Data Structure - Graph Data Structure

A graph is an abstract data type (ADT) that consists of a set of objects that are connected to each other via pnks. These objects are called vertices and the pnks are called edges.

Usually, a graph is represented as G = {V, E}, where G is the graph space, V is the set of vertices and E is the set of edges. If E is empty, the graph is known as a forest.

Before we proceed further, let s famiparize ourselves with some important terms −

Vertex − Each node of the graph is represented as a vertex. In the following example, the labelled circle represents vertices. Thus, A to G are vertices. We can represent them using an array as shown in the following image. Here A can be identified by index 0. B can be identified using index 1 and so on.

Edge − Edge represents a path between two vertices or a pne between two vertices. In the following example, the pnes from A to B, B to C, and so on represents edges. We can use a two-dimensional array to represent an array as shown in the following image. Here AB can be represented as 1 at row 0, column 1, BC as 1 at row 1, column 2 and so on, keeping other combinations as 0.

Adjacency − Two node or vertices are adjacent if they are connected to each other through an edge. In the following example, B is adjacent to A, C is adjacent to B, and so on.

Path − Path represents a sequence of edges between the two vertices. In the following example, ABCD represents a path from A to D.

Operations of Graphs

The primary operations of a graph include creating a graph with vertices and edges, and displaying the said graph. However, one of the most common and popular operation performed using graphs are Traversal, i.e. visiting every vertex of the graph in a specific order.

There are two types of traversals in Graphs −

Depth First Search Traversal

Breadth First Search Traversal

Depth First Search Traversal

Depth First Search is a traversal algorithm that visits all the vertices of a graph in the decreasing order of its depth. In this algorithm, an arbitrary node is chosen as the starting point and the graph is traversed back and forth by marking unvisited adjacent nodes until all the vertices are marked.

The DFS traversal uses the stack data structure to keep track of the unvisited nodes.

Breadth First Search Traversal

Breadth First Search is a traversal algorithm that visits all the vertices of a graph present at one level of the depth before moving to the next level of depth. In this algorithm, an arbitrary node is chosen as the starting point and the graph is traversed by visiting the adjacent vertices on the same depth level and marking them until there is no vertex left.

The DFS traversal uses the queue data structure to keep track of the unvisited nodes.

Representation of Graphs

While representing graphs, we must carefully depict the elements (vertices and edges) present in the graph and the relationship between them. Pictorially, a graph is represented with a finite set of nodes and connecting pnks between them. However, we can also represent the graph in other most commonly used ways, pke −

Adjacency Matrix

Adjacency List

Adjacency Matrix

The Adjacency Matrix is a V×V matrix where the values are filled with either 0 or 1. If the pnk exists between Vi and Vj, it is recorded 1; otherwise, 0.

For the given graph below, let us construct an adjacency matrix −

The adjacency matrix is −

Adjacency List

The adjacency pst is a pst of the vertices directly connected to the other vertices in the graph.

The adjacency pst is −

Types of graph

There are two basic types of graph −

Directed Graph

Undirected Graph

Directed graph, as the name suggests, consists of edges that possess a direction that goes either away from a vertex or towards the vertex. Undirected graphs have edges that are not directed at all.

Directed Graph

Undirected Graph

Spanning Tree

A spanning tree is a subset of an undirected graph that contains all the vertices of the graph connected with the minimum number of edges in the graph. Precisely, the edges of the spanning tree is a subset of the edges in the original graph.

If all the vertices are connected in a graph, then there exists at least one spanning tree. In a graph, there may exist more than one spanning tree.

Properties

A spanning tree does not have any cycle.

Any vertex can be reached from any other vertex.

Example

In the following graph, the highpghted edges form a spanning tree.

Minimum Spanning Tree

A Minimum Spanning Tree (MST) is a subset of edges of a connected weighted undirected graph that connects all the vertices together with the minimum possible total edge weight. To derive an MST, Prim’s algorithm or Kruskal’s algorithm can be used. Hence, we will discuss Prim’s algorithm in this chapter.

As we have discussed, one graph may have more than one spanning tree. If there are n number of vertices, the spanning tree should have ?−? number of edges. In this context, if each edge of the graph is associated with a weight and there exists more than one spanning tree, we need to find the minimum spanning tree of the graph.

Moreover, if there exist any duppcate weighted edges, the graph may have multiple minimum spanning tree.

In the above graph, we have shown a spanning tree though it’s not the minimum spanning tree. The cost of this spanning tree is (5+7+3+3+5+8+3+4)=38.

Shortest Path

The shortest path in a graph is defined as the minimum cost route from one vertex to another. This is most commonly seen in weighted directed graphs but are also apppcable to undirected graphs.

A popular real-world apppcation of finding the shortest path in a graph is a map. Navigation is made easier and simpler with the various shortest path algorithms where destinations are considered vertices of the graph and routes are the edges. The two common shortest path algorithms are −

Dijkstra’s Shortest Path Algorithm

Bellman Ford’s Shortest Path Algorithm

Example

Following are the implementations of this operation in various programming languages −

#include <stdio.h>
#include<stdpb.h>
#include <stdpb.h>
#define V 5 

// Maximum number of vertices in the graph
struct graph { 
   
   // declaring graph data structure
   struct vertex *point[V];
};
struct vertex { 
   
   // declaring vertices
   int end;
   struct vertex *next;
};
struct Edge { 
   
   // declaring edges
   int end, start;
};
struct graph *create_graph (struct Edge edges[], int x){
   int i;
   struct graph *graph = (struct graph *) malloc (sizeof (struct graph));
   for (i = 0; i < V; i++) {
      graph->point[i] = NULL;
   }
   for (i = 0; i < x; i++) {
      int start = edges[i].start;
      int end = edges[i].end;
      struct vertex *v = (struct vertex *) malloc (sizeof (struct vertex));
      v->end = end;
      v->next = graph->point[start];
      graph->point[start] = v;
   }
   return graph;
}
int main (){
   struct Edge edges[] = { {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 4}, {2, 4}, {2, 3}, {3, 1} };
   int n = sizeof (edges) / sizeof (edges[0]);
   struct graph *graph = create_graph (edges, n);
   int i;
   for (i = 0; i < V; i++) {
      struct vertex *ptr = graph->point[i];
      while (ptr != NULL) {
         printf ("(%d -> %d)	", i, ptr->end);
         ptr = ptr->next;
      }
      printf ("
");
   }
   return 0;
}

(1 -> 3)	(1 -> 0)	
(2 -> 1)	(2 -> 0)	
(3 -> 2)	(3 -> 0)	
(4 -> 2)	(4 -> 1)	

#include <bits/stdc++.h>
using namespace std;
#define V 5 

// Maximum number of vertices in the graph
struct graph { 
   
   // declaring graph data structure
   struct vertex *point[V];
};
struct vertex { 
   
   // declaring vertices
   int end;
   struct vertex *next;
};
struct Edge { 
   
   // declaring edges
   int end, start;
};
struct graph *create_graph (struct Edge edges[], int x){
   int i;
   struct graph *graph = (struct graph *) malloc (sizeof (struct graph));
   for (i = 0; i < V; i++) {
      graph->point[i] = NULL;
   }
   for (i = 0; i < x; i++) {
      int start = edges[i].start;
      int end = edges[i].end;
      struct vertex *v = (struct vertex *) malloc (sizeof (struct vertex));
      v->end = end;
      v->next = graph->point[start];
      graph->point[start] = v;
   }
   return graph;
}
int main (){
   struct Edge edges[] = { {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 4}, {2, 4}, {2, 3}, {3, 1} };
   int n = sizeof (edges) / sizeof (edges[0]);
   struct graph *graph = create_graph (edges, n);
   int i;
   for (i = 0; i < V; i++) {
      struct vertex *ptr = graph->point[i];
      while (ptr != NULL) {
         cout << "(" << i << " -> " << ptr->end << ")	";
         ptr = ptr->next;
      }
      cout << endl;
   }
   return 0;
}

(1 -> 3)	(1 -> 0)	
(2 -> 1)	(2 -> 0)	
(3 -> 2)	(3 -> 0)	
(4 -> 2)	(4 -> 1)

import java.util.*;

//class to store edges of the graph
class Edge {   
   int src, dest;
   Edge(int src, int dest) {
      this.src = src;
      this.dest = dest;
   }
}

// Graph class
pubpc class Graph {
   
   // node of adjacency pst
   static class vertex {
      int v;
      vertex(int v) {
        this.v = v;
      }
   };
   // define adjacency pst to represent the graph
   List<List<vertex>> adj_pst = new ArrayList<>();
   
   //Graph Constructor
   pubpc Graph(List<Edge> edges){
      
      // adjacency pst memory allocation
      for (int i = 0; i < edges.size(); i++)
      adj_pst.add(i, new ArrayList<>());
      
      // add edges to the graph
      for (Edge e : edges){
        
        // allocate new node in adjacency List from src to dest
        adj_pst.get(e.src).add(new vertex(e.dest));
      }
   }
   pubpc static void main (String[] args) {
      
      // define edges of the graph
      List<Edge> edges = Arrays.asList(new Edge(0, 1),new Edge(0, 2),
      new Edge(0, 3),new Edge(1, 2), new Edge(1, 4),
      new Edge(2, 4), new Edge(2, 3),new Edge(3, 1));
      
      // call graph class Constructor to construct a graph
      Graph graph = new Graph(edges);
      
      // print the graph as an adjacency pst
      int src = 0;
      int lsize = graph.adj_pst.size();
      System.out.println("The graph created is:");
      while (src < lsize) {
      
         //traverse through the adjacency pst and print the edges
         for (vertex edge : graph.adj_pst.get(src)) {
            System.out.print(src + " -> " + edge.v + "	");
         }
         System.out.println();
         src++;
      }
   }
}

The graph created is:
0 -> 1	0 -> 2	0 -> 3	
1 -> 2	1 -> 4	2 -> 4	2 -> 3	
3 -> 1