minimum-score-of-a-path-between-two-cities

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Algorithms

You are given a positive integer <code>n</code> representing <code>n</code> cities numbered from <code>1</code> to <code>n</code>. You are also given a 2D array <code>roads</code> where <code>roads[i] = [ai, bi, distancei]</code> indicates that there is a bidirectional road between cities <code>ai</code> and <code>bi</code> with a distance equal to <code>distancei</code>. The cities graph is not necessarily connected.

The score of a path between two cities is defined as the minimum distance of a road in this path.

Return the minimum possible score of a path between cities <code>1</code> and <code>n</code>.

Note:

<ul>
	<li>A path is a sequence of roads between two cities.</li>
	<li>It is allowed for a path to contain the same road multiple times, and you can visit cities <code>1</code> and <code>n</code> multiple times along the path.</li>
	<li>The test cases are generated such that there is at least one path between <code>1</code> and <code>n</code>.</li>
</ul>

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph11.png" style="width: 190px; height: 231px;" />
<pre>
Input: n = 4, roads = [[1,2,9],[2,3,6],[2,4,5],[1,4,7]]
Output: 5
Explanation: The path from city 1 to 4 with the minimum score is: 1 -&gt; 2 -&gt; 4. The score of this path is min(9,5) = 5.
It can be shown that no other path has less score.
</pre>

Example 2:
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph22.png" style="width: 190px; height: 231px;" />
<pre>
Input: n = 4, roads = [[1,2,2],[1,3,4],[3,4,7]]
Output: 2
Explanation: The path from city 1 to 4 with the minimum score is: 1 -&gt; 2 -&gt; 1 -&gt; 3 -&gt; 4. The score of this path is min(2,2,4,7) = 2.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>2 &lt;= n &lt;= 105</code></li>
	<li><code>1 &lt;= roads.length &lt;= 105</code></li>
	<li><code>roads[i].length == 3</code></li>
	<li><code>1 &lt;= ai, bi &lt;= n</code></li>
	<li><code>ai != bi</code></li>
	<li><code>1 &lt;= distancei &lt;= 104</code></li>
	<li>There are no repeated edges.</li>
	<li>There is at least one path between <code>1</code> and <code>n</code>.</li>
</ul>

minScore

Minimum Score of a Path Between Two Cities

Checking Existence of Edge Length Limited Paths

checking-existence-of-edge-length-limited-paths

检查边长度限制的路径是否存在

Checking Existence of Edge Length Limited Paths II

checking-existence-of-edge-length-limited-paths-ii

检查边长度限制的路径是否存在 II

Depth-First Search

Breadth-First Search

Union Find

Graph

给你一个正整数&nbsp;<code>n</code>&nbsp;，表示总共有&nbsp;<code>n</code>&nbsp;个城市，城市从&nbsp;<code>1</code>&nbsp;到&nbsp;<code>n</code>&nbsp;编号。给你一个二维数组&nbsp;<code>roads</code>&nbsp;，其中&nbsp;<code>roads[i] = [ai, bi, distancei]</code>&nbsp;表示城市&nbsp;<code>ai</code> 和&nbsp;<code>bi</code>&nbsp;之间有一条 双向&nbsp;道路，道路距离为&nbsp;<code>distancei</code>&nbsp;。城市构成的图不一定是连通的。

两个城市之间一条路径的 分数&nbsp;定义为这条路径中道路的 最小&nbsp;距离。

城市 <code>1</code> 和城市 <code>n</code> 之间的所有路径的 最小 分数。

注意：

<ul>
	<li>一条路径指的是两个城市之间的道路序列。</li>
	<li>一条路径可以 多次 包含同一条道路，你也可以沿着路径多次到达城市 <code>1</code>&nbsp;和城市 <code>n</code>&nbsp;。</li>
	<li>测试数据保证城市 <code>1</code>&nbsp;和城市<code>n</code>&nbsp;之间 至少&nbsp;有一条路径。</li>
</ul>

&nbsp;

示例 1：

<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph11.png" style="width: 190px; height: 231px;" />

<pre>
输入：n = 4, roads = [[1,2,9],[2,3,6],[2,4,5],[1,4,7]]
输出：5
解释：城市 1 到城市 4 的路径中，分数最小的一条为：1 -&gt; 2 -&gt; 4 。这条路径的分数是 min(9,5) = 5 。
不存在分数更小的路径。
</pre>

示例 2：

<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph22.png" style="width: 190px; height: 231px;" />

<pre>
输入：n = 4, roads = [[1,2,2],[1,3,4],[3,4,7]]
输出：2
解释：城市 1 到城市 4 分数最小的路径是：1 -&gt; 2 -&gt; 1 -&gt; 3 -&gt; 4 。这条路径的分数是 min(2,2,4,7) = 2 。
</pre>

&nbsp;

提示：

<ul>
	<li><code>2 &lt;= n &lt;= 105</code></li>
	<li><code>1 &lt;= roads.length &lt;= 105</code></li>
	<li><code>roads[i].length == 3</code></li>
	<li><code>1 &lt;= ai, bi &lt;= n</code></li>
	<li><code>ai != bi</code></li>
	<li><code>1 &lt;= distancei &lt;= 104</code></li>
	<li>不会有重复的边。</li>
	<li>城市 <code>1</code>&nbsp;和城市 <code>n</code>&nbsp;之间至少有一条路径。</li>
</ul>