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;
typedef long long ll;
#define INF 1e18

template <typename T>
struct Graph {
    int V, E;
    struct Edge {
        int from, to;
        T cost;
    };
    vector<vector<Edge>> edge;
    vector<ll> a;
    vector<Edge> es;

    Graph(int V, int E) : V(V), E(E) {
        edge.assign(V, vector<Edge>(0));
        a.resize(V);
    }

    void input(bool is_direct, bool inputc, bool index0 = false) {
        int a, b;
        T c = 0;
        for (int i = 0; i < E; i++) {
            cin >> a >> b;
            if (inputc) cin >> c;
            if (!index0) a--, b--;
            edge[a].push_back((Edge){a, b, c});
            es.push_back((Edge){a, b, c});
            if (!is_direct) {
                edge[b].push_back((Edge){b, a, c});
                es.push_back((Edge){b, a, c});
            }
        }
        if (!is_direct) E *= 2;
    }

    void add_edge(int from, int to, T cost, bool is_direct) {
        edge[from].push_back({from, to, cost});
        es.push_back({from, to, cost});
        if (!is_direct) {
            edge[to].push_back({to, from, cost});
            es.push_back({to, from, cost});
            E++;
        }
        E++;
    }

    vector<int> ans;
    vector<ll> dp;

    void dfs(int v, int from, int i) {
        if (from == -1) ans.assign(V, 0);
        for (const auto& e : edge[v]) {
            int to = e.to;
            if (to == from) continue;

            int change_point = lower_bound(dp.begin(), dp.begin() + i + 1, a[to]) - dp.begin();
            ll t1 = dp[change_point];

            dp[change_point] = a[to];
            ans[to] = lower_bound(dp.begin(), dp.begin() + i + 1, INF) - dp.begin();

            dfs(to, v, i + 1);
            dp[change_point] = t1;
        }
    }
};

int main() {
    int N;
    cin >> N;
    Graph<int> g(N, N - 1);
    for (int i = 0; i < N; i++) cin >> g.a[i];
    g.input(false, false);
    g.dp.assign(N + 10, INF);
    g.dfs(0, -1, 1);
    for (int i = 0; i < N; i++) cout << g.ans[i] + 1 << endl;
    return 0;
}