Author: [Author Name]<\/p>\n
Date: [Date]<\/p>\n<\/header>\n \n 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. \n For example, given N cities and M roads, and a starting city, we need to output the cities that are K distance away.\n <\/p>\n<\/section>\n \n Output the numbers of cities that are exactly K distance away in ascending order. If there are no such cities, output -1.\n <\/p>\n<\/section>\n Explanation: When starting from city 1, city 4 is the only city at distance 2.<\/p>\n<\/section>\n \n This problem can be solved using graph search algorithms. \n First, we will create a graph from the road information. \n We will use a queue for implementing BFS. \n 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.\n <\/p>\n<\/section>\n 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.<\/p>\n<\/section>\n \n The “Finding Cities at a Specific Distance” problem can be efficiently solved using graph search algorithms, particularly BFS.Problem Definition<\/h2>\n
\n Each city is represented by a number, and the distance information between two cities is provided in the form of bidirectional roads.
\n The goal is to find cities that are K distance away.\n <\/p>\nInput<\/h2>\n
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Output<\/h2>\n
Example<\/h2>\n
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\n Input:\n 4 4 2 1\n 1 2\n 1 3\n 2 3\n 2 4\n\n Output:\n 4\n <\/code>\n <\/pre>\nAlgorithm Approach<\/h2>\n
\n We will construct a graph that represents cities as nodes and roads as edges.
\n We will use the BFS (Breadth-First Search) algorithm to find cities that are K distance away from the starting city X.
\n BFS is useful for finding the shortest path and works efficiently even in large graphs, making it suitable for this problem.\n <\/p>\n<\/section>\nProblem Solving Process<\/h2>\n
1. Graph Creation<\/h3>\n
\n 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.\n <\/p>\n2. Implementing BFS<\/h3>\n
\n We add the starting city to the queue and keep track of visited cities.
\n For each city, we explore the cities that are K distance away.\n <\/p>\n3. Handling Output<\/h3>\n
Python Code<\/h2>\n
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\n from collections import deque\n\n def find_cities_with_distance(n, m, k, x, roads):\n # Create the graph\n graph = [[] for _ in range(n + 1)]\n for a, b in roads:\n graph[a].append(b)\n \n # Initialize BFS\n distance = [-1] * (n + 1)\n distance[x] = 0\n queue = deque([x])\n \n while queue:\n city = queue.popleft()\n for neighbor in graph[city]:\n if distance[neighbor] == -1: # Unvisited city\n distance[neighbor] = distance[city] + 1\n queue.append(neighbor)\n \n # Generate results\n result = [i for i in range(1, n + 1) if distance[i] == k]\n return result if result else [-1]\n \n # Test case\n n = 4\n m = 4\n k = 2\n x = 1\n roads = [(1, 2), (1, 3), (2, 3), (2, 4)]\n print(find_cities_with_distance(n, m, k, x, roads))\n <\/code>\n <\/pre>\nConclusion<\/h2>\n
\n Through this problem, we can enhance our understanding of the basic principles of BFS, how to construct graphs, and shortest path searches.
\n This will serve as a stepping stone from basics to more complex graph problems.\n <\/p>\n<\/section>\n