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 <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <functional>

using namespace std;

typedef long long ll;
typedef vector <int> vi;
typedef vector <vi> vvi;

const greater <int> gt;
vvi vex;
vi lis;
int A[2000000];
int H[2000000];

void dfs(int i, int p) {
    vi::iterator indx = lower_bound(lis.begin(), lis.end(), A[i]);
    bool lst = indx == lis.end();
    int ind = indx - lis.begin();
    
    if(lst) lis.push_back(A[i]);
    else swap(A[i], lis[ind]);
    
    H[i] = lis.size();
    
    for(int r : vex[i])
        if(r != p)
            dfs(r, i);
    
    if(lst) lis.pop_back();
    else swap(A[i], lis[ind]);
    
    return ;
}

int main() {
    cin.tie(NULL) -> sync_with_stdio(false);
    int n; cin >> n;
    vex.resize(n);
    for(int i = 0; i < n; ++i)
        cin >> A[i];
        
    int u, v;
    for(int i = 1; i < n; ++i) {
        cin >> u >> v;
        vex[--v].push_back(--u);
        vex[u].push_back(v);
    }
    
    dfs(0, 0);
    for(int i = 0; i < n; ++i)
        cout << H[i] << '\n';
    
    return 0;
}