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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define ADD(a, b) a = (a + ll(b)) % mod
#define MUL(a, b) a = (a * ll(b)) % mod
#define MAX(a, b) a = max(a, b)
#define MIN(a, b) a = min(a, b)
#define rep(i, a, b) for(int i = int(a); i < int(b); i++)
#define rer(i, a, b) for(int i = int(a) - 1; i >= int(b); i--)
#define all(a) (a).begin(), (a).end()
#define sz(v) (int)(v).size()
#define pb push_back
#define sec second
#define fst first
#define debug(fmt, ...) Debug(__LINE__, ":", fmt, ##__VA_ARGS__)
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef pair<int, pi> ppi;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<vl> mat;
typedef complex<double> comp;
void Debug() {cerr << '\n'; }
template<class FIRST, class... REST>void Debug(FIRST arg, REST... rest){
	cerr<<arg<<" ";Debug(rest...);}
template<class T>ostream& operator<<(ostream& out,const vector<T>& v) {
	out<<"[";if(!v.empty()){rep(i,0,sz(v)-1)out<<v[i]<<", ";out<<v.back();}out<<"]";return out;}
template<class S, class T>ostream& operator<<(ostream& out,const pair<S, T>& v){
	out<<"("<<v.first<<", "<<v.second<<")";return out;}
const int MAX_N = 500010;
const int MAX_V = 100010;
const double eps = 1e-6;
const ll mod = 1000000007;
const int inf = 1 << 30;
const ll linf = 1LL << 60;
const double PI = 3.14159265358979323846;
mt19937 rng; //use it by rng() % mod, shuffle(all(vec), rng)
///////////////////////////////////////////////////////////////////////////////////////////////////

int N;
int A[MAX_N];
bool used[MAX_N];

void solve() {
	cin >> N;
	rep(i, 0, (1 << N)) cin >> A[i];
	sort(A, A + (1 << N));
	reverse(A, A + (1 << N));
	used[0] = true;

	rep(i, 0, N) {
		deque<int> que;
		rep(j, 0, (1 << N)) {
			if(used[j]) que.push_back(A[j]);
			else {
				if(!que.empty() && A[j] < que.front()) {
					// debug(i, j);
					que.pop_front();
					used[j] = true;
				}
			}
		}
	}
	if(accumulate(used, used + (1 << N), 0) == (1 << N)) cout << "Yes\n";
	else cout << "No\n";
}

uint32_t rd() {
	uint32_t res;
#ifdef __MINGW32__
	asm volatile("rdrand %0" :"=a"(res) ::"cc");
#else
	res = std::random_device()();
#endif
	return res;
}

int main() {
#ifndef LOCAL
	ios::sync_with_stdio(false);
    cin.tie(0);
#endif
    cout << fixed;
	cout.precision(20);
    cerr << fixed;
	cerr.precision(6);
	rng.seed(rd());
#ifdef LOCAL
	//freopen("in.txt", "wt", stdout); //for tester
	if(!freopen("in.txt", "rt", stdin)) return 1;
#endif	
	solve();
    cerr << "Time: " << 1.0 * clock() / CLOCKS_PER_SEC << " s.\n";
	return 0;
}