**Algorithm 2.3: Graph Depth-First Search with a Stack**This algorithm demonstrates how to perform a depth-first search (DFS) on a graph using a stack data structure.**Procedure**: `StackDFS(G, node) → visited`- **Input**: - `G = (V, E)`: A graph where `V` is the set of vertices and `E` is the set of edges. - `node`: The starting vertex in the graph `G`.- **Output**: - `visited`: An array of size `|V|` such that `visited[i]` is TRUE if the node `i` has been visited, and FALSE otherwise.**Steps**:1. `S ← CreateStack()`: Initialize an empty stack `S`.2. `visited ← CreateArray(|V|)`: Create an array `visited` with a length equal to the number of vertices `|V|`.3. `for i ← 0 to |V| do`: Loop through all vertices.4. `visited[i] ← FALSE`: Mark all vertices as not visited.5. `Push(S, node)`: Push the starting node onto the stack.6. `while not IsStackEmpty(S) do`: Repeat while the stack `S` is not empty.7. `c ← Pop(S)`: Pop the top element from the stack and assign it to `c`.8. `visited[c] ← TRUE`: Mark the vertex `c` as visited.9. `foreach v in AdjacencyList(G, c) do`: For each vertex `v` adjacent to `c` in the graph.10. `if not visited[v] then`: If the vertex `v` has not been visited.11. `Push(S, v)`: Push `v` onto the stack.12. `return visited`: Return the array indicating the visited status of each vertex.This algorithm uses a stack to backtrack and explore all reachable vertices from the starting node, marking them as visited in the process. **REQUIREMENTS:**1. Using Python, you will write a program called `dfs-stack.py` that implements Algorithm 2.3 (p. 49): Graph depth-first search (DFS) with a stack.2. You will not use an adjacency list, as indicated in Algorithm 2.3. Instead, you will use an adjacency matrix (i.e., a two-dimensional array, or, in Python, a list of lists).3. You may use ANY list method you wish (e.g., append, pop, etc.).**IMPLEMENTATION DETAILS:**1. Based upon the REQUIREMENTS above, along with the IMPLEMENTATION DETAILS (i.e., this section), you MUST first develop an algorithmic solution using pseudocode. This includes both your logic (in pseudocode) and the logic presented in the pseudocode indicated in Algorithm 2.3.2. Be sure to include your name, along with the Certificate of Authenticity, as comments at the very beginning of your Python code. Also, if you collaborated with others, be sure to state their names as well.3. Your program should begin by prompting the user for the number of vertices, \( V \), in the graph, \( G \).4. Your program will represent the graph \( G \) using an adjacency matrix, which is a square matrix with a row and a column for each vertex. Thus, your program will need to create a matrix \( M \) that consists of a \( V \times V \) two-dimensional array -- in Python, a list of lists. (I recommend that your program initialize each element of the matrix equal to zero.)5. Next, your program should prompt the user to indicate which elements in the matrix should be assigned the value of 1 (i.e., information about vertices). Recall that each element in the matrix is the intersection of a row and a column.6. The result of steps 3, 4, and 5 should be an adjacency matrix representation of a graph, \( G \).7. At this point, your program should print the newly-created adjacency matrix on the screen.8. Next, your program should prompt the user to specify node -- i.e., the starting vertex in \( G \).9. From here, you then proceed with the implementation of Algorithm 2.3, with the following enhancements: - Be sure to use the newly-created adjacency matrix, instead of an adjacency list.

The solution is given in the next step

Algorithm 2.3: Graph depth-first search with a stack. StackDFS (G, node) →visited Input: G = (V, E), a graph 1 2 3 4 5 6 7 8 9 10 11 12 node, the starting vertex in G Output: visited, an array of size |V| such that visited[i] is TRUE if we have visited node i, FALSE otherwise S CreateStack() visited CreateArray (IV) for i0 to V| do visited[i] ← FALSE Push (S, node) while not IsStackEmpty(S) do c← Pop (s) visited[c] ← TRUE foreach v in AdjacencyList (G, c) do if not visited[v] then Push (S, v) return visited

Algorithm 2.3: Graph depth-first search with a stack. StackDFS (G, node) →visited Input: G = (V, E), a graph 1 2 3 4 5 6 7 8 9 10 11 12 node, the starting vertex in G Output: visited, an array of size |V| such that visited[i] is TRUE if we have visited node i, FALSE otherwise S CreateStack() visited CreateArray (IV) for i0 to V| do visited[i] ← FALSE Push (S, node) while not IsStackEmpty(S) do c← Pop (s) visited[c] ← TRUE foreach v in AdjacencyList (G, c) do if not visited[v] then Push (S, v) return visited

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

**Algorithm 2.3: Graph Depth-First Search with a Stack**

This algorithm demonstrates how to perform a depth-first search (DFS) on a graph using a stack data structure.

**Procedure**: `StackDFS(G, node) → visited`

- **Input**:
- `G = (V, E)`: A graph where `V` is the set of vertices and `E` is the set of edges.
- `node`: The starting vertex in the graph `G`.

- **Output**:
- `visited`: An array of size `|V|` such that `visited[i]` is TRUE if the node `i` has been visited, and FALSE otherwise.

**Steps**:

1. `S ← CreateStack()`: Initialize an empty stack `S`.
2. `visited ← CreateArray(|V|)`: Create an array `visited` with a length equal to the number of vertices `|V|`.
3. `for i ← 0 to |V| do`: Loop through all vertices.
4. `visited[i] ← FALSE`: Mark all vertices as not visited.
5. `Push(S, node)`: Push the starting node onto the stack.
6. `while not IsStackEmpty(S) do`: Repeat while the stack `S` is not empty.
7. `c ← Pop(S)`: Pop the top element from the stack and assign it to `c`.
8. `visited[c] ← TRUE`: Mark the vertex `c` as visited.
9. `foreach v in AdjacencyList(G, c) do`: For each vertex `v` adjacent to `c` in the graph.
10. `if not visited[v] then`: If the vertex `v` has not been visited.
11. `Push(S, v)`: Push `v` onto the stack.
12. `return visited`: Return the array indicating the visited status of each vertex.

This algorithm uses a stack to backtrack and explore all reachable vertices from the starting node, marking them as visited in the process.

**REQUIREMENTS:**

1. Using Python, you will write a program called `dfs-stack.py` that implements Algorithm 2.3 (p. 49): Graph depth-first search (DFS) with a stack.
2. You will not use an adjacency list, as indicated in Algorithm 2.3. Instead, you will use an adjacency matrix (i.e., a two-dimensional array, or, in Python, a list of lists).
3. You may use ANY list method you wish (e.g., append, pop, etc.).

**IMPLEMENTATION DETAILS:**

1. Based upon the REQUIREMENTS above, along with the IMPLEMENTATION DETAILS (i.e., this section), you MUST first develop an algorithmic solution using pseudocode. This includes both your logic (in pseudocode) and the logic presented in the pseudocode indicated in Algorithm 2.3.

2. Be sure to include your name, along with the Certificate of Authenticity, as comments at the very beginning of your Python code. Also, if you collaborated with others, be sure to state their names as well.

3. Your program should begin by prompting the user for the number of vertices, \( V \), in the graph, \( G \).

4. Your program will represent the graph \( G \) using an adjacency matrix, which is a square matrix with a row and a column for each vertex. Thus, your program will need to create a matrix \( M \) that consists of a \( V \times V \) two-dimensional array -- in Python, a list of lists. (I recommend that your program initialize each element of the matrix equal to zero.)

5. Next, your program should prompt the user to indicate which elements in the matrix should be assigned the value of 1 (i.e., information about vertices). Recall that each element in the matrix is the intersection of a row and a column.

6. The result of steps 3, 4, and 5 should be an adjacency matrix representation of a graph, \( G \).

7. At this point, your program should print the newly-created adjacency matrix on the screen.

8. Next, your program should prompt the user to specify node -- i.e., the starting vertex in \( G \).

9. From here, you then proceed with the implementation of Algorithm 2.3, with the following enhancements:

- Be sure to use the newly-created adjacency matrix, instead of an adjacency list.

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