Orig Description
Depth First Search
Depth-first search (DFS) follows the strategy to search ”deeper” in the graph whenever possible. In DFS, edges are recursively explored out of the most recently discovered vertex $v$ that still has unexplored edges leaving it. When all of $v$'s edges have been explored, the search ”backtracks” to explore edges leaving the vertex from which $v$ was discovered.
This process continues until all the vertices that are reachable from the original source vertex have been discovered. If any undiscovered vertices remain, then one of them is selected as a new source and the search is repeated from that source.
DFS timestamps each vertex as follows:
$d[v]$ records when $v$ is first discovered.
$f[v]$ records when the search finishes examining $v$’s adjacency list.
Write a program which reads a directed graph $G = (V, E)$ and demonstrates DFS on the graph based on the following rules:
$G$ is given in an adjacency-list. Vertices are identified by IDs $1, 2,... n$ respectively.
IDs in the adjacency list are arranged in ascending order.
The program should report the discover time and the finish time for each vertex.
When there are several candidates to visit during DFS, the algorithm should select the vertex with the smallest ID.
The timestamp starts with 1.
Input
In the first line, an integer $n$ denoting the number of vertices of $G$ is given. In the next $n$ lines, adjacency lists of $u$ are given in the following format:
$u$ $k$ $v_1$ $v_2$ ... $v_k$
$u$ is ID of the vertex and $k$ denotes its degree. $v_i$ are IDs of vertices adjacent to $u$.
Output
For each vertex, print $id$, $d$ and $f$ separated by a space character in a line. $id$ is ID of the vertex, $d$ and $f$ is the discover time and the finish time respectively. Print in order of vertex IDs.
Constraints
$1 \leq n \leq 100$
Sample Input 1
4
1 1 2
2 1 4
3 0
4 1 3
Sample Output 1
1 1 8
2 2 7
3 4 5
4 3 6
Sample Input 2
6
1 2 2 3
2 2 3 4
3 1 5
4 1 6
5 1 6
6 0
Sample Output 2
1 1 12
2 2 11
3 3 8
4 9 10
5 4 7
6 5 6
This is example for Sample Input 2 (discover/finish)
Reference
Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.
Extracted Specification
An integer n (1 ≤ n ≤ 100), representing some quantity or size.
Next n lines contain:
An integer u (1 ≤ u ≤ n), representing some identifier.
An integer k (0 ≤ k ≤ n - 1), representing some degree.
Followed by k integers, each integer v (1 ≤ v ≤ n), representing some identifiers.
### Example Input:
```
4
1 1 2
2 1 4
3 0
4 1 3
```
### Function Signature:
Write a function f(inputs) that takes in the input.
```python
def f(inputs: List[Tuple[int, int, List[int]]]):
'''
inputs: a list of tuples, each containing an integer, an integer, and a list of integers
'''
```
Test Case Generator
Here's a Python function to generate test cases for the DFS problem described:
```python
import random
def generate_adjacency_list(n):
adjacency_list = []
for u in range(1, n + 1):
k = random.randint(0, n - 1) # Degree of vertex u
if k == 0:
adjacency_list.append((u, 0, []))
else:
neighbors = sorted(random.sample(range(1, n + 1), k))
adjacency_list.append((u, k, neighbors))
return adjacency_list
def dfs_tcgen():
n = random.randint(1, 100) # Number of vertices
adjacency_list = generate_adjacency_list(n)
input_str = f"{n}\n"
for u, k, neighbors in adjacency_list:
input_str += f"{u} {k} {' '.join(map(str, neighbors))}\n"
return input_str
# Example of generating a test case
print(dfs_tcgen())
```
### Explanation:
1. **`generate_adjacency_list(n)`**:
- This helper function generates an adjacency list for a graph with `n` vertices.
- Each vertex `u` has a degree `k` which is a random integer between `0` and `n-1`.
- If `k == 0`, it means vertex `u` has no neighbors.
- Otherwise, `k` neighbors are randomly selected from the range `[1, n]` and are sorted to ensure they are in ascending order.
2. **`dfs_tcgen()`**:
- Generates a random number `n` representing the number of vertices.
- Calls `generate_adjacency_list(n)` to generate the adjacency list for these vertices.
- Constructs the input string for the DFS problem based on the adjacency list.
- The input string is then returned, which can be used for testing.
This function should provide a wide variety of test cases to check the correctness and efficiency of a DFS implementation.
Extract Arguments
def extract_arguments(fh):
n = int(fh.readline().strip())
adjacency_list = []
for _ in range(n):
line = list(map(int, fh.readline().strip().split()))
u = line[0]
k = line[1]
v_list = line[2:]
adjacency_list.append((u, k, v_list))
return n, adjacency_list
# Sample usage
# if __name__ == "__main__":
# input_path = sys.argv[1]
# with open(input_path, 'r') as fh:
# n, adjacency_list = extract_arguments(fh)
# print(n, adjacency_list)