Python Coding Test Course, Finding Cities at a Specific Distance

Author: [Author Name]

Date: [Date]

Problem Definition

This is a problem of finding a list of cities that can be reached exactly at a specific distance K based on the given city and road information.
Each city is represented by a number, and the distance information between two cities is provided in the form of bidirectional roads.
The goal is to find cities that are K distance away.

For example, given N cities and M roads, and a starting city, we need to output the cities that are K distance away.

Input

  • N: Number of cities (1 ≤ N ≤ 300,000)
  • M: Number of roads (1 ≤ M ≤ 1,000,000)
  • K: Distance information (0 ≤ K ≤ 300,000)
  • X: Starting city number (1 ≤ X ≤ N)
  • Road information: A, B (road from A to B)

Output

Output the numbers of cities that are exactly K distance away in ascending order. If there are no such cities, output -1.

Example

                
                Input:
                4 4 2 1
                1 2
                1 3
                2 3
                2 4

                Output:
                4

Explanation: When starting from city 1, city 4 is the only city at distance 2.

Algorithm Approach

This problem can be solved using graph search algorithms.
We will construct a graph that represents cities as nodes and roads as edges.
We will use the BFS (Breadth-First Search) algorithm to find cities that are K distance away from the starting city X.
BFS is useful for finding the shortest path and works efficiently even in large graphs, making it suitable for this problem.

Problem Solving Process

1. Graph Creation

First, we will create a graph from the road information.
We will represent the graph using a list and use a dictionary where each city is a key and the connected cities are the values.

2. Implementing BFS

We will use a queue for implementing BFS.
We add the starting city to the queue and keep track of visited cities.
For each city, we explore the cities that are K distance away.

3. Handling Output

After the BFS search, we will either sort and output the city numbers that match the K distance or output -1 if there are no cities.

Python Code

                
                from collections import deque

                def find_cities_with_distance(n, m, k, x, roads):
                    # Create the graph
                    graph = [[] for _ in range(n + 1)]
                    for a, b in roads:
                        graph[a].append(b)
                    
                    # Initialize BFS
                    distance = [-1] * (n + 1)
                    distance[x] = 0
                    queue = deque([x])
                    
                    while queue:
                        city = queue.popleft()
                        for neighbor in graph[city]:
                            if distance[neighbor] == -1:  # Unvisited city
                                distance[neighbor] = distance[city] + 1
                                queue.append(neighbor)
                    
                    # Generate results
                    result = [i for i in range(1, n + 1) if distance[i] == k]
                    return result if result else [-1]
                
                # Test case
                n = 4
                m = 4
                k = 2
                x = 1
                roads = [(1, 2), (1, 3), (2, 3), (2, 4)]
                print(find_cities_with_distance(n, m, k, x, roads))

This code uses BFS to find the cities that can be reached at a specific distance K. The graph is represented using an adjacency list and the connectivity between nodes is defined based on the edge information.

Conclusion

The “Finding Cities at a Specific Distance” problem can be efficiently solved using graph search algorithms, particularly BFS.
Through this problem, we can enhance our understanding of the basic principles of BFS, how to construct graphs, and shortest path searches.
This will serve as a stepping stone from basics to more complex graph problems.

Thank you!