Problem:
Score : 600 points

Problem Statement

We have one slime.

You can set the health of this slime to any integer value of your choice.

A slime reproduces every second by spawning another slime that has strictly less health. You can freely choose the health of each new slime. The first reproduction of our slime will happen in one second.

Determine if it is possible to set the healths of our first slime and the subsequent slimes spawn so that the multiset of the healths of the 2^N slimes that will exist in N seconds equals a multiset S.

Here S is a multiset containing 2^N (possibly duplicated) integers: S_1,~S_2,~...,~S_{2^N}.

Constraints

All values in input are integers.

1 \leq N \leq 18

1 \leq S_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
S_1 S_2 ... S_{2^N}

Output

If it is possible to set the healths of the first slime and the subsequent slimes spawn so that the multiset of the healths of the 2^N slimes that will exist in N seconds equals S, print Yes; otherwise, print No.

Sample Input 1

2
4 2 3 1

Sample Output 1

Yes

We will show one way to make the multiset of the healths of the slimes that will exist in 2 seconds equal to S.

First, set the health of the first slime to 4.

By letting the first slime spawn a slime whose health is 3, the healths of the slimes that exist in 1 second can be 4,~3.

Then, by letting the first slime spawn a slime whose health is 2, and letting the second slime spawn a slime whose health is 1, the healths of the slimes that exist in 2 seconds can be 4,~3,~2,~1, which is equal to S as multisets.

Sample Input 2

2
1 2 3 1

Sample Output 2

Yes

S may contain multiple instances of the same integer.

Sample Input 3

1
1 1

Sample Output 3

No

Sample Input 4

5
4 3 5 3 1 2 7 8 7 4 6 3 7 2 3 6 2 7 3 2 6 7 3 4 6 7 3 4 2 5 2 3

Sample Output 4

No

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.BufferedReader;
import java.io.File;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.PriorityQueue;
import java.util.Random;
import java.util.StringTokenizer;
import java.util.TreeMap;
 
public class Main {
	
	public static void main(String[] args) {
		FastReader scan = new FastReader();
		PrintWriter out = new PrintWriter(System.out);
		Task solver = new Task();
		int t = 1;
		while(t-->0) solver.solve(1, scan, out);
		out.close();
	}
	
	static class Task {
		public void solve(int testNumber, FastReader scan, PrintWriter out) {
			int n = 1 << scan.nextInt();
			int max = 0;
			TreeMap<Integer, Integer> t = new TreeMap<>();
			PriorityQueue<Integer> q = new PriorityQueue<>();
			for(int i = 0; i < n; i++) {
				int a = scan.nextInt();
				max = Math.max(max, a);
				add(t, a);
			}
			remove(t, max);
			q.add(max);
			while(q.size() != n) {
				ArrayList<Integer> res = new ArrayList<>();
				for(int i : q) {
					Integer g = t.lowerKey(i);
					if(g == null) {
						out.println("No");
						return;
					}
					remove(t, g);
					res.add(g);
				}
				for(int i : res) q.add(i);
			}
			out.println("Yes");
		}
		
		static void remove(TreeMap<Integer, Integer> t, int x) {
			int g = t.get(x);
			if(g == 1) t.remove(x);
			else t.put(x, g-1);
		}
		
		static void add(TreeMap<Integer, Integer> t, int x) {
			t.put(x, t.getOrDefault(x, 0)+1);
		}
		
	}
	
	static void shuffle(int[] a) {
		Random get = new Random();
		for(int i = 0; i < a.length; i++) {
			int r = get.nextInt(a.length);
			int temp = a[i];
			a[i] = a[r];
			a[r] = temp;
		}
	}
	
	static void shuffle(long[] a) {
		Random get = new Random();
		for(int i = 0; i < a.length; i++) {
			int r = get.nextInt(a.length);
			long temp = a[i];
			a[i] = a[r];
			a[r] = temp;
		}
	}
	
 
	static class FastReader {
		BufferedReader br;
		StringTokenizer st;
 
		public FastReader() {
			br = new BufferedReader(new InputStreamReader(System.in));
		}
		public FastReader(String s) throws FileNotFoundException {
			br = new BufferedReader(new FileReader(new File(s)));
		}
 
		String next() {
			while (st == null || !st.hasMoreElements()) {
				try {
					st = new StringTokenizer(br.readLine());
				} catch (IOException e) {
					e.printStackTrace();
				}
			}
			return st.nextToken();
		}
 
		int nextInt() {
			return Integer.parseInt(next());
		}
 
		long nextLong() {
			return Long.parseLong(next());
		}
 
		double nextDouble() {
			return Double.parseDouble(next());
		}
 
		String nextLine() {
			String str = "";
			try {
				str = br.readLine();
			} catch (IOException e) {
				e.printStackTrace();
			}
			return str;
		}
	}
 
}