Understanding the Problem
We are given n
cities, numbered from 0
to n - 1
, and a list of edges that represent bidirectional roads between cities. Each edge contains a source city, a destination city, and the distance between them. We are also given a distanceThreshold
, which defines the maximum distance we can travel from any city.
Our goal is to find the city with the fewest number of reachable cities within this threshold. If there are multiple cities with the same minimum count, we return the city with the largest index.
Step-by-Step Solution with Example
Step 1: Understand the Input
Suppose we have n = 4
cities, the following edges:
[
[0, 1, 3],
[1, 2, 1],
[2, 3, 4],
[0, 3, 7]
]
And a
distanceThreshold = 4
.
Step 2: Represent the Graph
We create an adjacency matrix to represent the shortest distance between all pairs of cities. Initially, set the distance between each pair as Infinity
, and set dist[i][i] = 0
. Then fill in the given edges.
Step 3: Use Floyd-Warshall to Compute All-Pairs Shortest Paths
Floyd-Warshall is ideal when n
is small (like ≤ 100). It updates the shortest distance between every pair of cities using a dynamic programming approach. For each intermediate city k
, update:
if (dist[i][j] > dist[i][k] + dist[k][j]) {
dist[i][j] = dist[i][k] + dist[k][j];
}
Step 4: Count Reachable Cities
For each city, count how many other cities are reachable such that dist[i][j] ≤ distanceThreshold
. Do not count the city itself.
Step 5: Track the Best Candidate
As we compute the reachable count for each city, track the city with the fewest reachable cities. If there's a tie, choose the city with the higher index.
Step 6: Final Output
From our example, city 3
can reach only one city within distance 4, so it will be the answer.
Edge Cases
- No reachable neighbors: A city completely disconnected from others within the threshold will have a count of 0.
- Multiple cities with same count: We must return the one with the highest index.
- Disconnected components: Floyd-Warshall handles this by keeping unreachable distances as Infinity.
- Zero threshold: Only self-reachability is allowed, so all cities will have 0 reachable neighbors.
Finally
This problem is a great use case for the Floyd-Warshall algorithm, which efficiently handles all-pairs shortest paths for small graphs. By applying it, we can easily evaluate reachability for every city and determine the one that meets the problem's criteria. Always keep in mind the edge conditions, such as tie-breaking and unreachable nodes, to ensure a complete and accurate solution.
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