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: Accepted)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using static System.Math;

namespace CSharpSpiral {
	class Program {
		static void Main(string[] args) {
			var sw = new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush=
#if DEBUG
			true
#else
			false
#endif
			};
			Console.SetOut(sw);
			new Program().Solve(new ConsoleInput(
#if DEBUG
			new StreamReader("../../../inputData.txt")
#else
			Console.In
#endif
			, ' '));
			Console.Out.Flush();
		}
		public void Solve(ConsoleInput cin) {
			var N = cin.ReadInt;
			var A_nodeValues = cin.ReadIntArray(N);
			var G = new Graph<int>(N);
			for(int i = 0; i<N; i++) {
				G.Nodes[i]=A_nodeValues[i];
			}
			for(int i = 0; i<N-1; i++) {
				var edge = cin.ReadIntArray(2);
				G.AddEdge(edge[0]-1, edge[1]-1, 0);
				G.AddEdge(edge[1]-1, edge[0]-1, 0);
			}
			var Res = new int[N];
			var LISTable = new int[N+1];
			for(int i = 0; i<N+1; i++) {
				LISTable[i]=int.MaxValue;
			}
			var UnReplaces = new (int j, int x)[N];
			for(int i = 0; i<N; i++) {
				UnReplaces[i]=(-1, -1);
			}
			G.PointDFS(0,
			j => {
				var k = LISUpdate(j, LISTable, G.Nodes[j], out int old);
				var r = Array.BinarySearch(LISTable, int.MaxValue-1);
				Res[j]=-r-1;
				UnReplaces[j]=(k, old);
			},
			j => {
				var u = UnReplaces[j];
				if(u.j>=0) LISTable[u.j]=u.x;
			});
			Console.WriteLine(Res.Join("\n"));
		}
		public int LISUpdate(int i, int[] LIS, int value, out int oldValue) {
			var j = Array.BinarySearch(LIS, value);
			if(j<0) {
				oldValue=LIS[-j-1];
				LIS[-j-1]=value;
				return -j-1;
			}
			oldValue=-1;
			return -1;
		}
		class Graph<T> {
			public readonly int N;
			public T[] Nodes { get; }
			private List<int>[] Nexts { get; }
			private Dictionary<int, T>[] Edges { get; set; }
			public Graph(int _N) {
				N=_N;
				Nodes=new T[_N];
				Nexts=new List<int>[_N];
				Edges=new Dictionary<int, T>[_N];
				for(int i = 0; i<_N; i++) {
					Nexts[i]=new List<int>();
					Edges[i]=new Dictionary<int, T>();
				}
			}
			public void Update(int i, T value) {
				Nodes[i]=value;
			}
			public void AddEdge(int from, int to, T value) {
				Edges[from].Add(to, value);
				Nexts[from].Add(to);
			}
			public void PointDFS(int from, Action<int> onEnter, Action<int> onLeave) {
				var S = new Stack<int>();
				S.Push(from);
				var entered = new bool[N];
				var departed = new bool[N];//dep<=ent
				var childCount = new int[N];
				while(S.Any()) {
					var v = S.Peek();
					if(!entered[v]) {
						onEnter?.Invoke(v);
						entered[v]=true;
					}
					for(int i = childCount[v]; i<Nexts[v].Count; i++) {
						if(!entered[Nexts[v][i]]) {
							S.Push(Nexts[v][i]);
							childCount[v]=i+1;
							goto NextLoop;
						}
					}
					S.Pop();
					childCount[v]=Nexts[v].Count;
					if(!departed[v]) {
						onLeave?.Invoke(v);
						departed[v]=true;
					}
				NextLoop:;
				}
			}
		}
	}
	static class ExtendClass {
		public static string Join<T>(this IEnumerable<T> X, string separator = " ") => string.Join(separator, X);
		
	}
	class ConsoleInput {
		private readonly TextReader _stream;
		private readonly char _separator = ' ';
		private readonly Queue<string> inputStream;
		public ConsoleInput(TextReader stream, char separator = ' ') {
			this._separator=separator;
			this._stream=stream;
			inputStream=new Queue<string>();
		}
		public string Read {
			get {
				if(inputStream.Count!=0) return inputStream.Dequeue();
				string[] tmp = _stream.ReadLine().Split(_separator);
				for(int i = 0; i<tmp.Length; ++i)
					inputStream.Enqueue(tmp[i]);
				return inputStream.Dequeue();
			}
		}
		public string ReadLine { get { return _stream.ReadLine(); } }
		public int ReadInt { get { return int.Parse(Read); } }
		public long ReadLong { get { return long.Parse(Read); } }
		public double ReadDouble { get { return double.Parse(Read); } }
		public string[] ReadStrArray(long N) {
			var ret = new string[N];
			for(long i = 0; i<N; ++i)
				ret[i]=Read;
			return ret;
		}
		public int[] ReadIntArray(long N) { var ret = new int[N]; for(long i = 0; i<N; ++i) ret[i]=ReadInt; return ret; }
		public long[] ReadLongArray(long N) { var ret = new long[N]; for(long i = 0; i<N; ++i) ret[i]=ReadLong; return ret; }
		public double[] ReadDoubleArray(long N) { var ret = new double[N]; for(long i = 0; i<N; ++i) ret[i]=ReadDouble; return ret; }
		public bool isEOF {
			get { return _stream.Peek()==-1; }
		}
	}

}