Introduction
The objective of this article is to provide a basic introduction about graphs and the commonly used algorithms used for traversing the graph, BFS and DFS. Breadth First Search (BFS) and Depth First Search (DFS) are the two popular algorithms asked in most of the programming interviews. I was not able to find a simple, precise explanation for beginners on this topic. So, I decided to write an article for graph. This article will help any beginner to get some basic understanding about what graphs are, how they are represented, graph traversals using BFS and DFS.What is a Graph?
Graphs are one of the most interesting data structures in computer science. Graphs and the trees are somewhat similar by their structure. In fact, tree is derived from the graph data structure. However there are two important differences between trees and graphs.- Unlike trees, in graphs, a node can have many parents.
- The link between the nodes may have values or weights.
In the above graph, A,B,C,D,E,F are called nodes and the connecting
lines between these nodes are called edges. The edges can be directed
edges which are shown by arrows; they can also be weighted edges in
which some numbers are assigned to them. Hence, a graph can be a
directed/undirected and weighted/un-weighted graph. In this article, we
will discuss undirected and un-weighted graphs. Graph Representation in Programming Language
Every graph has two components, Nodes and Edges. Let’s see how these two components are implemented in a programming language like JAVA.1. Nodes
Nodes are implemented by class, structures or as Link-List nodes. As an example in JAVA, we will represent node for the above graph as follows:
//
Class Node
{
Char data;
Public Node(char c)
{
this.data=c;
}
}
//
2. Edges
Edges represent the connection between nodes. There are two ways to represent edges.Adjacency Matrix
It is a two dimensional array with Boolean flags. As an example, we can represent the edges for the above graph using the following adjacency matrix.
In the given graph, A is connected with B, C and D nodes, so
adjacency matrix will have 1s in the ‘A’ row for the ‘B’, ‘C’ and ‘D’
column. The advantages of representing the edges using adjacency matrix are:
- Simplicity in implementation as you need a 2-dimensional array
- Creating edges/removing edges is also easy as you need to update the Booleans
- Increased memory as you need to declare N*N matrix where N is the total number of nodes.
- Redundancy of information, i.e. to represent an edge between A to B and B to A, it requires to set two Boolean flag in an adjacency matrix.
Adjacency List
It is an array of linked list nodes. In other words, it is like a list whose elements are a linked list. For the given graph example, the edges will be represented by the below adjacency list:
Graph Traversal
The breadth first search (BFS) and the depth first search (DFS) are the two algorithms used for traversing and searching a node in a graph. They can also be used to find out whether a node is reachable from a given node or not.Depth First Search (DFS)
The aim of DFS algorithm is to traverse the graph in such a way that it tries to go far from the root node. Stack is used in the implementation of the depth first search. Let’s see how depth first search works with respect to the following graph:
As stated before, in DFS, nodes are visited by going through the
depth of the tree from the starting node. If we do the depth first
traversal of the above graph and print the visited node, it will be “A B
E F C D”. DFS visits the root node and then its children nodes until it
reaches the end node, i.e. E and F nodes, then moves up to the parent
nodes.Algorithmic Steps
- Step 1: Push the root node in the Stack.
- Step 2: Loop until stack is empty.
- Step 3: Peek the node of the stack.
- Step 4: If the node has unvisited child nodes, get the unvisited child node, mark it as traversed and push it on stack.
- Step 5: If the node does not have any unvisited child nodes, pop the node from the stack.
Based upon the above steps, the following Java code shows the implementation of the DFS algorithm:
// public void dfs() { //DFS uses Stack data structure Stack s=new Stack(); s.push(this.rootNode); rootNode.visited=true; printNode(rootNode); while(!s.isEmpty()) { Node n=(Node)s.peek(); Node child=getUnvisitedChildNode(n); if(child!=null) { child.visited=true; printNode(child); s.push(child); } else { s.pop(); } } //Clear visited property of nodes clearNodes(); } //
Breadth First Search (BFS)
This is a very different approach for traversing the graph nodes. The aim of BFS algorithm is to traverse the graph as close as possible to the root node. Queue is used in the implementation of the breadth first search. Let’s see how BFS traversal works with respect to the following graph:
If we do the breadth first traversal of the above graph and print the
visited node as the output, it will print the following output. “A B C D
E F”. The BFS visits the nodes level by level, so it will start with
level 0 which is the root node, and then it moves to the next levels
which are B, C and D, then the last levels which are E and F. Algorithmic Steps
- Step 1: Push the root node in the Queue.
- Step 2: Loop until the queue is empty.
- Step 3: Remove the node from the Queue.
- Step 4: If the removed node has unvisited child nodes, mark them as visited and insert the unvisited children in the queue.
Based upon the above steps, the following Java code shows the implementation of the BFS algorithm:
// public void bfs() { //BFS uses Queue data structure Queue q=new LinkedList(); q.add(this.rootNode); printNode(this.rootNode); rootNode.visited=true; while(!q.isEmpty()) { Node n=(Node)q.remove(); Node child=null; while((child=getUnvisitedChildNode(n))!=null) { child.visited=true; printNode(child); q.add(child); } } //Clear visited property of nodes clearNodes(); } //
NIL then
2n
n - 1