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: Wrong Answer)

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

#define SZ(v) ((int)(v).size())
using ll = long long;

const int MAXN = 2e5;

int ans[MAXN];
vector<int> G[MAXN];
vector<int> best[MAXN];
int val[MAXN];

void dfs(int u, int p)
{
	int lo(0), up(MAXN-1);
	while (lo < up)
	{
		int mid = (lo+up+1)/2;
		if (SZ(best[mid]) == 0 or best[mid].back() >= val[u])
			up = mid-1;
		else
			lo = mid;
	}
	if (SZ(best[lo]) and best[lo].back() < val[u])
		++lo;
	ans[u] = lo;
	best[lo].push_back(val[u]);
	for (auto v : G[u])
		if (v != p)
			dfs(v, u);
	best[ans[u]].pop_back();
}

int main(void)
{
	ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0);

	int N;
	cin >> N;
	for (int i(0); i < N; ++i)
		cin >> val[i];
	for (int i(1); i < N; ++i)
	{
		int _u, _v;
		cin >> _u >> _v;
		--_u, --_v;
		G[_u].push_back(_v);
		G[_v].push_back(_u);
	}

	dfs(0,0);
	for (int i(0); i < N; ++i)
		cout << ans[i]+1 << '\n';
}