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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)

import java.util.*;
import java.io.*;

	
public class Main {
	public static void main(String[] args) throws Exception {
		FastScanner sc = new FastScanner();
		PrintWriter out = new PrintWriter(System.out);
		
		int vn = sc.nextInt();
		long[] written_num = new long[vn];
		
		for(int i=0; i<vn; i++){
			written_num[i] = sc.nextLong();
		}
		
		ArrayList<ArrayList<Integer>> plain_graph = new ArrayList<>();
        
        for(int i=0; i<vn; i++){
            ArrayList<Integer> null_arr = new ArrayList<>(0);
            plain_graph.add(null_arr);
        }
        
        int a = -1, b = -1;
        for(int i=1; i<vn; i++){
            a = sc.nextInt()-1;
            b = sc.nextInt()-1;
            plain_graph.get(a).add(b);
            plain_graph.get(b).add(a);
        }
        
        int[] parent = new int[vn];
        parent[0]=-1;
        
        ArrayDeque<Integer> know_pare = new ArrayDeque<>();
        know_pare.add(0);
        
		ArrayList<ArrayList<Long>> dp = new ArrayList<>();//頂点iまでで、長さjを作る時の最後の値
		
        for(int i=0; i<vn; i++){
            ArrayList<Long> null_arr = new ArrayList<>(0);
            dp.add(null_arr);
        }
		
		dp.get(0).add(written_num[0]);
		
		int[] anss = new int[vn];
		anss[0]=1;
        
        int dw=0, ad=0;
        while(know_pare.size()>0){
            dw = know_pare.poll();
            for(int i: plain_graph.get(dw)){
                if(i != parent[dw]){
                    know_pare.add(i);
                    parent[i]=dw;
                    
                    ArrayList<Long> al1 = new ArrayList<>();
			        for(int j=0; j<dp.get(dw).size(); j++){
			            long long1 = dp.get(dw).get(j);
			            al1.add(long1);
			        }
                    
                    dp.set(i, al1);
                    anss[i] = anss[dw];
                    
                    ad = binarySearchMaxAL(dp.get(i), written_num[i]);
                    if(ad<0){
                        dp.get(i).set(0, written_num[i]);
                    }else if((ad+1) == dp.get(i).size()){
                        dp.get(i).add(written_num[i]);
                        anss[i]++;
                    }else{
                        dp.get(i).set(ad+1, written_num[i]);
                    }
                }
            }
        }
		
	   	for(int i=0; i<vn; i++){
	   	    out.println(anss[i]);
	   	}
		
		
		/*sc.next();
		int n_machine = sc.nextInt();
		int kinds_bread = sc.nextInt();
		int n_order = sc.nextInt();
		
		int ln_bom = sc.nextInt();
		int ln_combi = sc.nextInt();
		
		int[] machine_size = new int[n_machine];
		ArrayList<ArrayList<Integer>> holiday_dt = new ArrayList<>();
	   	for(int i=0; i<n_machine; i++){
	   	    ArrayList<Integer> al1 = new ArrayList<>();
			holiday_dt.add(al1);
	   	}
	   	
	   	for(int i=0; i<n_machine; i++){
		    sc.next();
	   	    int machine_num1 = sc.nextInt();
	   	    machine_size[i] = sc.nextInt();
	   	    int n_holiday1 = sc.nextInt();
			for(int j=0; j<n_holiday1; j++){
			    int holiday_a1 = sc.nextInt();
			    holiday_dt.get(machine_num1).add(holiday_a1);
			}
	   	}
	   	
	   	int[][] time2bake = new int[n_machine][kinds_bread];
	   	fill(time2bake,INF);
	   	int[][] ele_cost = new int[n_machine][kinds_bread];
	   	fill(ele_cost,INF);
		
	   	for(int i=0; i<ln_bom; i++){
		    sc.next();
	   	    int machine_num1 = sc.nextInt();
	   	    int bread_num1 = sc.nextInt();
	   	    time2bake[machine_num1][bread_num1] = sc.nextInt();
	   	    ele_cost[machine_num1][bread_num1] = sc.nextInt();
	   	}
		
		int[][][] time2change = new int[n_machine][kinds_bread][kinds_bread];
		fill(time2change,INF);
		int[][][] cost2change = new int[n_machine][kinds_bread][kinds_bread];
		fill(cost2change,INF);
		
	   	for(int i=0; i<ln_combi; i++){
		    sc.next();
	   	    int machine_num1 = sc.nextInt();
	   	    int bread_pre = sc.nextInt();
	   	    int bread_nex = sc.nextInt();
	   	    time2change[machine_num1][bread_pre][bread_nex] = sc.nextInt();
	   	    cost2change[machine_num1][bread_pre][bread_nex] = sc.nextInt();
	   	}
		
		int[] ordered_bread = new int[n_order];
		int[] startable_time = new int[n_order];
		int[] order_limit = new int[n_order];
		int[] ordered_amount = new int[n_order];
		int[] spiltable_amount = new int[n_order];
		int[] paid = new int[n_order];
		int[] awaited_time = new int[n_order];
		
	   	for(int i=0; i<n_order; i++){
		    sc.next();
	   	    int order_code1 = sc.nextInt();
	   	    ordered_bread[order_code1] = sc.nextInt();
	   	    startable_time[order_code1] = sc.nextInt();
	   	    order_limit[order_code1] = sc.nextInt();
	   	    ordered_amount[order_code1] = sc.nextInt();
	   	    spiltable_amount[order_code1] = sc.nextInt();
	   	    paid[order_code1] = sc.nextInt();
	   	    awaited_time[order_code1] = sc.nextInt();
	   	}*/
		
		
		out.flush();
	}
	
    private static ArrayList<boolean[]> combi_enum(int ln, boolean[] former, int now_pos, int rest){
        ArrayList<boolean[]> ans = new ArrayList<>();
        boolean[] ans1 = new boolean[former.length+1];
	   	for(int i=0; i<former.length; i++){
	   	    ans1[i] = former[i];
	   	}
        if(ln==(now_pos+1)){
            switch(rest){
                case 0:
                    ans1[former.length] = false;
                    ans.add(ans1);
                    break;
                case 1:
                    ans1[former.length] = true;
                    ans.add(ans1);
                    break;
            }
        }else{
            if(rest == 0){
                ans1[former.length] = false;
                ans.addAll(combi_enum(ln, ans1, now_pos+1, rest));
            }else{
                ans1[former.length] = false;
                ans.addAll(combi_enum(ln, ans1.clone(), now_pos+1, rest));
                ans1[former.length] = true;
                ans.addAll(combi_enum(ln, ans1.clone(), now_pos+1, rest-1));
            }
        }
        return ans;
    }

	private static int INF = (int)1e8;
	private static long INFL = (long)1e17;
	private static int e97 = (int)1e9 + 7;
	
	
	private static int abs(int a){
	    return (a>=0) ? a: -a;
	}
	private static long abs(long a){
	    return (a>=0) ? a: -a;
	}
	
	private static int min(int a, int b){
	    return (a>b) ? b : a;
	}
	private static long min(long a, long b){
	    return (a>b) ? b : a;
	}
	
	private static int max(int a, int b){
		return (a>b) ? a : b;
	}
	private static long max(long a, long b){
		return (a>b) ? a : b;
	}
	
	private static int minN(int... ins){
		int min = ins[0];
		for(int i=1; i<ins.length; i++){
			if(ins[i] < min) min = ins[i];
		}
		return min;
	}
	private static int maxN(int... ins){
		int max = ins[0];
		for(int i=1; i<ins.length; i++){
			if(ins[i] > max) max = ins[i];
		}
		return max;
	}
	private static long minN(long... ins){
		long min = ins[0];
		for(int i=1; i<ins.length; i++){
			if(ins[i] < min) min = ins[i];
		}
		return min;
	}
	private static long maxN(long... ins){
		long max = ins[0];
		for(int i=1; i<ins.length; i++){
			if(ins[i] > max) max = ins[i];
		}
		return max;
	}
	
	private static int minExAd(int[] dt, int ad){
	int min=Integer.MAX_VALUE;
		for(int i=0; i<dt.length; i++){
			if((i != ad) && (dt[i] < min)) min = dt[i];
		}
		return min;
	}
	private static long minExAd(long[] dt, int ad){
	long min=Long.MAX_VALUE;
		for(int i=0; i<dt.length; i++){
			if((i != ad) && (dt[i] < min)) min = dt[i];
		}
		return min;
	}
	private static int minExVal(int[] dt, int ex_val){
	int min=Integer.MAX_VALUE;
		for(int i=0; i<dt.length; i++){
			if((dt[i] != ex_val) && (dt[i] < min)) min = dt[i];
		}
		return min;
	}
	private static long minExVal(long[] dt, long ex_val){
	long min=Long.MAX_VALUE;
		for(int i=0; i<dt.length; i++){
			if((dt[i] != ex_val) && (dt[i] < min)) min = dt[i];
		}
		return min;
	}
	
	private static boolean same3(long a, long b, long c){
	if(a!=b) return false;
	if(b!=c) return false;
	if(c!=a) return false;
	return true;
	}
	private static boolean dif3(long a, long b, long c){
	if(a==b) return false;
	if(b==c) return false;
	if(c==a) return false;
	return true;
	}
	
	private static int factorial(int n) {
		int ans=1;
		for(int i=n; i>0; i--){
			ans*=i;
		}
		return ans;
	}
	
	private static long gcd(long m, long n) {
		if(m < n) return gcd(n, m);
		if(n == 0) return m;
		return gcd(n, m % n);
	}
	
	private static int pow2(int in){
		return in*in;
	}
	private static long pow2(long in){
		return in*in;
	}
	
	private static int getDigit2(long num){
		long cf = 1;
		int d=0;
		while(num >= cf){
			d++;
			cf = 1<<d;
		}
		return d; //numはd桁の数で、2^dより小さい
	}
	
	private static int getDigit10L(long num){
		long cf = 1;
		int d=0;
		while(num >= cf){
			d++;
			cf*=10;
		}
		return d; //numはd桁の数で、10^dより小さい
	}
	
	private static boolean isINF(int in){
	if((in*20)>INF) return true;
	else return false;
	}
	private static boolean isINFL(long in){
	if((in*10000)>INFL) return true;
	else return false;
	}
	
	private static long pow10E97(long ob, int soeji){
		if(soeji==2) return (ob*ob)%e97;
		
		int d = getDigit2(soeji);
		long[] ob_pow_2pow = new long[d];
		
		ob_pow_2pow[0] = ob;
		for(int i=1; i<d; i++){
			ob_pow_2pow[i] = (ob_pow_2pow[i-1]*ob_pow_2pow[i-1])%e97;
		}
		
		long ans=1;
		for(int i=d-1; i>=0; i--){
			if(soeji >= (1<<i)){
				soeji -= (1<<i);
				ans = (ans*ob_pow_2pow[i])%e97;
			}
 		}
		
		return ans;
	}
	
	
	private static int flag(int pos){
	    return (1<<pos);
	}
	private static boolean isFlaged(int bit, int pos){
    	if((bit&(1<<pos)) > 0) return true;
    	else return false;
	}
	private static int deflag(int bit, int pos){
	    return bit&~(1<<pos);
	}
	private static void showflag(int bit){
    	for(int i=0; i<getDigit2(bit); i++){
        	if(isFlaged(bit,i)) System.out.print("O");
        	else System.out.print(".");
    	}
	}
	
	
	private static void fill(boolean[] ob, boolean res){
		for(int i=0; i<ob.length; i++){
			ob[i] = res;
		}
		return;
	}
	private static void fill(int[] ob, int res){
		for(int i=0; i<ob.length; i++){
			ob[i] = res;
		}
		return;
	}
	private static void fill(long[] ob, long res){
		for(int i=0; i<ob.length; i++){
			ob[i] = res;
		}
		return;
	}
	private static void fill(char[] ob, char res){
		for(int i=0; i<ob.length; i++){
			ob[i] = res;
		}
		return;
	}
	private static void fill(int[][] ob, int res){
		for(int i=0; i<ob.length; i++){
			for(int j=0; j<ob[0].length; j++){
				ob[i][j] = res;
			}
		}
		return;
	}
	private static void fill(long[][] ob, long res){
		for(int i=0; i<ob.length; i++){
			for(int j=0; j<ob[0].length; j++){
				ob[i][j] = res;
			}
		}
		return;
	}
	private static void fill(char[][] ob, char res){
		for(int i=0; i<ob.length; i++){
			for(int j=0; j<ob[0].length; j++){
				ob[i][j] = res;
			}
		}
		return;
	}
	private static void fill(int[][][] ob, int res){
		for(int i=0; i<ob.length; i++){
			for(int j=0; j<ob[0].length; j++){
				for(int k=0; k<ob[0][0].length; k++){
			ob[i][j][k] = res;
				}
			}
		}
		return;
	}
	private static void fill(long[][][][] ob, long res){
		for(int i=0; i<ob.length; i++){
			for(int j=0; j<ob[0].length; j++){
				for(int k=0; k<ob[0][0].length; k++){
				for(int l=0; l<ob[0][0][0].length; l++){
				ob[i][j][k][l]=res;
				}
				}
			}
		}
		return;
	}
	
	static void show_b2d_array(boolean[][] dt, int lit_x, int lit_y){
		for(int i=0; i<dt[0].length; i++){
			for(int j=0; j<dt.length; j++){
				if((i==lit_y) && (j==lit_x)) System.out.print("X");
				else if(dt[j][i]) System.out.print("O");
				else System.out.print(".");
			}
			System.out.println();
		}
	}
	static void show_L2(long[][] dt, String cmnt){
		PrintWriter out = new PrintWriter(System.out);
		for(int i=0; i<dt[0].length; i++){
			for(int j=0; j<dt.length; j++){
				out.print(dt[j][i]+",");
			}
			out.println("<-"+cmnt+":"+i);
		}
		out.flush();
	}
	static void disp_que(ArrayDeque<Integer> dt){		//上手くいかなかった時用
		int a=0;
		while(dt.size()>0){
			a=dt.removeLast();
			System.out.print(a);
		}
		System.out.println("\n");
	}
	
	
	private static void nextPerm(ArrayList<Integer> former){
		int pivot=-1;
		int ln = former.size();
		for(int i = ln; i>1; i--){
			if(former.get(i-2) < former.get(i-1)){
				pivot = i-2;
				break;
			}
		}
		if(pivot==-1){
		former = null;
		return;
		}
		int pivot_fig = former.get(pivot);
		
		int min_pos=pivot+1;
		int min=former.get(min_pos);
		for(int i=ln-1; i > pivot+1; i--){
			if((former.get(i) < min)&&(former.get(i) > pivot_fig)){
				min_pos=i;
				min=former.get(min_pos);
			}
		}
		Collections.swap(former, pivot, min_pos);
		Collections.sort(former.subList(pivot+1, ln));
		return;
	}
	
	public static int binarySearch(int[] dt, int target){
		int left=0, right=dt.length-1;
		int mid=-1;
		while(left<=right){
			mid = (right+left)/2;
			if(dt[mid] == target) return mid;
			if(dt[mid] < target) left=mid+1;
			else right=mid-1;
		}
		return -1;
	}
	public static int binarySearchMax(long[] dt, long target){
		int left=-1, right=dt.length;
		int mid=-1;
		
		while((right-left)>1){
			mid = left + (right-left)/2;
			if(dt[mid] < target) left=mid;
			else right=mid;
		}
		return left;//target未満の最大のaddress
	}
	public static int binarySearchMaxAL(ArrayList<Long> dt, long target){
		int left=-1, right=dt.size();
		int mid=-1;
		
		while((right-left)>1){
			mid = left + (right-left)/2;
			if(dt.get(mid) < target) left=mid;
			else right=mid;
		}
		return left;//target未満の最大のaddress
	}
	
	
	private static int dfs2(boolean[][] state, int x, int y, int depth){		//薄氷渡り
		int[] tmpltX = {1,-1,0,0};
		int[] tmpltY = {0,0,1,-1};
		int newX = -1, newY = -1;
		
		state[x][y] = false;
		depth++;
		
		int maxDunder_this_brunch = depth, tmpD=0;
		for(int i=0; i < 4; i++){
			newX = x+tmpltX[i];
			newY = y+tmpltY[i];
			if((newX >= 0) && (newY >= 0) && (newX < state.length) && (newY < state[0].length) && state[newX][newY]) tmpD = dfs2(state, newX, newY, depth);
			if(tmpD > maxDunder_this_brunch) maxDunder_this_brunch=tmpD;
		}
		state[x][y] = true;
		return maxDunder_this_brunch;
	}
	
    private static int areaSum(int[][] acum_plus1, int bY_notInclude, int aY_include, int bX_notInclude, int aX_include){
        int ans = acum_plus1[aY_include+1][aX_include+1];
        ans -= acum_plus1[aY_include+1][bX_notInclude+1];
        ans -= acum_plus1[bY_notInclude+1][aX_include+1];
        ans += acum_plus1[bY_notInclude+1][bX_notInclude+1];
        return ans;
    }



static class Edge{
    int v2;
    long weight;
    public Edge(int vn, long w){
        this.v2 = vn;
        this.weight = w;
    }
}
static class Edge2{
    int v2;
    long cost1,cost2;
    public Edge2(int vn, long w1, long w2){
        this.v2 = vn;
        this.cost1 = w1;
        this.cost2 = w2;
    }
}

static class Comparator_Edge implements Comparator<Edge>{
    public int compare(Edge a, Edge b){
        if(a.weight>b.weight){
            return 1;
        }else if(a.weight<b.weight){
            return -1;
        }
        return 0;
    }
}
static class FastScanner {
	private final InputStream in = System.in;
	private final byte[] buffer = new byte[1024];
	private int ptr = 0;
	private int buflen = 0;
	private boolean hasNextByte() {
		if (ptr < buflen) {
			return true;
		}else{
			ptr = 0;
			try {
				buflen = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			if (buflen <= 0) {
				return false;
			}
		}
		return true;
	}
	
	private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
	private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
	public boolean hasNext() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++; return hasNextByte();}
	public String next() {
		if (!hasNext()) throw new NoSuchElementException();
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while(isPrintableChar(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	public long nextLong() {
		if (!hasNext()) throw new NoSuchElementException();
		long n = 0;
		boolean minus = false;
		int b = readByte();
		if (b == '-') {
			minus = true;
			b = readByte();
		}
		if (b < '0' || '9' < b) {
			throw new NumberFormatException();
		}
		while(true){
			if ('0' <= b && b <= '9') {
				n *= 10;
				n += b - '0';
			}else if(b == -1 || !isPrintableChar(b)){
				return minus ? -n : n;
			}else{
				throw new NumberFormatException();
			}
			b = readByte();
		}
	}
	public int nextInt() {
		long nl = nextLong();
		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
		return (int) nl;
	}
	public double nextDouble() { return Double.parseDouble(next());}
}
}