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)

//! Framework <https://github.com/vain0x/procon>

#![allow(non_snake_case)]
#![allow(unused_imports)]

use std::collections::*;

pub struct Scan(Box<dyn Iterator<Item = &'static str>>); // '

impl Scan {
    fn new() -> Self {
        let mut buf = String::new();
        let read_line = move || {
            std::io::stdin().read_line(&mut buf).unwrap();
            Box::leak(buf.split_off(0).into_boxed_str()).split_whitespace()
        };
        Scan(Box::new(std::iter::repeat_with(read_line).flatten()))
    }

    pub fn word<T: std::str::FromStr>(&mut self) -> T {
        self.0.next().unwrap().parse().ok().unwrap()
    }

    pub fn list<T: std::str::FromStr>(&mut self, len: usize) -> Vec<T> {
        std::iter::repeat_with(|| self.word()).take(len).collect()
    }
}

#[derive(PartialEq, Clone, Debug)]
struct Interval<T> {
    l: T,
    r: T,
}

impl<T: Ord> Interval<T> {
    fn new(l: T, r: T) -> Interval<T> {
        Interval { l: l, r: r }
    }

    fn disjoint(&self, other: &Self) -> bool {
        self.r <= other.l || other.r <= self.l
    }

    fn covers(&self, other: &Self) -> bool {
        self.l <= other.l && other.r <= self.r
    }
}

#[derive(Debug)]
pub struct SegTree<T, F> {
    len: usize,

    /// Number of leaf nodes.
    width: usize,

    /// len = `2w-1` for `w-1` inners and `w` leaves,
    /// where `w` is the smallest `2^p` (`>= len`).
    node: Vec<T>,

    mempty: T,
    mappend: F,
}

impl<T, F> SegTree<T, F>
where
    T: Clone,
    F: Fn(T, T) -> T,
{
    pub fn new(items: Vec<T>, mempty: T, mappend: F) -> SegTree<T, F> {
        let len = items.len();

        let mut w = 1;
        while w < len {
            w *= 2;
        }
        debug_assert!(w >= len);

        let mut node = vec![mempty.clone(); 2 * w - 1];

        for (ei, item) in items.into_iter().enumerate() {
            node[w - 1 + ei] = item;
        }

        for ni in (0..w - 1).rev() {
            node[ni] = mappend(node[2 * ni + 1].clone(), node[2 * ni + 2].clone());
        }

        SegTree {
            len: len,
            width: w,
            node: node,
            mempty: mempty,
            mappend: mappend,
        }
    }

    pub fn len(&self) -> usize {
        self.len
    }

    pub fn set(&mut self, ei: usize, value: T) {
        let mut ni = self.width - 1 + ei;
        self.node[ni] = value;

        while ni > 0 {
            ni = (ni - 1) / 2;
            self.node[ni] =
                (self.mappend)(self.node[2 * ni + 1].clone(), self.node[2 * ni + 2].clone());
        }
    }

    pub fn sum(&self, ql: usize, qr: usize) -> T {
        let q = Interval::new(ql, qr);
        if q.disjoint(&Interval::new(0, self.len())) {
            self.mempty.clone()
        } else {
            self.sum_core(0, Interval::new(0, self.width), &q)
        }
    }

    fn sum_core(&self, ni: usize, e: Interval<usize>, q: &Interval<usize>) -> T {
        if e.disjoint(&q) {
            self.mempty.clone()
        } else if q.covers(&e) {
            self.node[ni].clone()
        } else {
            let m = (e.l + e.r) / 2;
            let vl = self.sum_core(2 * ni + 1, Interval::new(e.l, m), q);
            let vr = self.sum_core(2 * ni + 2, Interval::new(m, e.r), q);
            (self.mappend)(vl.clone(), vr.clone())
        }
    }
}

struct Solve {
    graph: Vec<Vec<usize>>,
    a: Vec<usize>,
    seg: SegTree<i64, &'static dyn Fn(i64, i64) -> i64>,
    ans: Vec<i64>,
}

impl Solve {
    fn new(graph: Vec<Vec<usize>>, a: Vec<usize>) -> Self {
        let N = graph.len();

        Solve {
            graph,
            a,
            seg: SegTree::new(vec![0; N + 1], 0, &std::cmp::max),
            ans: vec![0; N],
        }
    }

    fn solve(mut self) {
        let N = self.graph.len();

        self.dfs(0, N);

        for k in 0..N {
            println!("{}", self.ans[k]);
        }
    }

    fn dfs(&mut self, u: usize, p: usize) {
        let N = self.graph.len();

        let a = self.a[u];
        let prev = self.seg.sum(a, a + 1);

        let m = self.seg.sum(0, a) + 1;
        self.seg.set(a, m);
        self.ans[u] = self.seg.sum(0, N + 1);

        for i in 0..self.graph[u].len() {
            let v = self.graph[u][i];
            if v == p {
                continue;
            }

            self.dfs(v, u);
        }

        self.seg.set(a, prev);
    }
}

fn main() {
    let mut scan = Scan::new();

    let N = scan.word::<usize>();
    let a = scan.list::<i64>(N);
    let mut graph = vec![vec![]; N];

    for _ in 0..N - 1 {
        let u = scan.word::<usize>() - 1;
        let v = scan.word::<usize>() - 1;
        graph[u].push(v);
        graph[v].push(u);
    }

    let a = {
        let mut num = a.clone();
        num.sort();
        num.dedup();
        let compress = num
            .into_iter()
            .enumerate()
            .map(|(i, x)| (x, i))
            .collect::<HashMap<_, _>>();
        a.into_iter().map(|x| compress[&x]).collect::<Vec<_>>()
    };

    Solve::new(graph, a).solve();
}