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: Time Limit Exceeded)

import java.util.*;

import java.io.*;
import java.nio.charset.StandardCharsets;

public class Main {
	public static void main(String[] args) throws IOException {
		InputStreamReader reader = new InputStreamReader(System.in, StandardCharsets.UTF_8);
		BufferedReader in = new BufferedReader(reader);

		int N = Integer.parseInt(in.readLine());
		String[] tokens = in.readLine().split(" ");
		Long[] S = new Long[tokens.length];
		for (int i = 0; i < tokens.length; ++i) {
			S[i] = Long.parseLong(tokens[i]);
		}
		Arrays.sort(S, new Comparator<Long>() {
			@Override
			public int compare(Long o1, Long o2) {
				return -1 * Long.compare(o1, o2);
			}
		});
		// baseとなるqueを作成
		Deque<Long> que = new ArrayDeque<>(S.length);
		for (int i = 0; i < S.length; ++i) {
			que.addLast(S[i]);
		}
		Long[] array = new Long[1];
		array[0] = que.pollFirst();
		for (int i = 0; i < N; ++i) {
			array = step(array, que);
			if (array == null) {
				break;
			}
		}
		// System.out.println(toStr(array));
		// arrayの検証
		/*
		for (int i = 0; i < N; ++i) {
			array = revStep(array);
			if (array == null) {
				break;
			}
		}*/
		if (array != null) {
			System.out.println("Yes");
		} else {
			System.out.println("No");
		}
	}

	static Long[] revStep(Long[] base) {
		// 条件チェック
		for (int i = 0; i < base.length; i += 2) {
			if (base[i] <= base[i + 1]) {
				return null;
			}
		}
		Long[] nexts = new Long[base.length / 2];
		for (int i = 0; i < nexts.length; ++i) {
			nexts[i] = base[i * 2];
		}

		return nexts;
	}

	static Long[] step(Long[] base, Deque<Long> que) {
		Long[] nexts = new Long[base.length * 2];
		// base要素のコピー
		for (int i = 0; i < base.length; ++i) {
			nexts[i * 2] = base[i];
		}
		// 追加
		for (int i = 1; i < nexts.length; i += 2) {
			Long val = search(que, nexts[i - 1]);
			if (val < 0) {
				return null;
			}
			nexts[i] = val;
		}
		return nexts;
	}

	/*
	 * 
	 */
	static Long search(Deque<Long> que, Long val) {
		Deque<Long> temp = new ArrayDeque<>(que.size());
		Long result = -1L;
		while (!que.isEmpty()) {
			if (que.peekFirst() < val) {
				result = que.pollFirst();
				break;
			} else {
				temp.addLast(que.pollFirst());
			}
		}
		// 残りのものを戻す
		while (!temp.isEmpty()) {
			que.addFirst(temp.pollLast());
		}
		return result;
	}

	static String toStr(Long[] array) {
		StringBuilder strBuilder = new StringBuilder();
		for (int i = 0; i < array.length; ++i) {
			strBuilder.append(" " + array[i]);
		}
		return strBuilder.toString();
	}
}