.id = id
self.next = nil
self.weight = weight
end
end
class Vertices
# Define the accessor and reader of class Vertices
attr_accessor :data, :next

def initialize(data)
self.data = data
self.next = nil
end
end
class MyGraph
# Define the accessor and reader of class MyGraph
attr_accessor :size, :node
# number of Vertices
def initialize(size)
self.size = size
self.node = Array.new(size) {nil}
self.set_data()
end
# Set initial node value
def set_data()
if (@node == nil)
print("\nEmpty Graph")
else
index = 0
while (index < @size)
@node[index] = Vertices.new(index)
index += 1
end
end
end
# Connect two node
newEdge = AjlistNode.new(last, weight)
if (@node[start].next == nil)
# Include first adjacency list node of location start
@node[start].next = newEdge
else
temp = @node[start].next
# Add new node at the end of edge
while (temp.next != nil)
temp = temp.next
end
temp.next = newEdge
end
end
# Display graph elements
def print_graph()
if (@size > 0 &&
@node != nil)
index = 0
while (index < @size)
print("\nAdjacency list of vertex ", index ,"  : ")
temp = @node[index].next
while (temp != nil)
print(@node[temp.id].data ," ")
temp = temp.next
end
index += 1
end
end
end
def view_path(path, location)
if (path[location] == -1)
return
end
self.view_path(path, path[location])
print(" ", location)
end
def bellman_ford(source)
if (source < 0 &&
source >= self.size)
return
end
if (@node != nil)
distance = Array.new(@size) {0}
path = Array.new(@size) {0}
i = 0
while (i < @size)
# Initial distance is Integer.MAX_VALUE is infinity
distance[i] = (2 ** (0. size * 8 - 2))
path[i] = -1
i += 1
end
distance[source] = 0
temp = nil
print("\n Source Node ", source ," \n")
i = 1
while (i < @size)
j = 0
while (j < @size)
temp = @node[j].next
# compare node (J) outgoing edges
while (temp != nil)
if (distance[j] != (2 ** (0. size * 8 - 2)) &&
distance[j] + temp.weight < distance[temp.id])
distance[temp.id] = distance[j] + temp.weight
path[temp.id] = j
end
temp = temp.next
end
j += 1
end
i += 1
end
i = 1
while (i < @size)
j = 0
while (j < @size)
temp = @node[j].next
# compare node (J) outgoing edges
while (temp != nil)
if (distance[j] != (2 ** (0. size * 8 - 2)) &&
distance[j] + temp.weight < distance[temp.id])
print("\n Negative Cycle exist node (", temp.id ," ", j ,")\n")
return
end
temp = temp.next
end
j += 1
end
i += 1
end
print("\nVertex Distance Path\n")
i = 0
while (i < @size)
if (distance[i] == (2 ** (0. size * 8 - 2)))
print(" ", i ," \t \t Integer MAX VALUE\n")
else
print(" ", i ," \t\t", distance[i] ,"\t\t  ", source)
end
self.view_path(path, i)
print("\n")
i += 1
end
else
print("Empty Graph")
end
end
end
def main()
# Number of nodes
totalNode = 5
g = MyGraph.new(totalNode)
# Connected two node with Edges
# Third parameter is weight
g.print_graph()
# Start location
start = 3
g.bellman_ford(start)
end
main()```
```

#### Output

``````Adjacency list of vertex 0  : 1 2 4
Adjacency list of vertex 1  : 4
Adjacency list of vertex 2  : 1
Adjacency list of vertex 3  : 0 2 4
Adjacency list of vertex 4  : 2
Source Node 3

Vertex Distance Path
0 		4		  3 0
1 		-4		  3 0 2 1
2 		1		  3 0 2
3 		0		  3
4 		1		  3 0 2 1 4
``````
``````class AjlistNode(var id: Int,
var next: AjlistNode,
var weight: Int) {

def this(id: Int, weight: Int) {
//Set value
this(id,null,weight);
}
}
class Vertices(var data: Int,
var next: AjlistNode) {

def this(data: Int) {
this(data,null);
}
}
class MyGraph(var size: Int,
var node: Array[Vertices]) {
def this(value: Int) {
//set value
this(value,Array.fill[Vertices](value)(null));
//set initial values of graph node
this.set_data();
}

//Set initial node value
def set_data(): Unit = {
if (this.node == null) {
print("\nEmpty Graph");
} else {
var index: Int = 0;
while (index < this.size) {
this.node(index) = new Vertices(index);
index += 1;
}
}
}
//Connect two node
def add_edge(start: Int, last: Int, weight: Int): Unit = {
var newEdge: AjlistNode = new AjlistNode(last, weight);

if (node(start).next == null) {
//Include first adjacency list node of location start
node(start).next = newEdge;
} else {
var temp: AjlistNode = node(start).next;

//Add new node at the end of edge
while (temp.next != null) {
temp = temp.next;
}
temp.next = newEdge;
}
}
//Display graph elements
def print_graph(): Unit = {
if (size > 0 &&
node != null) {
var index: Int = 0;
while (index < size) {
print("\nAdjacency list of vertex " + index + " : ");
var temp: AjlistNode = node(index).next;
while (temp != null) {
print(" "+node(temp.id).data );
temp = temp.next;
}
index += 1;
}
}
}
def view_path(path: Array[Int], location: Int): Unit = {
if (path(location) == -1) {
return;
}
view_path(path, path(location));
print(" " + location);
}
def bellman_ford(source: Int ): Unit = {
if (source < 0 &&
source >= this.size) {
return;
}
if (node != null) {
var distance: Array[Int] = Array.fill[Int](this.size)(0);
var path: Array[Int] = Array.fill[Int](this.size)(0);
var i: Int = 0;
while (i < size) {
//Initial distance is Integer.MAX_VALUE is infinity
distance(i) = Int.MaxValue;
path(i) = -1;
i += 1;
}
distance(source) = 0;
var temp: AjlistNode = null;
print("\n Source Node " + source + " \n");
i = 1;
var j: Int = 0;
while (i < size) {
j = 0;
while (j < size) {
temp = node(j).next;

//compare node (J) outgoing edges
while (temp != null) {
if (distance(j) != Int.MaxValue &&
distance(j) + temp.weight < distance(temp.id)) {
distance(temp.id) = distance(j) + temp.weight;
path(temp.id) = j;
}
temp = temp.next;
}
j += 1;
}
i += 1;
}
i = 1;
while (i < size) {
j = 0;
while (j < size) {
temp = node(j).next;

//compare node (J) outgoing edges
while (temp != null) {
if (distance(j) != Int.MaxValue &&
distance(j) + temp.weight < distance(temp.id)) {
print("\n Negative Cycle exist node (" + temp.id + " " + j + ")\n");

return;
}
temp = temp.next;
}
j += 1;
}
i += 1;
}
print("\nVertex Distance Path\n");
i = 0;
while (i < size) {
if (distance(i) == Int.MaxValue) {
print(" " + i + " \t \t Integer MAX VALUE\n");
} else {
print(" " + i + " \t " + distance(i) + "\t " + source);
}
view_path(path, i);
print("\n");
i += 1;
}
} else {
print("Empty Graph");
}
}
}
object Main {
def main(args: Array[String]): Unit = {
//Number of nodes
var totalNode: Int = 5;
var g: MyGraph = new MyGraph(totalNode);

//Connected two node with Edges
//Third parameter is weight
g.print_graph();

//Start location
var start: Int = 3;
g.bellman_ford(start);
}
}```
```

#### Output

``````Adjacency list of vertex 0 :  1 2 4
Adjacency list of vertex 1 :  4
Adjacency list of vertex 2 :  1
Adjacency list of vertex 3 :  0 2 4
Adjacency list of vertex 4 :  2
Source Node 3

Vertex Distance Path
0 	 4	 3 0
1 	 -4	 3 0 2 1
2 	 1	 3 0 2
3 	 0	 3
4 	 1	 3 0 2 1 4``````
``````// Swift program
// Show path in Bellman Ford algorithm
class AjlistNode {
//Vertices node key
var id: Int;
var next: AjlistNode? ;
var weight: Int;
init(_ id: Int, _ weight: Int) {
//Set value of node key
self.id = id;
self.next = nil;
self.weight = weight;
}
}
class Vertices {
var data: Int;
var next: AjlistNode? ;
init(_ data: Int) {
self.data = data;
self.next = nil;
}
}
class MyGraph {
//number of Vertices
var size: Int;
var node: [Vertices]? = [Vertices]() ;
init(_ size: Int) {
self.size = size;
var i = 0;
while (i<size) {
self.node!.append(Vertices(i));
i+=1;
}
}
//Connect two node
func add_edge(_ start: Int, _ last: Int, _ weight: Int) {
let newEdge: AjlistNode? = AjlistNode(last, weight);
if (self.node![start].next == nil) {
//Include first adjacency list node of location start
self.node![start].next = newEdge;
} else {
var temp: AjlistNode? = self.node![start].next;
//Add new node at the end of edge
while (temp!.next != nil) {
temp = temp!.next;
}
temp!.next = newEdge;
}
}
//Display graph elements
func print_graph() {
if (self.size > 0 &&
self.node != nil) {
var index: Int = 0;
while (index < self.size) {
print("\nAdjacency list of vertex ", index ," : ", terminator: "");
var temp: AjlistNode? = self.node![index].next;
while (temp != nil) {
print(self.node![temp!.id].data ," ", terminator: "");
temp = temp!.next;
}
index += 1;
}
}
}
func view_path(_ path: [Int], _ location: Int) {
if (path[location] == -1) {
return;
}
self.view_path(path, path[location]);
print(" ", location, terminator: "");
}
func bellman_ford(_ source: Int) {
if (source < 0 &&
source >= self.size) {
return;
}
if (self.node != nil) {
var distance: [Int] = Array(repeating: 0, count: self.size);
var path: [Int] = Array(repeating: 0, count: self.size);
var i: Int = 0;
while (i < self.size) {
//Initial distance is Integer.MAX_VALUE is infinity
distance[i] = Int.max;
path[i] = -1;
i += 1;
}
distance[source] = 0;
var temp: AjlistNode? = nil;
print("\n Source Node ", source ," \n", terminator: "");
i = 1;
var j: Int = 0;
while (i < self.size) {
j = 0;
while (j < self.size) {
temp = self.node![j].next;
//compare node (J) outgoing edges
while (temp != nil) {
if (distance[j] != Int.max &&
distance[j] + temp!.weight < distance[temp!.id]) {
distance[temp!.id] = distance[j] + temp!.weight;
path[temp!.id] = j;
}
temp = temp!.next;
}
j += 1;
}
i += 1;
}
i = 1;
while (i < self.size) {
j = 0;
while (j < self.size) {
temp = self.node![j].next;
//compare node (J) outgoing edges
while (temp != nil) {
if (distance[j] != Int.max &&
distance[j] + temp!.weight < distance[temp!.id]) {
print("\n Negative Cycle exist node (", temp!.id ," ", j ,")\n", terminator: "");
return;
}
temp = temp!.next;
}
j += 1;
}
i += 1;
}
print("\nVertex Distance Path\n", terminator: "");
i = 0;
while (i < self.size) {
if (distance[i] == Int.max) {
print(" ", i ," \t \t Integer MAX VALUE\n", terminator: "");
} else {
print(" ", i ," \t ", distance[i] ,"\t ", source, terminator: "");
}
self.view_path(path, i);
print("\n", terminator: "");
i += 1;
}
} else {
print("Empty Graph", terminator: "");
}
}
}
func main() {
//Number of nodes
let totalNode: Int = 5;
let g: MyGraph? = MyGraph(totalNode);
//Connected two node with Edges
//Third parameter is weight
g!.print_graph();
//Start location
let start: Int = 3;
g!.bellman_ford(start);
}
main();```
```

#### Output

``````Adjacency list of vertex  0  : 1  2  4
Adjacency list of vertex  1  : 4
Adjacency list of vertex  2  : 1
Adjacency list of vertex  3  : 0  2  4
Adjacency list of vertex  4  : 2
Source Node  3

Vertex Distance Path
0  	  4 	  3  0
1  	  -4 	  3  0  2  1
2  	  1 	  3  0  2
3  	  0 	  3
4  	  1 	  3  0  2  1  4``````

### Understanding the Code:

The provided code is an implementation of the Bellman-Ford algorithm in C. Let's break down the code into its main components:

#### Data Structures:

• `struct AjlistNode`: Represents a node in the adjacency list. It contains the ID of the connected vertex and the weight of the edge.
• `struct Graph`: Represents a graph. It contains a data field (node key value) and a pointer to the first node in the adjacency list.

#### Functions:

• `set_data`: Sets the data (node key value) for each node in the graph.
• `connect_edge`: Connects two nodes by creating an adjacency node and adding it to the adjacency list.
• `add_edge`: Adds an edge between two given nodes with a specified weight.
• `print_graph`: Prints the adjacency list representation of the graph.
• `view_path`: Recursive function to view the shortest path from the source node to a given node.
• `bellman_ford`: Implements the Bellman-Ford algorithm to find the shortest path from the source node to all other nodes.

#### Walkthrough of the Bellman-Ford Algorithm:

Now, let's understand how the Bellman-Ford algorithm works based on the provided code:

##### Initializing the Graph:
• The `size` variable represents the number of nodes in the graph.
• The `set_data` function is called to set the key value for each node in the graph.

The add_edge function is used to add edges between nodes along with their corresponding weights. This creates the adjacency list representation of the graph.

##### Bellman-Ford Algorithm:
• The `bellman_ford` function implements the Bellman-Ford algorithm.
• It initializes the distance array with infinity values except for the source node, which is set to 0.
• The algorithm iterates `size - 1` times to relax all the edges in the graph.
• For each iteration, it checks all the outgoing edges from each node and updates the distance if a shorter path is found.
• After the iterations, the algorithm performs one more pass to check for negative cycles. If a shorter path is found, it indicates the presence of a negative cycle.
• Finally, the algorithm prints the vertex, its distance from the source, and the shortest path to reach that vertex using the `view_path` function.

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