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)

#![allow(non_snake_case)]

use proconio::{input, marker::Usize1, fastout};
use superslice::*;

fn dfs(
  v: usize,
  dp: &mut Vec<usize>,
  hasVisited: &mut Vec<bool>,
  res: &mut Vec<usize>,
  graph: &Vec<Vec<usize>>,
  a: &[usize],
) {
  hasVisited[v] = true;
  let t = dp.lower_bound(&a[v]);
  let snapshot = dp[t];
  dp[t] = a[v];
  res[v] = dp.lower_bound(&std::usize::MAX);
  for &w in &graph[v] {
    if hasVisited[w] {
      continue;
    }
    dfs(w, dp, hasVisited, res, graph, a);
  }
  dp[t] = snapshot;
}

#[fastout]
fn main() {
  input! {
    N: usize,
    a: [usize; N],
    edges: [(Usize1, Usize1); N-1],
  }
  let mut graph = vec![vec![0; 0]; N];
  for &(u, v) in &edges {
    graph[u].push(v);
    graph[v].push(u);
  }
  let mut dp = vec![std::usize::MAX; N];
  let mut hasVisited = vec![false; N];
  let mut res = vec![0; N];
  dfs(0, &mut dp, &mut hasVisited, &mut res, &graph, &a);
  for r in res {
    println!("{}", r);
  }
}