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: Wrong Answer)

#include<bits/stdc++.h>
using namespace std;
const int N=4e5+5;
int n,num,ans[N],rt[N],t[N*50],ls[N*50],rs[N*50],tot,a[N],hs[N],head[N],nex[N<<1],to[N<<1];
void add(int u,int v){to[++tot]=v;nex[tot]=head[u];head[u]=tot;}
int newnode(){return ++num;}
void Insert(int l,int r,int x,int v,int &now,int pre)
{
    if(!now) now=newnode(),t[now]=t[pre];
    t[now]=max(t[now],v);
    if(l==r) return;
    int m=l+r>>1;
    if(x<=m) Insert(l,m,x,v,ls[now],ls[pre]),rs[now]=rs[pre];
    else Insert(m+1,r,x,v,rs[now],rs[pre]),ls[now]=ls[pre];
}
int query(int l,int r,int x,int now)
{
    if(!now||l>x) return 0;
    if(r<=x) return t[now];
    int m=l+r>>1;
    return max(query(l,m,x,ls[now]),query(m+1,r,x,rs[now]));
}
void dfs(int u,int p)
{
    ans[u]=query(1,n,a[u]-1,rt[p])+1;
    Insert(1,n,a[u],ans[u],rt[u],rt[p]);
    for(int i=head[u];i;i=nex[i])
    {
        int v=to[i];if(v==p) continue;
        dfs(v,u);
    }
}
void build(int l,int r,int &now)
{
    now=newnode();
    if(l==r) return;
    int m=l+r>>1;
    build(l,m,ls[now]);build(m+1,r,rs[now]);
}
int main()
{
    scanf("%d",&n);
    for(int i=1;i<=n;i++) scanf("%d",&a[i]),hs[i]=a[i];
    sort(hs+1,hs+1+n);
    for(int i=1;i<=n;i++) a[i]=lower_bound(hs+1,hs+1+n,a[i])-hs;
    build(1,n,rt[0]);
    for(int i=1;i<n;i++)
    {
        int u,v;scanf("%d%d",&u,&v);add(u,v);add(v,u);
    }
    dfs(1,0);
    for(int i=1;i<=n;i++) printf("%d\n",ans[i]);
}