3203. Find Minimum Diameter After Merging Two Trees

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Tree Depth-First Search Breadth-First Search Graph Theory

There exist two undirected trees with n and m nodes, numbered from 0 to n - 1 and from 0 to m - 1, respectively. You are given two 2D integer arrays edges1 and edges2 of lengths n - 1 and m - 1, respectively, where edges1[i] = [a_i, b_i] indicates that there is an edge between nodes a_i and b_i in the first tree and edges2[i] = [u_i, v_i] indicates that there is an edge between nodes u_i and v_i in the second tree.

You must connect one node from the first tree with another node from the second tree with an edge.

Return the minimum possible diameter of the resulting tree.

The diameter of a tree is the length of the longest path between any two nodes in the tree.

Example 1:

Input: edges1 = [[0,1],[0,2],[0,3]], edges2 = [[0,1]]

Output: 3

Explanation:

We can obtain a tree of diameter 3 by connecting node 0 from the first tree with any node from the second tree.

Example 2:

Input: edges1 = [[0,1],[0,2],[0,3],[2,4],[2,5],[3,6],[2,7]], edges2 = [[0,1],[0,2],[0,3],[2,4],[2,5],[3,6],[2,7]]

Output: 5

Explanation:

We can obtain a tree of diameter 5 by connecting node 0 from the first tree with node 0 from the second tree.

Constraints:

1 <= n, m <= 10⁵
edges1.length == n - 1
edges2.length == m - 1
edges1[i].length == edges2[i].length == 2
edges1[i] = [a_i, b_i]
0 <= a_i, b_i < n
edges2[i] = [u_i, v_i]
0 <= u_i, v_i < m
The input is generated such that edges1 and edges2 represent valid trees.

Hints

Hint 1

Suppose that we connected node a in tree1 with node b in tree2. The diameter length of the resulting tree will be the largest of the following 3 values:

The diameter of tree 1.
The diameter of tree 2.
The length of the longest path that starts at node a and that is completely within Tree 1 + The length of the longest path that starts at node b and that is completely within Tree 2 + 1.

The added one in the third value is due to the additional edge that we have added between trees 1 and 2.

Hint 2

Values 1 and 2 are constant regardless of our choice of a and b. Therefore, we need to pick a and b in such a way that minimizes value 3.

Hint 3

If we pick a and b optimally, they will be in the diameters of Tree 1 and Tree 2, respectively. Exactly which nodes of the diameter should we pick?

Hint 4

a is the center of the diameter of tree 1, and b is the center of the diameter of tree 2.

3203. Find Minimum Diameter After Merging Two Trees

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