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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
 
using namespace std;
 
typedef long long int ll;
typedef pair<int, int> pii;
 
#define PB push_back
#define MP make_pair
 
#define FOR(prom, a, b) for(int prom = (a); prom < (b); prom++)
#define FORDE(prom, a, b) for(int prom = (a); prom >= (b); prom--)
#define R1(a) do{scanf("%d", &(a));}while(0)
#define R2(a, b) do{scanf("%d%d", &(a), &(b));}while(0)
#define R3(a, b, c) do{scanf("%d%d%d", &(a), &(b), &(c));}while(0)
#define SV(vec) do{int __sv;scanf("%d", &(__sv));vec.PB(__sv);}while(0)
#define ALL(vec) (vec).begin(), (vec).end()
#define MM(co, cim) memset((co), (cim), sizeof((co)))
#define DEB(x) cerr << ">>> " << #x << " : " << x << endl;
#define INF   1000000007
#define INFLL 1000000000000000007LL
 
int n, a[200014], res[200014];
vector<int> g[200014];
multiset<int> s;
 
void go (int x, int p)
{
  s.insert(a[x]);
  auto it = s.lower_bound(a[x]);
  ++it;
  int erased = -1;
  if (it != s.end()) 
  {
    erased = *it;
    s.erase(it);
  }
  res[x] = s.size();
  //handle neighbors
  for (auto y : g[x]) if (y != p) go(y, x);
  //remove lis stuff
  if (erased != -1) s.insert(erased);
  s.erase(s.find(a[x]));
}
 
int main ()
{
  R1(n);
  FOR(i, 0, n) R1(a[i]);
  FOR(i, 0, n - 1)
  {
    int from, to;
    R2(from, to);--from;--to;
    g[from].PB(to);
    g[to].PB(from);
  }
  go(0, -1);
  FOR(i, 0, n) printf("%d\n", res[i]);
  return 0;
}