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# (Mono-csc 3.5.0) (judge verdict: Runtime Error)

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    class Leaf
    {
        /// <summary>
        /// 整数
        /// </summary>
        public long a;

        /// <summary>
        /// 辺
        /// </summary>
        public List<Leaf> children;

        /// <summary>
        /// 長さ
        /// </summary>
        public long len;

        public Leaf(long _a)
        {
            a = _a;
            children = new List<Leaf>();
        }
    }

    long n;

    Leaf[] leaf;

    long[] dp;

    static void Main(string[] args)
    {
        new Program();
    }

    Program()
    {
        n = long.Parse(Console.ReadLine());

        var a = (from x in Console.ReadLine().Split(' ') select long.Parse(x)).ToArray();

        leaf = new Leaf[n + 1];
        dp = new long[n + 1];

        for (var i = 0; i <= n; i++)
        {
            leaf[i] = new Leaf(a[i]);
            dp[i] = long.MaxValue;
        }

        for (var i = 1; i < n; i++)
        {
            var uv = Console.ReadLine().Split(' ');
            var u = leaf[long.Parse(uv[0]) - 1];
            var v = leaf[long.Parse(uv[1]) - 1];
            u.children.Add(v);
            v.children.Add(u);
        }

        var root = leaf[0];
        dp[0] = root.a;

        Sub1(root, null);

        foreach (var le in leaf)
        {
            Console.WriteLine(le.len);
        }
    }

    void Sub1(Leaf parent, Leaf c)
    {
        var i = 0L;
        var s1 = new Stack<long>();
        var s2 = new Stack<long>();

        for (; i < n; i++)
        {
            if (dp[i] == long.MaxValue) break;
            if (dp[i] < parent.a)
            {
                if (dp[i + 1] > parent.a)
                {
                    s1.Push(i + 1);
                    s2.Push(dp[i + 1]);
                    dp[i + 1] = parent.a;
                }
            }
        }

        parent.len = i;

        foreach (var child in parent.children)
        {
            if (child == c) continue;

            Sub1(child, parent);
        }

        while (s1.Count != 0)
        {
            dp[s1.Pop()] = s2.Pop();
        }
    }
}