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.util.*;
import java.io.*;

public class f {
	
	static int n;
	static long[] as, bs, ans;
	static ArrayList<Integer>[] adjs;
	
	public static void main(String[] args) throws IOException {
		FastScanner in = new FastScanner(System.in);
		PrintWriter out = new PrintWriter(System.out);
		
		n = in.nextInt();
		as = new long[n];
		bs = new long[n];
		ans = new long[n];
		adjs = new ArrayList[n];
		for (int i = 0; i < n; i++) {
			as[i] = in.nextLong();
			adjs[i] = new ArrayList<>();
		}
		for (int i = 0; i < n; i++)
			bs[i] = in.nextLong();
		for (int i = 0; i < n - 1; i++) {
			int u = in.nextInt() - 1;
			int v = in.nextInt() - 1;
			adjs[u].add(v);
			adjs[v].add(u);
		}
		
		go(0, -1);
		
		for (long a : ans)
			out.print(a + " ");
		out.println();
		
		out.close();
	}
	
	static CHT go(int cur, int par) {
		CHT cht = new CHT(-1);
		boolean isLeaf = true;
		for (int next : adjs[cur]) {
			if (next == par) continue;
			isLeaf = false;
			CHT other = go(next, cur);
			if (other.hull.size() > cht.size()) {
				CHT tmp = other;
				other = cht;
				cht = tmp;
			}
			for (CHT.Line l : other.hull)
				cht.add(l);
		}
		if (!isLeaf) {
			ans[cur] = cht.query(as[cur]);
		}
		cht.add(bs[cur], ans[cur]);
		return cht;
	}
	
	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 FastScanner {
		BufferedReader br;
		StringTokenizer st;
		
		public FastScanner(InputStream i) {
			br = new BufferedReader(new InputStreamReader(i));
			st = new StringTokenizer("");
		}
				
		public String next() throws IOException {
			if(st.hasMoreTokens())
				return st.nextToken();
			else
				st = new StringTokenizer(br.readLine());
			return next();
		}

		public int nextInt() throws IOException {
			return Integer.parseInt(next());
		}
		public long nextLong() throws IOException {
			return Long.parseLong(next());
		}
		public double nextDouble() throws IOException {
			return Double.parseDouble(next());
		}
	}
}