1. DFS of Graph

The problem can be found at the following link: Question Link

Problem Description

Given a connected undirected graph represented by a 2D adjacency list adj[][], where adj[i] represents the list of vertices connected to vertex i. Perform a Depth First Search (DFS) traversal starting from vertex 0, visiting vertices from left to right as per the given adjacency list.

Return a list containing the DFS traversal of the graph.

Note: Traverse nodes in the same order as given in the adjacency list.

Note: Sorry for uploading late, my exam is going on.

Examples

Example 1:

Input:

adj = [[2, 3, 1],
       [0],
       [0, 4],
       [0],
       [2]]

Output:

[0, 2, 4, 3, 1]

Explanation:

The DFS traversal proceeds as follows:

• Start at vertex 0 → Output: [0]

• Visit 2 (first neighbor of 0) → Output: [0, 2]

• Visit 4 (first neighbor of 2) → Output: [0, 2, 4]

• Backtrack to 2, then backtrack to 0, visit 3 → Output: [0, 2, 4, 3]

• Backtrack to 0 and visit 1 → Final Output: [0, 2, 4, 3, 1]

Example 2:

Input:

adj = [[1, 2],
       [0, 2],
       [0, 1, 3, 4],
       [2],
       [2]]

Output:

[0, 1, 2, 3, 4]

Explanation:

The DFS traversal proceeds as follows:

• Start at vertex 0 → Output: [0]

• Visit 1 (first neighbor of 0) → Output: [0, 1]

• Visit 2 (first neighbor of 1) → Output: [0, 1, 2]

• Visit 3 (first neighbor of 2) → Output: [0, 1, 2, 3]

• Backtrack to 2 and visit 4 → Final Output: [0, 1, 2, 3, 4]

Constraints:

• $1 \leq$ adj.size() $\leq 10^4$

• $1 \leq$ adj[i][j] $\leq 10^4$

My Approach

Recursive DFS (Using Lambda Function)

Algorithm Steps:

1. Maintain a visited array to track visited nodes.

2. Implement DFS using recursion and a lambda function.

3. Start DFS traversal from node 0 and recursively visit neighbors in the given order.

4. If a node is unvisited, continue DFS.

5. Store the DFS traversal sequence in a list.

Time and Auxiliary Space Complexity

• Expected Time Complexity: O(V + E), since each vertex and edge is visited once.

• Expected Auxiliary Space Complexity: O(V), as we store the visited array and recursive function calls.

Code (C++)

class Solution {
  public:
    vector<int> dfs(vector<vector<int>>& adj) {
        int V = adj.size();
        vector<int> res, vis(V, 0);
        function<void(int)> traverse = [&](int v) {
            vis[v] = 1;
            res.push_back(v);
            for (int u : adj[v])
                if (!vis[u]) traverse(u);
        };
        for (int i = 0; i < V; i++)
            if (!vis[i]) traverse(i);
        return res;
    }
};

⚡ Alternative Approaches

📊 2️⃣ Iterative DFS Approach (Using Stack)

Algorithm Steps:

1. Use a stack to perform Depth-First Search iteratively.

2. For each unvisited vertex, push it to the stack and mark it visited.

3. Process the top element and push its unvisited neighbors in reverse order to maintain DFS order.

4. Repeat the process until the stack is empty.

class Solution {
public:
    vector<int> dfs(vector<vector<int>>& adj) {
        int V = adj.size();
        vector<int> res;
        vector<bool> vis(V, false);
        for (int i = 0; i < V; i++) {
            if (!vis[i]) {
                stack<int> st;
                st.push(i);
                while (!st.empty()) {
                    int v = st.top();
                    st.pop();
                    if (!vis[v]) {
                        vis[v] = true;
                        res.push_back(v);
                        for (int j = adj[v].size() - 1; j >= 0; j--) {
                            int u = adj[v][j];
                            if (!vis[u])
                                st.push(u);
                        }
                    }
                }
            }
        }
        return res;
    }
};

📝 Complexity Analysis:

• ✅ Time Complexity: O(V + E) - Each vertex and edge are processed once.

• ✅ Space Complexity: O(V) - Due to the stack used in the iterative approach.

✅ Why This Approach?

It eliminates the risk of stack overflow due to recursion, making it suitable for graphs with a large depth.

🔄 3️⃣ Recursive DFS without Lambda (Traditional Approach)

Algorithm Steps:

1. Use a helper function for recursion.

2. Start from an unvisited vertex, mark it as visited, and add to the result.

3. Recursively call the function for each unvisited neighbor.

class Solution {
public:
    void dfsUtil(int v, vector<vector<int>>& adj, vector<int>& res, vector<bool>& vis) {
        vis[v] = true;
        res.push_back(v);
        for (int u : adj[v]) {
            if (!vis[u])
                dfsUtil(u, adj, res, vis);
        }
    }

    vector<int> dfs(vector<vector<int>>& adj) {
        int V = adj.size();
        vector<int> res;
        vector<bool> vis(V, false);
        for (int i = 0; i < V; i++) {
            if (!vis[i])
                dfsUtil(i, adj, res, vis);
        }
        return res;
    }
};

📝 Complexity Analysis:

• ✅ Time Complexity: O(V + E) - Each vertex and edge are processed once.

• ✅ Space Complexity: O(V) - Due to the recursive call stack.

✅ Why This Approach?

The traditional recursive approach is simple and intuitive, but it risks stack overflow for deep recursion.

🆚 Comparison of Approaches

Approach	⏱️ Time Complexity	🗂️ Space Complexity	✅ Pros	⚠️ Cons
Recursive DFS (Lambda)	🟢 O(V + E)	🟡 O(V)	Compact code with lambda functions	Potential stack overflow for deep recursion
Iterative DFS (Stack)	🟢 O(V + E)	🟡 O(V)	No recursion issues, avoids stack overflow	Slightly more complex than recursive
Recursive DFS (Traditional)	🟢 O(V + E)	🟡 O(V)	Simple and intuitive recursive approach	Risk of stack overflow for large graphs

✅ Best Choice?

• Use Recursive DFS with Lambda for compact and readable code when graph depth is manageable.

• Use Iterative DFS to avoid recursion issues when the graph has a large depth.

• The Traditional Recursive DFS is good for simple cases but should be avoided for deep recursion.

Code (Java)

class Solution {
    public ArrayList<Integer> dfs(ArrayList<ArrayList<Integer>> adj) {
        ArrayList<Integer> r = new ArrayList<>();
        boolean[] v = new boolean[adj.size()];
        for (int i = 0; i < adj.size(); i++) if (!v[i]) go(i, adj, v, r);
        return r;
    }
    void go(int i, ArrayList<ArrayList<Integer>> a, boolean[] v, ArrayList<Integer> r) {
        v[i] = true;
        r.add(i);
        for (int j : a.get(i)) if (!v[j]) go(j, a, v, r);
    }
}

Code (Python)

class Solution:
    def dfs(self, adj):
        r, v = [], [False] * len(adj)
        def go(i):
            v[i] = True
            r.append(i)
            for j in adj[i]:
                if not v[j]:
                    go(j)
        for i in range(len(adj)):
            if not v[i]:
                go(i)
        return r

Contribution and Support:

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