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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::{fastout, input};
use proconio::marker::*;

const INF: usize = 1001001001001001001;

#[fastout]
fn main() {
   input! {
      n: usize,
      a: [usize; n],
      uv: [(Usize1, Usize1); n-1]
   }

   let mut to = vec![vec![]; n];
   for (u, v) in uv {
      to[u].push(v);
      to[v].push(u);
   }

   let mut lis = vec![INF; n+1];
   let mut ans = vec![0;n];
   dfs(0, 0, &mut lis, &mut ans, &to, &a);
   for x in ans {
      println!("{}", x);
   }
}

fn dfs(v: usize, p: usize, lis: &mut [usize], ans: &mut [usize], to: &[Vec<usize>], a: &[usize]) {
   let pos = lis.lower_bound(&a[v]);
   let res = lis[pos];
   lis[pos] = a[v];
   ans[v] = lis.lower_bound(&INF);
   for u in to[v].iter().copied() {
      if u == p {
         continue;
      }
      dfs(u, v, lis, ans, to, a);
   }
   lis[pos] = res;
}

use std::cmp::Ordering;
/// Equivalent to std::lowerbound and std::upperbound in c++
pub trait BinarySearch<T> {
   fn lower_bound(&self, x: &T) -> usize;
   fn upper_bound(&self, x: &T) -> usize;
}
impl<T: Ord> BinarySearch<T> for [T] {
   fn lower_bound(&self, x: &T) -> usize {
      let mut low = 0;
      let mut high = self.len();
      while low != high {
         let mid = (low + high) / 2;
         match self[mid].cmp(x) {
            Ordering::Less => {
               low = mid + 1;
            }
            Ordering::Equal | Ordering::Greater => {
               high = mid;
            }
         }
      }
      low
   }
   fn upper_bound(&self, x: &T) -> usize {
      let mut low = 0;
      let mut high = self.len();
      while low != high {
         let mid = (low + high) / 2;
         match self[mid].cmp(x) {
            Ordering::Less | Ordering::Equal => {
               low = mid + 1;
            }
            Ordering::Greater => {
               high = mid;
            }
         }
      }
      low
   }
}