Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i.
Vertex i has an integer a_i written on it.
For every integer k from 1 through N, solve the following problem:

We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.

Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i \leq 10^9

1 \leq u_i , v_i \leq N

u_i \neq v_i

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Sample Input 1

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in C++ (GCC 9.2.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <vector>
#include <deque>
#include <queue>

struct node
{
    bool visited;
    std::uint32_t a;
    std::vector<std::size_t> edges;
    std::vector<std::uint32_t> lis;
};

int main()
{
    std::size_t N;
    std::cin >> N;

    std::vector<node> tree(N, node{false, 0, {}, {}});
    for(auto& n : tree)
    {
        std::cin >> n.a;
    }
    for(std::size_t i=0; i<N-1; ++i)
    {
        std::size_t u, v;
        std::cin >> u >> v;
        u -= 1; // zero-based index
        v -= 1;
        tree.at(u).edges.push_back(v);
        tree.at(v).edges.push_back(u);
    }

    std::queue<std::size_t> q;
    q.push(0);
    tree.at(0).visited = true;
    while(!q.empty())
    {
        auto& current = tree.at(q.front());
        q.pop();

        const auto it = std::lower_bound(current.lis.begin(), current.lis.end(), current.a);
        if(it == current.lis.end())
        {
            current.lis.push_back(current.a);
        }
        else
        {
            *it = current.a;
        }
        for(const auto child : current.edges)
        {
            if(not tree.at(child).visited)
            {
                tree.at(child).visited = true;
                tree.at(child).lis = current.lis;
                q.push(child);
            }
        }
    }
    std::size_t current_i = 1;
    for(const auto& n : tree)
    {
        std::cout << n.lis.size() << std::endl;
    }
    return 0;
}