maximum-total-importance-of-roads

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Algorithms

You are given an integer <code>n</code> denoting the number of cities in a country. The cities are numbered from <code>0</code> to <code>n - 1</code>.

You are also given a 2D integer array <code>roads</code> where <code>roads[i] = [ai, bi]</code> denotes that there exists a bidirectional road connecting cities <code>ai</code> and <code>bi</code>.

You need to assign each city with an integer value from <code>1</code> to <code>n</code>, where each value can only be used once. The importance of a road is then defined as the sum of the values of the two cities it connects.

Return the maximum total importance of all roads possible after assigning the values optimally.

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2022/04/07/ex1drawio.png" style="width: 290px; height: 215px;" />
<pre>
Input: n = 5, roads = [[0,1],[1,2],[2,3],[0,2],[1,3],[2,4]]
Output: 43
Explanation: The figure above shows the country and the assigned values of [2,4,5,3,1].
- The road (0,1) has an importance of 2 + 4 = 6.
- The road (1,2) has an importance of 4 + 5 = 9.
- The road (2,3) has an importance of 5 + 3 = 8.
- The road (0,2) has an importance of 2 + 5 = 7.
- The road (1,3) has an importance of 4 + 3 = 7.
- The road (2,4) has an importance of 5 + 1 = 6.
The total importance of all roads is 6 + 9 + 8 + 7 + 7 + 6 = 43.
It can be shown that we cannot obtain a greater total importance than 43.
</pre>

Example 2:
<img alt="" src="https://assets.leetcode.com/uploads/2022/04/07/ex2drawio.png" style="width: 281px; height: 151px;" />
<pre>
Input: n = 5, roads = [[0,3],[2,4],[1,3]]
Output: 20
Explanation: The figure above shows the country and the assigned values of [4,3,2,5,1].
- The road (0,3) has an importance of 4 + 5 = 9.
- The road (2,4) has an importance of 2 + 1 = 3.
- The road (1,3) has an importance of 3 + 5 = 8.
The total importance of all roads is 9 + 3 + 8 = 20.
It can be shown that we cannot obtain a greater total importance than 20.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>2 &lt;= n &lt;= 5 * 104</code></li>
	<li><code>1 &lt;= roads.length &lt;= 5 * 104</code></li>
	<li><code>roads[i].length == 2</code></li>
	<li><code>0 &lt;= ai, bi &lt;= n - 1</code></li>
	<li><code>ai != bi</code></li>
	<li>There are no duplicate roads.</li>
</ul>