Question

Discuss graph representation, Breadth-first search and Depth-first search.  Use examples to highlight pros and cons.  

Answer 1

Graph Representation : You can represent a graph in many ways. The two most common ways of representing a graph is as follows:

Adjacency matrix

An adjacency matrix is a VxV binary matrix A. Element Ai,j is 1 if there is an edge from vertex i to vertex j else  Ai,jis 0.

Note: A binary matrix is a matrix in which the cells can have only one of two possible values - either a 0 or 1.

The adjacency matrix can also be modified for the weighted graph in which instead of storing 0 or 1 in Ai,j, the weight or cost of the edge will be stored.

In an undirected graph, if  Ai,j = 1, then Aj,i = 1. In a directed graph, if  Ai,j = 1, then Aj,i may or may not be 1.

Adjacency matrix provides constant time access (O(1) ) to determine if there is an edge between two nodes. Space complexity of the adjacency matrix is O(V2).

Adjacency list

The other way to represent a graph is by using an adjacency list. An adjacency list is an array A of separate lists. Each element of the array A_i is a list, which contains all the vertices that are adjacent to vertex i.

For a weighted graph, the weight or cost of the edge is stored along with the vertex in the list using pairs. In an undirected graph, if vertex j is in list  Ai then vertex i will be in list  Aj.

The space complexity of adjacency list is O(V + E) because in an adjacency list information is stored only for those edges that actually exist in the graph. In a lot of cases, where a matrix is sparse using an adjacency matrix may not be very useful. This is because using an adjacency matrix will take up a lot of space where most of the elements will be 0, anyway. In such cases, using an adjacency list is better.

A sparse matrix is a matrix in which most of the elements are zero, whereas a dense matrix is a matrix in which most of the elements are non-zero.

If a matrix is dense it's optimal to use adjacency matrix whereas if the graph is sparse adjacency list is preferred over adjacency matrix

Breadth first search - BFS is a traversing algorithm where you should start traversing from a selected node (source or starting node) and traverse the graph layerwise thus exploring the neighbour nodes (nodes which are directly connected to source node). You must then move towards the next-level neighbour nodes.

As the name BFS suggests, you are required to traverse the graph breadthwise as follows:

1. First move horizontally and visit all the nodes of the current layer
2. Move to the next layer

Depth First search - The DFS algorithm is a recursive algorithm that uses the idea of backtracking. It involves exhaustive searches of all the nodes by going ahead, if possible, else by backtracking.

Here, the word backtrack means that when you are moving forward and there are no more nodes along the current path, you move backwards on the same path to find nodes to traverse. All the nodes will be visited on the current path till all the unvisited nodes have been traversed after which the next path will be selected.

This recursive nature of DFS can be implemented using stacks. The basic idea is as follows:
Pick a starting node and push all its adjacent nodes into a stack.
Pop a node from stack to select the next node to visit and push all its adjacent nodes into a stack.
Repeat this process until the stack is empty. However, ensure that the nodes that are visited are marked. This will prevent you from visiting the same node more than once. If you do not mark the nodes that are visited and you visit the same node more than once, you may end up in an infinite loop.

1 a) BFS(Breadth First Search) uses Queue data structure for finding the shortest path.

b) DFS(Depth First Search) uses Stack data structure.

2. a) BFS can be used to find single source shortest path in an unweighted graph, because in BFS, we reach a vertex with minimum number of edges from a source vertex.

b) In DFS, we might traverse through more edges to reach a destination vertex from a source

3. a) BFS is more suitable for searching verteces which are closer to the given source

b) DFS is more suitable when there are solutions away from source

4. a) BFS considers all neighbors first and therefore not suitable for decision making trees used in games or puzzles.

b) DFS is more suitable for game or puzzle problems. We make a decision, then explore all paths through this decision. And if this decision leads to win situation, we stop.