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 PHP (7.4.4) (judge verdict: Wrong Answer)

<?php
[$N] = ints();
$A = ints();
$G = loadTree();
//print_r($G);

dfs(1, -1, 0, [1]);
echo implode("\n", $result);
function dfs($current, $from = -1, $d = 0,$path = []){
  global $G, $distance,$A,$result;
  $distance[$current] = $d;
  if(count($G[$current])==1){
      $score = [];
      foreach($path as $p){
          $score[] = $A[$p-1];
      }
      $ret = lis($score);
      foreach($path as $i => $p){
        $result[$p] = $ret[$i];
      }
  }
  foreach($G[$current] as $to){
    $path2 = $path;
    $path2[] = $to;
    if($from != $to){
      dfs($to, $current, $d + 1, $path2);
    }
  }
}
/**
 * 単調増加列
 * @param $A 配列
 * @param $kougi true：広義 false：狭義
 * @param $inv true：減少列
 **/
function lis($A, $kougi = false, $inv = false){
    if($inv)foreach($A as $i => $a)$A[$i] = -$a;
    $dp = array_fill(0, count($A)+1, PHP_INT_MAX);
    $dp[0] = PHP_INT_MIN;
    $longest = 0;
    foreach($A as $a){
        $b = lower_bound($dp, $a+$kougi);
        $dp[$b] = $a;
        $longest = max($longest, $b);
        $ret[] = $longest;
    }
    return $ret;
}

function lower_bound($array, $key){
  $min = -1;
  $max = count($array);
  while($max - $min > 1){
    $c = intdiv($min + $max, 2);
    if($array[$c] >= $key){
      $max = $c;
    }else{
      $min = $c;
    }
  }
  return $max;
}

function str(){return trim(fgets(STDIN));}
function ints(){return array_map('intval',explode(' ',trim(fgets(STDIN))));}
function int(){return intval(trim(fgets(STDIN)));}
function chmax(&$a,$b){if($a<$b){$a=$b;return 1;}return 0;}
function chmin(&$a,$b){if($a>$b){$a=$b;return 1;}return 0;}
function popcount($x){$c=0;while($x){$x&=$x-1;++$c;}return$c;}
function o(...$val){
    if(count($val)==1)$val=array_shift($val);
    echo debug_backtrace()[0]['line'].")";
    if(is_array($val)){
        if(count($val) == 0)echo "empty array";
        elseif(!is_array(current($val)))echo "array: ",implode(" ", addIndex($val)),"\n";
        else{
            echo "array:array\n";
            if(isCleanArray($val))foreach($val as $row)echo implode(" ", addIndex($row)),"\n";
            else foreach($val as $i => $row)echo "[".$i."] ".implode(" ", addIndex($row)),"\n";
        }
    }else echo $val."\n";
}
function addIndex($val){if(!isCleanArray($val)){$val = array_map(function($k, $v){return $k.":".$v;}, array_keys($val), $val);}return $val;}
function isCleanArray($array){$clean=true;$i = 0;foreach($array as $k=>$v){if($k != $i++)$clean = false;}return $clean;}
// 座圧対象の配列 -> [圧縮された配列,復元用のMap,圧縮用のMap]
function atsu($array){$a = array_flip($array);$fuku=array_flip($a);sort($fuku);$atsu = array_flip($fuku);foreach($array as $i=>$val)$array[$i]=$atsu[$val];return [$array, $fuku, $atsu];}
function gcdAll($array){$gcd=$array[0];for($i=1,$I=count($array);$i<$I;++$i){$gcd=gcd($gcd,$array[$i]);}return $gcd;}
function gcd($m, $n){if(!$n)return $m;return gcd($n, $m % $n);}
function lcmAll($array){$lcm=$array[0];for($i=1,$I=count($array);$i<$I;++$i){$lcm=lcm($lcm,$array[$i]);}return $lcm;}
function lcm($a, $b) {return $a / gcd($a, $b) * $b;}


function loadTree($N = false){
  if($N === false)$N = $GLOBALS['N'];
  return loadGraph($N, $N-1);
}
function loadGraph($N = false, $M = false, $both = true){
  if($N === false)$N = $GLOBALS['N'];
  if($M === false)$M = $GLOBALS['M'];
  $G = array_fill(1, $N, []);
  for($i = 0; $i < $M; $i++){
    $values = ints();
    if(count($values) == 2){
      list($a, $b) = $values;
      $G[$a][] = $b;
      if($both)$G[$b][] = $a;
    }else{
      list($a, $b, $d) = $values;
      $G[$a][] = [$b, $d];
      if($both)$G[$b][] = [$a, $d];
    }
  }
  return $G;
}