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.*;
import java.util.*;

public class F implements Runnable{
	public static void main (String[] args) {new Thread(null, new F(), "_cf", 1 << 28).start();}

	int n, a[] , b[];
	long[] res;
	ArrayList<Integer>[] adj;
	
	public void run() {
		FastScanner fs = new FastScanner();
		PrintWriter out = new PrintWriter(System.out);
		System.err.println("Go!");

		n = fs.nextInt();
		a = fs.nextIntArray(n);
		b = fs.nextIntArray(n);
		adj = new ArrayList[n];
		for(int i = 0; i < n; i++)
			adj[i] = new ArrayList<>();
		for(int i = 0; i < n - 1; i++) {
			int u = fs.nextInt() - 1;
			int v = fs.nextInt() - 1;
			adj[u].add(v);
			adj[v].add(u);
		}
		
		res = new long[n];
		dfs(0, -1);
		for(int i = 0; i < n; i++)
			out.print(res[i] + " ");

		out.close();
	}
	
	CHO dfs(int u, int p) {
		int kids = 0;
		CHO res = new CHO(-1);
		for(int v : adj[u]) {
			if(v == p) continue;
			kids++;
			CHO ret = dfs(v, u);
			res = smallToBig(res, ret);
		}
		
		if(kids == 0) {
			//leaf
			CHO ret = new CHO(-1);
			ret.add(b[u], 0);
			this.res[u] = 0;
			return ret;
		}
		
		this.res[u] = res.query(a[u]);
		res.add(b[u], this.res[u]);
		return res;
	}
	
	CHO smallToBig(CHO one, CHO two) {
		CHO big = one.size() >= two.size() ? one : two;
		CHO small = one.size() >= two.size() ? two : one;
		for(CHO.Line ll : small.hull)
			big.add(ll);
		return big;
	}
	
	static class CHO {
		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);
			}
		};
		// -1 for min; +1 for max
		public CHO(int typee) { type = typee;  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 mm, long x, long y) { m = mm; b = -m * x + y; }
			public Line(long mm, long bb) { m = mm; b = bb; }
			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);
			}
			public String toString() {return "Slope = " + m + " intercept = " + b;}
		}
	}

	void sort (int[] a) {
		int n = a.length;
		for(int i = 0; i < 50; i++) {
			Random r = new Random();
			int x = r.nextInt(n), y = r.nextInt(n);
			int temp = a[x];
			a[x] = a[y];
			a[y] = temp;
		}
		Arrays.sort(a);
	}

	class FastScanner {
		public int BS = 1<<16;
		public char NC = (char)0;
		byte[] buf = new byte[BS];
		int bId = 0, size = 0;
		char c = NC;
		double num = 1;
		BufferedInputStream in;

		public FastScanner() {
			in = new BufferedInputStream(System.in, BS);
		}

		public FastScanner(String s) throws FileNotFoundException {
			in = new BufferedInputStream(new FileInputStream(new File(s)), BS);
		}

		public char nextChar(){
			while(bId==size) {
				try {
					size = in.read(buf);
				}catch(Exception e) {
					return NC;
				}                
				if(size==-1)return NC;
				bId=0;
			}
			return (char)buf[bId++];
		}

		public int nextInt() {
			return (int)nextLong();
		}

		public long nextLong() {
			num=1;
			boolean neg = false;
			if(c==NC)c=nextChar();
			for(;(c<'0' || c>'9'); c = nextChar()) {
				if(c=='-')neg=true;
			}
			long res = 0;
			for(; c>='0' && c <='9'; c=nextChar()) {
				res = (res<<3)+(res<<1)+c-'0';
				num*=10;
			}
			return neg?-res:res;
		}

		public double nextDouble() {
			double cur = nextLong();
			return c!='.' ? cur:cur+nextLong()/num;
		}

		public String next() {
			StringBuilder res = new StringBuilder();
			while(c<=32)c=nextChar();
			while(c>32) {
				res.append(c);
				c=nextChar();
			}
			return res.toString();
		}

		public String nextLine() {
			StringBuilder res = new StringBuilder();
			while(c<=32)c=nextChar();
			while(c!='\n') {
				res.append(c);
				c=nextChar();
			}
			return res.toString();
		}

		public boolean hasNext() {
			if(c>32)return true;
			while(true) {
				c=nextChar();
				if(c==NC)return false;
				else if(c>32)return true;
			}
		}
		
		public int[] nextIntArray(int n) {
			int[] res = new int[n];
			for(int i = 0; i < n; i++) res[i] = nextInt();
			return res;
		}
		
	}

}