Rewrite Y16x64 3 to Make It Easy to Graph Using a Trnaslation Desribe the Graph

A graph is a data structure that consists of the following two components:
1. A finite set of vertices also called as nodes.
2. A finite set of ordered pair of the form (u, v) called as edge. The pair is ordered because (u, v) is not the same as (v, u) in case of a directed graph(di-graph). The pair of the form (u, v) indicates that there is an edge from vertex u to vertex v. The edges may contain weight/value/cost.
Graphs are used to represent many real-life applications: Graphs are used to represent networks. The networks may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like linkedIn, Facebook. For example, in Facebook, each person is represented with a vertex(or node). Each node is a structure and contains information like person id, name, gender, and locale. See this for more applications of graph.
Following is an example of an undirected graph with 5 vertices.

The following two are the most commonly used representations of a graph.
1. Adjacency Matrix
2. Adjacency List
There are other representations also like, Incidence Matrix and Incidence List. The choice of graph representation is situation-specific. It totally depends on the type of operations to be performed and ease of use.
Adjacency Matrix:
Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w.

The adjacency matrix for the above example graph is:

Pros: Representation is easier to implement and follow. Removing an edge takes O(1) time. Queries like whether there is an edge from vertex 'u' to vertex 'v' are efficient and can be done O(1).
Cons: Consumes more space O(V^2). Even if the graph is sparse(contains less number of edges), it consumes the same space. Adding a vertex is O(V^2) time.  Computing all neighbors of a vertex takes O(V) time (Not efficient).
Please see this for a sample Python implementation of adjacency matrix.

Implementation of taking input for adjacency matrix

C++

#include <iostream>

using namespace std;

int main()

{

int n, m;

cin >> n >> m ;

int adjMat[n + 1][n + 1];

for ( int i = 0; i < m; i++){

int u , v ;

cin >> u >> v ;

adjMat[u][v] = 1 ;

adjMat[v][u] = 1 ;

}

return 0;

}

Adjacency List:
An array of lists is used. The size of the array is equal to the number of vertices. Let the array be an array[]. An entry array[i] represents the list of vertices adjacent to the i th vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is the adjacency list representation of the above graph.

Note that in the below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of the linked list. The vector implementation has advantages of cache friendliness.

C++

#include <bits/stdc++.h>

using namespace std;

void addEdge(vector< int > adj[], int u, int v)

{

adj[u].push_back(v);

adj[v].push_back(u);

}

void printGraph(vector< int > adj[], int V)

{

for ( int v = 0; v < V; ++v) {

cout << "\n Adjacency list of vertex " << v

<< "\n head " ;

for ( auto x : adj[v])

cout << "-> " << x;

printf ( "\n" );

}

}

int main()

{

int V = 5;

vector< int > adj[V];

addEdge(adj, 0, 1);

addEdge(adj, 0, 4);

addEdge(adj, 1, 2);

addEdge(adj, 1, 3);

addEdge(adj, 1, 4);

addEdge(adj, 2, 3);

addEdge(adj, 3, 4);

printGraph(adj, V);

return 0;

}

C

#include <stdio.h>

#include <stdlib.h>

struct AdjListNode {

int dest;

struct AdjListNode* next;

};

struct AdjList {

struct AdjListNode* head;

};

struct Graph {

int V;

struct AdjList* array;

};

struct AdjListNode* newAdjListNode( int dest)

{

struct AdjListNode* newNode

= ( struct AdjListNode*) malloc (

sizeof ( struct AdjListNode));

newNode->dest = dest;

newNode->next = NULL;

return newNode;

}

struct Graph* createGraph( int V)

{

struct Graph* graph

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

graph->V = V;

graph->array = ( struct AdjList*) malloc (

V * sizeof ( struct AdjList));

int i;

for (i = 0; i < V; ++i)

graph->array[i].head = NULL;

return graph;

}

void addEdge( struct Graph* graph, int src, int dest)

{

struct AdjListNode* check = NULL;

struct AdjListNode* newNode = newAdjListNode(dest);

if (graph->array[src].head == NULL) {

newNode->next = graph->array[src].head;

graph->array[src].head = newNode;

}

else {

check = graph->array[src].head;

while (check->next != NULL) {

check = check->next;

}

check->next = newNode;

}

newNode = newAdjListNode(src);

if (graph->array[dest].head == NULL) {

newNode->next = graph->array[dest].head;

graph->array[dest].head = newNode;

}

else {

check = graph->array[dest].head;

while (check->next != NULL) {

check = check->next;

}

check->next = newNode;

}

}

void printGraph( struct Graph* graph)

{

int v;

for (v = 0; v < graph->V; ++v) {

struct AdjListNode* pCrawl = graph->array[v].head;

printf ( "\n Adjacency list of vertex %d\n head " , v);

while (pCrawl) {

printf ( "-> %d" , pCrawl->dest);

pCrawl = pCrawl->next;

}

printf ( "\n" );

}

}

int main()

{

int V = 5;

struct Graph* graph = createGraph(V);

addEdge(graph, 0, 1);

addEdge(graph, 0, 4);

addEdge(graph, 1, 2);

addEdge(graph, 1, 3);

addEdge(graph, 1, 4);

addEdge(graph, 2, 3);

addEdge(graph, 3, 4);

printGraph(graph);

return 0;

}

Java

import java.util.*;

class Graph {

static void addEdge(ArrayList<ArrayList<Integer> > adj,

int u, int v)

{

adj.get(u).add(v);

adj.get(v).add(u);

}

static void

printGraph(ArrayList<ArrayList<Integer> > adj)

{

for ( int i = 0 ; i < adj.size(); i++) {

System.out.println( "\nAdjacency list of vertex"

+ i);

System.out.print( "head" );

for ( int j = 0 ; j < adj.get(i).size(); j++) {

System.out.print( " -> "

+ adj.get(i).get(j));

}

System.out.println();

}

}

public static void main(String[] args)

{

int V = 5 ;

ArrayList<ArrayList<Integer> > adj

= new ArrayList<ArrayList<Integer> >(V);

for ( int i = 0 ; i < V; i++)

adj.add( new ArrayList<Integer>());

addEdge(adj, 0 , 1 );

addEdge(adj, 0 , 4 );

addEdge(adj, 1 , 2 );

addEdge(adj, 1 , 3 );

addEdge(adj, 1 , 4 );

addEdge(adj, 2 , 3 );

addEdge(adj, 3 , 4 );

printGraph(adj);

}

}

Python3

class AdjNode:

def __init__( self , data):

self .vertex = data

self . next = None

class Graph:

def __init__( self , vertices):

self .V = vertices

self .graph = [ None ] * self .V

def add_edge( self , src, dest):

node = AdjNode(dest)

node. next = self .graph[src]

self .graph[src] = node

node = AdjNode(src)

node. next = self .graph[dest]

self .graph[dest] = node

def print_graph( self ):

for i in range ( self .V):

print ( "Adjacency list of vertex {}\n head" . format (i), end = "")

temp = self .graph[i]

while temp:

print ( " -> {}" . format (temp.vertex), end = "")

temp = temp. next

print ( " \n" )

if __name__ = = "__main__" :

V = 5

graph = Graph(V)

graph.add_edge( 0 , 1 )

graph.add_edge( 0 , 4 )

graph.add_edge( 1 , 2 )

graph.add_edge( 1 , 3 )

graph.add_edge( 1 , 4 )

graph.add_edge( 2 , 3 )

graph.add_edge( 3 , 4 )

graph.print_graph()

C#

using System;

using System.Collections.Generic;

class Graph {

static void addEdge(LinkedList< int >[] adj, int u, int v)

{

adj[u].AddLast(v);

adj[v].AddLast(u);

}

static void printGraph(LinkedList< int >[] adj)

{

for ( int i = 0; i < adj.Length; i++) {

Console.WriteLine( "\nAdjacency list of vertex "

+ i);

Console.Write( "head" );

foreach ( var item in adj[i])

{

Console.Write( " -> " + item);

}

Console.WriteLine();

}

}

public static void Main(String[] args)

{

int V = 5;

LinkedList< int >[] adj = new LinkedList< int >[ V ];

for ( int i = 0; i < V; i++)

adj[i] = new LinkedList< int >();

addEdge(adj, 0, 1);

addEdge(adj, 0, 4);

addEdge(adj, 1, 2);

addEdge(adj, 1, 3);

addEdge(adj, 1, 4);

addEdge(adj, 2, 3);

addEdge(adj, 3, 4);

printGraph(adj);

Console.ReadKey();

}

}

Javascript

<script>

function addEdge(adj,u,v)

{

adj[u].push(v);

adj[v].push(u);

}

function printGraph(adj)

{

for (let i = 0; i < adj.length; i++) {

document.write( "<br>Adjacency list of vertex" + i+ "<br>" );

document.write( "head" );

for (let j = 0; j < adj[i].length; j++) {

document.write( " -> " +adj[i][j]);

}

document.write( "<br>" );

}

}

let V = 5;

let adj= [];

for (let i = 0; i < V; i++)

adj.push([]);

addEdge(adj, 0, 1);

addEdge(adj, 0, 4);

addEdge(adj, 1, 2);

addEdge(adj, 1, 3);

addEdge(adj, 1, 4);

addEdge(adj, 2, 3);

addEdge(adj, 3, 4);

printGraph(adj);

</script>

Output

            Adjacency list of vertex 0  head -> 1-> 4   Adjacency list of vertex 1  head -> 0-> 2-> 3-> 4   Adjacency list of vertex 2  head -> 1-> 3   Adjacency list of vertex 3  head -> 1-> 2-> 4   Adjacency list of vertex 4  head -> 0-> 1-> 3

Pros: Saves space O(|V|+|E|) . In the worst case, there can be C(V, 2) number of edges in a graph thus consuming O(V^2) space. Adding a vertex is easier. Computing all neighbors of a vertex takes optimal time.
Cons: Queries like whether there is an edge from vertex u to vertex v are not efficient and can be done O(V).
 In Real-life problems,  graphs are sparse(|E| <<|V|²). That's why adjacency lists Data structure is commonly used for storing graphs. Adjacency matrix will enforce (|V|²) bound on time complexity for such algorithms.

Reference:
http://en.wikipedia.org/wiki/Graph_%28abstract_data_type%29
Related Post:
Graph representation using STL for competitive programming | Set 1 (DFS of Unweighted and Undirected)
Graph implementation using STL for competitive programming | Set 2 (Weighted graph)
This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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Source: https://www.geeksforgeeks.org/graph-and-its-representations/

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