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The Floyd-Warshall algorithm is a classical dynamic programming algorithm to solve the all-pairs shortest path problem. It works well for graphs that have both weighted and unweighted edges. The algorithm iteratively updates the distance matrix for each possible intermediate node and provides the shortest distances between any two nodes.
The algorithm has a time complexity of O(n^3) and a space complexity of O(n^2), making it suitable for graphs with up to 100 nodes as given in the problem constraints.
Time Complexity: O(n^3), where n is the number of cities.
Space Complexity: O(n^2) for storing the distance matrix.
1#include <stdio.h>
2#include <limits.h>
3
4#define MAX 100
5
6void findCity(int n
This solution implements the Floyd-Warshall algorithm to compute shortest paths between all city pairs. It then counts the number of neighboring cities within the given distance threshold for each city and identifies the city with the smallest number of reachable neighbors.
Dijkstra's Algorithm can be employed to find the shortest path from one city to another in a graph with non-negative weights. We can utilize this algorithm to determine the shortest distance from every city to all other cities consecutively, assessing the count of reachable cities within the threshold afterward.
Time Complexity: O(n^2) for each node multiplied by n trips resulting in O(n^3).
Space Complexity: O(n^2) for graph storage.
Here, the Java solution applies Dijkstra's algorithm creatively using a priority queue facilitating minimized computations. It results in determining city reachabilities under a specified distance threshold.