You are given a tree with n nodes (numbered from 1 to n) rooted at node 1. Also, each node has two values associated with it. The values for i-th node are ai and bi.

You can jump from a node to any node in its subtree. The cost of one jump from node x to node y is the product of ax and by. The total cost of a path formed by one or more jumps is sum of costs of individual jumps. For every node, calculate the minimum total cost to reach any leaf from that node. Pay attention, that root can never be leaf, even if it has degree 1.

Note that you cannot jump from a node to itself.

Input

The first line of input contains an integer n (2 ≤ n ≤ 105) — the number of nodes in the tree.

The second line contains n space-separated integers a1, a2, ..., an( - 105 ≤ ai ≤ 105).

The third line contains n space-separated integers b1, b2, ..., bn( - 105 ≤ bi ≤ 105).

Next n - 1 lines contains two space-separated integers ui and vi (1 ≤ ui, vi ≤ n) describing edge between nodes ui and vi in the tree.

Output

Output n space-separated integers, i-th of which denotes the minimum cost of a path from node i to reach any leaf.

Examples

Input

3
2 10 -1
7 -7 5
2 3
2 1


Output

10 50 0 

Input

4
5 -10 5 7
-8 -80 -3 -10
2 1
2 4
1 3


Output

-300 100 0 0 

Note

In the first example, node 3 is already a leaf, so the cost is 0. For node 2, jump to node 3 with cost a2 × b3 = 50. For node 1, jump directly to node 3 with cost a1 × b3 = 10.

In the second example, node 3 and node 4 are leaves, so the cost is 0. For node 2, jump to node 4 with cost a2 × b4 = 100. For node 1, jump to node 2 with cost a1 × b2 = - 400 followed by a jump from 2 to 4 with cost a2 × b4 = 100.
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.StringTokenizer;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Comparator;
import java.util.TreeSet;
public class F932 
{
    
static class CHT 
{

    TreeSet<Line> hull;
    int type;
    boolean query = false;
    Comparator<Line> comp = new Comparator<Line>() 
    {
        public int compare(Line a, Line b) 
        {
            if (!query) 
            {
                return type * Long.compare(a.m, b.m);
            }
            if (a.left == b.left) 
            {
                return Long.compare(a.m, b.m);
            }
            return Double.compare(a.left, b.left);
        }
    };

    public CHT(final int type) 
    {
        this.type = type;
        hull = new TreeSet<>(comp);
    }

    public void add(long m, long b) 
    {
        add(new Line(m, b));
    }

    public void add(Line a) 
    {
        Line[] LR = {hull.lower(a), hull.ceiling(a)};
        for (int i = 0; i < 2; i++) 
        {
            if (LR[i] != null && LR[i].m == a.m) {
                if (type == 1 && LR[i].b >= a.b) {
                    return;
                }
                if (type == -1 && LR[i].b <= a.b) {
                    return;
                }
                remove(LR[i]);
            }
        }

        hull.add(a);
        Line L = hull.lower(a), R = hull.higher(a);
        if (L != null && R != null && a.inter(R) <= R.left) 
        {
            hull.remove(a);
            return;
        }
        Line LL = (L != null) ? hull.lower(L) : null;
        Line RR = (R != null) ? hull.higher(R) : null;
        if (L != null) 
        {
            a.left = a.inter(L);
        }
        if (R != null) 
        {
            R.left = a.inter(R);
        }
        while (LL != null && L.left >= a.inter(LL)) 
        {
            remove(L);
            a.left = a.inter(L = LL);
            LL = hull.lower(L);
        }
        while (RR != null && R.inter(RR) <= a.inter(RR)) 
        {
            remove(R);
            RR.left = a.inter(R = RR);
            RR = hull.higher(R);
        }
    }

    public long query(long x) 
    {
        Line temp = new Line(0, 0, 0);
        temp.left = x;
        query = true;
        long ans = (long) hull.floor(temp).eval(x);
        query = false;
        return ans;
    }

    private void remove(Line x) 
    {
        hull.remove(x);
    }

    public int size() 
    {
        return hull.size();
    }

    static class Line 
    {

        long m, b;
        double left = Long.MIN_VALUE;

        public Line(long m, long x, long y) 
        {
            this.m = m;
            this.b = -m * x + y;
        }

        public Line(long m, long b) 
        {
            this.m = m;
            this.b = b;
        }

        public long eval(long x) 
        {
            return m * x + b;
        }

        public double inter(Line x) 
        {
            return (double) (x.b - this.b) / (double) (this.m - x.m);
        }
    }
}

    static class Scanner
    {
        BufferedReader br;
        StringTokenizer tk=new StringTokenizer("");
        public Scanner(InputStream is) 
        {
            br=new BufferedReader(new InputStreamReader(is));
        }
        public int nextInt() throws IOException
        {
            if(tk.hasMoreTokens())
                return Integer.parseInt(tk.nextToken());
            tk=new StringTokenizer(br.readLine());
            return nextInt();
        }
        public long nextLong() throws IOException
        {
            if(tk.hasMoreTokens())
                return Long.parseLong(tk.nextToken());
            tk=new StringTokenizer(br.readLine());
            return nextLong();
        }
        public String next() throws IOException
        {
            if(tk.hasMoreTokens())
                return (tk.nextToken());
            tk=new StringTokenizer(br.readLine());
            return next();
        }
        public String nextLine() throws IOException
        {
            tk=new StringTokenizer("");
            return br.readLine();
        }
        public double nextDouble() throws IOException
        {
            if(tk.hasMoreTokens())
                return Double.parseDouble(tk.nextToken());
            tk=new StringTokenizer(br.readLine());
            return nextDouble();
        }
        public char nextChar() throws IOException
        {
            if(tk.hasMoreTokens())
                return (tk.nextToken().charAt(0));
            tk=new StringTokenizer(br.readLine());
            return nextChar();
        }
        public int[] nextIntArray(int n) throws IOException
        {
            int a[]=new int[n];
            for(int i=0;i<n;i++)
                a[i]=nextInt();
            return a;
        }
        public long[] nextLongArray(int n) throws IOException
        {
            long a[]=new long[n];
            for(int i=0;i<n;i++)
                a[i]=nextLong();
            return a;
        }
        public int[] nextIntArrayOneBased(int n) throws IOException
        {
            int a[]=new int[n+1];
            for(int i=1;i<=n;i++)
                a[i]=nextInt();
            return a;
        }
        public long[] nextLongArrayOneBased(int n) throws IOException
        {
            long a[]=new long[n+1];
            for(int i=1;i<=n;i++)
                a[i]=nextLong();
            return a;
        }
    
    
    }
    
    public static void main(String args[]) throws IOException
    {
        new Thread(null, new Runnable() {
            public void run() {
                try
                {
                    solve();
                }
                catch(Exception e)
                {
                    
                }
            }
        }, "1", 1 << 26).start();
        
    }
    
    static int a[],b[];
    static ArrayList<Integer> g[];
    static long ans[];
    static CHT dfs(int parent,int node)
    {
        CHT c=new CHT(-1);
        for(int i=0;i<g[node].size();i++)
        {
            int v=g[node].get(i);
            if(v==parent)
                continue;
            CHT x=dfs(node,v);
            for(CHT.Line l:x.hull)
                c.add(l);
        }
        if(!c.hull.isEmpty())
            ans[node]=c.query(a[node]);
        c.add(b[node],ans[node]);
        return c;
    }
    static void solve() throws IOException
    {
        Scanner in=new Scanner(System.in);
        PrintWriter out=new PrintWriter(System.out);
        int n=in.nextInt();
        a=in.nextIntArrayOneBased(n);
        b=in.nextIntArrayOneBased(n);
        ans=new long[n+1];
        g=new ArrayList[n+1];
        for(int i=1;i<=n;i++)
            g[i]=new ArrayList<>();
        for(int i=1;i<n;i++)
        {
            int u=in.nextInt(),v=in.nextInt();
            g[u].add(v);
            g[v].add(u);
        }
        dfs(0,1);
        for(int i=1;i<=n;i++)
            out.print(ans[i]+" ");
        out.println();
        out.close();
    }

}
