Difficulty : Medium
Pre-requisites : Lowest Common Ancestor, Trees
Problem : Given a tree, you have to answer the question of the format “r u v” which means which is the LCA of u and v if the root of the tree is at r.
At first, let’s consider some partial solutions.
How to get 20 points
For this sub-task you an find the LCA in any way you want as long as the complexity is not slower than O(N). For example, by DFS from the root, you can number the vertices so that given two arbitrary vertices, you can check whether they are ancestor and descendant(for this, you can store T_in and T_out for each node. T_in is the time when the DFS for that node was begun. T_out is the time when the DFS was over).
How to get 60 points
Here there are not more than 10 different roots, but the queries are quite high, so you should know the fast way to find LCA. More specifically O(Nlog(N)) is enough. Notice that there will be no more than 10 different roots so your complexity will be O(10 × Nlog(N)).
How to get 100 points
There are two interesting observations that you can make:
Given the query “r u v” what can be the answer? The possible answers are r, u, v, LCA(r, u), LCA(r, v), LCA(u, v) where LCA(x, y) is LCA of x and y when the tree is rooted at 1.
The LCA of u and v when the root is at r is the vertex x such that the sum of shortest path from x to u, x to v and x to r is smallest.
With this two observations you need to implement two function: finding LCA and distance of the two vertices in the tree. Proof for these two observation is not hard but too long to be mentioned here. It is left as an exercise for you.