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 Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
	"strconv"
)

func getScanner(fp *os.File) *bufio.Scanner {
	scanner := bufio.NewScanner(fp)
	scanner.Split(bufio.ScanWords)
	scanner.Buffer(make([]byte, 1000005), 1000005)
	return scanner
}

func getNextString(scanner *bufio.Scanner) string {
	scanner.Scan()
	return scanner.Text()
}

func getNextInt(scanner *bufio.Scanner) int {
	i, _ := strconv.Atoi(getNextString(scanner))
	return i
}

func getNextInt64(scanner *bufio.Scanner) int64 {
	i, _ := strconv.ParseInt(getNextString(scanner), 10, 64)
	return i
}

func getNextUint64(scanner *bufio.Scanner) uint64 {
	i, _ := strconv.ParseUint(getNextString(scanner), 10, 64)
	return i
}

func getNextFloat64(scanner *bufio.Scanner) float64 {
	i, _ := strconv.ParseFloat(getNextString(scanner), 64)
	return i
}

func main() {
	fp := os.Stdin
	wfp := os.Stdout
	cnt := 0

	if os.Getenv("MASPY") == "ますピ" {
		fp, _ = os.Open(os.Getenv("BEET_THE_HARMONY_OF_PERFECT"))
		cnt = 0
	}
	if os.Getenv("MASPYPY") == "ますピッピ" {
		wfp, _ = os.Create(os.Getenv("NGTKANA_IS_GENIUS10"))
	}

	scanner := getScanner(fp)
	writer := bufio.NewWriter(wfp)
	solve(scanner, writer)
	for i := 0; i < cnt; i++ {
		fmt.Fprintln(writer, "-----------------------------------")
		solve(scanner, writer)
	}

	writer.Flush()
}

func solve(scanner *bufio.Scanner, writer *bufio.Writer) {
	n := getNextInt(scanner)
	aa := make([]int, n)
	for i := 0; i < n; i++ {
		aa[i] = getNextInt(scanner)
	}
	ee := make([][]edge, n)
	for i := 0; i < n-1; i++ {
		u := getNextInt(scanner) - 1
		v := getNextInt(scanner) - 1
		appendEdge(u, v, i, ee)
		appendEdge(v, u, i, ee)
	}

	yokubalis := make([]int, n)
	for i := 0; i < n; i++ {
		yokubalis[i] = math.MaxInt32
	}

	ans := make([]int, n)
	dfs(0, -1, aa, yokubalis, ans, ee)
	for i := 0; i < n; i++ {
		fmt.Fprintln(writer, ans[i])
	}
}
func dfs(c, p int, aa, yokubalis, ans []int, ee [][]edge) {
	l := -1
	r := len(yokubalis)
	for l+1 < r {
		m := (l + r) >> 1
		if yokubalis[m] >= aa[c] {
			r = m
			continue
		}
		l = m
	}

	old := yokubalis[r]
	yokubalis[r] = aa[c]
	ll := -1
	rr := len(yokubalis)
	for ll+1 < rr {
		mm := (ll + rr) >> 1
		if yokubalis[mm] == math.MaxInt32 {
			rr = mm
			continue
		}
		ll = mm
	}
	ans[c] = rr

	for _, e := range ee[c] {
		if e.to == p {
			continue
		}
		dfs(e.to, c, aa, yokubalis, ans, ee)
	}
	yokubalis[r] = old
}

func appendEdge(from, to, id int, ee [][]edge) {
	ee[from] = append(ee[from], edge{
		to: to,
		id: id,
	})
}

type edge struct {
	to, id int
}