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# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using Pi = Pair<int, int>;

class Solver
{
    List<int>[] g;
    int[] A;
    PersistentArray<int>[] lis;
    int[] res;
    public void Solve(Scanner sc)
    {
        var N = sc.Int;
        A = sc.ArrInt;
        lis = new PersistentArray<int>[N];
        g = Create(N, () => new List<int>());
        for (int i = 0; i < N - 1; i++)
        {
            int u, v;
            sc.Make(out u, out v); u--; v--;
            g[u].Add(v);
            g[v].Add(u);
        }
        lis[0] = new PersistentArray<int>(N);
        for (int i = 1; i < N; i++) lis[0] = lis[0].Set(i, int.MaxValue);
        res = new int[N];
        lis[0]=lis[0].Set(0, A[0]);
        dfs(0, -1, 0);
        Console.WriteLine(string.Join("\n", res));
    }
    void dfs(int i, int p, int last)
    {
        /*Write($"lis[{i}] = ");
        for (int j = 0; j < A.Length; j++)
        {
            Write(lis[i][j] + " ");
        }
        WriteLine();*/
        res[i] = last + 1;
        foreach (var e in g[i])
        {
            if (e == p) continue;
            if (lis[i][last] < A[e])
            {
                lis[e] = lis[i].Set(last + 1, A[e]);
                dfs(e, i, last + 1);
            }
            else
            {
                int l = -1, r = A.Length;
                while (r - l > 1)
                {
                    var m = (r + l) / 2;
                    if (lis[i][m] >= A[i]) r = m;
                    else l = m;
                }
                lis[e] = lis[i].Set(r, A[e]);
                dfs(e, i, last);
            }
        }
    }
}
public class PersistentArray<T>
{
    public int Length { get; private set; }
    Node root;
    public PersistentArray(int sz)
    {
        Length = sz;
        root = new Node { idx = 1 };
        var Q = new Queue<Node>();
        if (2 <= sz)
            Q.Enqueue(root);
        while (Q.Any())
        {
            var p = Q.Dequeue();
            p.lft = new Node() { idx = 1 << p.idx };
            if ((p.lft.idx << 1) <= sz) Q.Enqueue(p.lft);
            if (((p.idx << 1) | 1) <= sz)
            {
                p.rgt = new Node() { idx = (p.idx << 1) | 1 };
                if ((p.rgt.idx << 1) <= sz) Q.Enqueue(p.rgt);
            }
        }
    }
    PersistentArray(Node r, int sz) { root = r; Length = sz; }
    public T this[int i]
    {
        get
        {
            var mask = MSB(++i) - 1;
            var nd = root;
            while (mask >= 0) { if ((1 & i >> mask) == 0) nd = nd.lft; else nd = nd.rgt; mask--; }
            return nd.dat;
        }
    }
    public PersistentArray<T> Set(int i, T v)
    {
        var nd = root.DeepCopy();
        var mask = MSB(++i) - 1;
        var r = new PersistentArray<T>(nd, Length);
        while (mask >= 0)
        {
            if ((1 & i >> mask) == 0) nd = nd.lft = nd.lft.DeepCopy();
            else nd = nd.rgt = nd.rgt.DeepCopy();
            mask--;
        }
        nd.dat = v;
        return r;
    }
    class Node
    {
        public int idx;
        public T dat;
        public Node lft, rgt;
        public Node DeepCopy() => new Node() { idx = idx, dat = dat, lft = lft, rgt = rgt };
    }

    int MSB(int v)
    {
        var rt = 0;
        if ((v >> 16) > 0) { rt |= 16; v >>= 16; }
        if ((v >> 8) > 0) { rt |= 8; v >>= 8; }
        if ((v >> 4) > 0) { rt |= 4; v >>= 4; }
        if ((v >> 2) > 0) { rt |= 2; v >>= 2; }
        return rt | (v >> 1);
    }
}

#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver().Solve(new Scanner());
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) > 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) < 0) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static IEnumerable<T> Shuffle<T>(this IEnumerable<T> A) => A.OrderBy(v => Guid.NewGuid());
    public static int CompareTo<T>(this T[] A, T[] B, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; for (var i = 0; i < Min(A.Length, B.Length); i++) { int c = cmp(A[i], B[i]); if (c > 0) return 1; else if (c < 0) return -1; } if (A.Length == B.Length) return 0; if (A.Length > B.Length) return 1; else return -1; }
    public static int MaxElement<T>(this IList<T> A, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; T max = A[0]; int rt = 0; for (int i = 1; i < A.Count; i++) if (cmp(max, A[i]) < 0) { max = A[i]; rt = i; } return rt; }
    public static T PopBack<T>(this List<T> A) { var v = A[A.Count - 1]; A.RemoveAt(A.Count - 1); return v; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}
public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString() => $"{v1.ToString()} {v2.ToString()}";
    public void Deconstruct(out T1 a, out T2 b) { a = v1; b = v2; }
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString() => $"{base.ToString()} {v3.ToString()}";
    public void Deconstruct(out T1 a, out T2 b, out T3 c) { Deconstruct(out a, out b); c = v3; }
}
#endregion