Depth-First-Search(DFS) Explained With Visualization

DFS Overview

The Depth First Search(DFS) is the most fundamental search algorithm used to explore the nodes and edges of a graph. It runs with time complexity of O(V+E), where V is the number of nodes, and E is the number of edges in a graph.

DFS is often used as a building block in other algorithms; it can be used to:

A naive solution for any searching problem.
Finding connected components or strongly connected components.
Topological sorting.
Finding the bridges of a graph.
Detect cycles in a graph.

DFS Visualization on Maze

The source is at the position of left-up, and the target is the position of right-bottom.

As we can see, DFS explores as far as possible along each branch before backtracking:

file:img/dfs.gif

The maze is generated by disjoint set.

The recursive implementation

#include <list>
#include <vector>
#include <iostream>
using namespace std;

class Graph {
  int V;
  list<int> *adj;
  bool DFS_rec(int v, int t, vector<bool>& visited);

public:
  Graph(int V);
  void addEdge(int v, int w);
  bool DFS(int v, int t);
};

Graph::Graph(int V) {
  this->V = V;
  adj = new list<int>[V];
}

void Graph::addEdge(int v, int w) {
  adj[v].push_back(w);
}

bool Graph::DFS_rec(int v, int t, vector<bool>& visited) {
  visited[v] = true;
  cout << v << " ";
  if(v == t) return true; // Find a path
  
  for (list<int>::iterator it; it = adj[v].begin(); it != adj[v].end(); ++it) {
    if (!visited[*it] && DFS_rec(*it, t, visited))
      return true;
  }
  return false;
}

bool Graph::DFS(int v, int t) {
  vector<bool> visited(V, false);
  return DFS_rec(v, t, visited);
}

int main() {
  Graph g(4);
  g.addEdge(0, 1);
  g.addEdge(0, 2);
  g.addEdge(1, 2);
  g.addEdge(2, 0);
  g.addEdge(2, 3);
  g.addEdge(3, 3);

  cout << "Following is Depth First Traversal (0 -> 3): \n";
  if(g.DFS(0, 3)) cout << "\nFind a Path!" << std::endl;
  else cout << "\nNo Path!" << std::endl;
  return 0;
}

The iterative implementation

A non-recursive implementation of DFS needs the data-structure of stack.

The worst-case space complexity is O(E).

bool Graph::DFS(int v, int t) {
  vector<bool> marked(V, false);

  stack<int> S;
  S.push(v);
  marked[v] = true;
  while(!S.empty()) {
    int n = S.top(); S.pop();
    cout << n << " ";
    if(n == t) //Find a path to target
      return true;
    for(list<int>::iterator it = adj[n].begin(); it != adj[n].end(); ++it) {
      if(!marked[*it]) {
        marked[*it] = true;
        S.push(*it);
      }
    }
  }
  return false;
}