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: Accepted)

#include <bits/stdc++.h>
#define MAX 200000
using namespace std;

int n;
vector<int> v[MAX];
int num[MAX];
int res[MAX];

void dfs( int cur, int par, set<int>& st ){
    auto f = st.upper_bound(num[cur] - 1);
    int rev = -1, ins = 0;
    if(f == st.end() ){
        st.insert(num[cur]), ins = 1;
    }
    else{
        rev = (*f);
        st.erase(f);
        st.insert(num[cur]);
        ins = 1;
    }
    res[cur] = st.size();
    for( int u : v[cur] ){
        if(u != par )
            dfs(u, cur, st);
    }
    if(ins)
        st.erase(num[cur]);
    if(rev != -1)
        st.insert(rev);
}

int main()
{
    cin >> n;
    for( int i = 0; i < n; i++ ){
        cin >> num[i];
    }
    for( int i = 0; i < n - 1; i++ ){
        int a,b;
        cin >> a >> b;
        a--, b--;
        v[a].push_back(b);
        v[b].push_back(a);
    }
    set<int> st;
    dfs(0, -1, st);
    for( int i = 0; i < n; i++ )
        cout << res[i] << endl;
    return 0;
}