Problem: A positive integer x is called a power of two if it can be represented as x = 2^y, where y is a non-negative integer. So, the powers of two are 1, 2, 4, 8, 16, ....

You are given two positive integers n and k. Your task is to represent n as the sum of exactly k powers of two.

Input

The only line of the input contains two integers n and k (1 ≤ n ≤ 10^9, 1 ≤ k ≤ 2 ⋅ 10^5).

Output

If it is impossible to represent n as the sum of k powers of two, print NO.

Otherwise, print YES, and then print k positive integers b_1, b_2, ..., b_k such that each of b_i is a power of two, and ∑ _{i = 1}^{k} b_i = n. If there are multiple answers, you may print any of them.

Examples

Input


9 4


Output


YES
1 2 2 4 


Input


8 1


Output


YES
8 


Input


5 1


Output


NO


Input


3 7


Output


NO

Solution (in JAVA):

Solution:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.NoSuchElementException;
import java.util.PriorityQueue;
import java.util.Random;
import java.util.Set;
import java.util.Stack;
import java.util.StringTokenizer;
import java.util.TreeMap;
import java.util.TreeSet;

// CFPS -> CodeForcesProblemSet
public final class CFPS {
	static FastReader fr = new FastReader();
	static PrintWriter out = new PrintWriter(System.out);
	static final int gigamod = 1000000007;
	static int t = 1;
	public static void main(String[] args) {
		OUTER: 
		for (int tc = 0; tc < t; tc++) {
			
			long n = fr.nextLong();
			long ncpy = n;
			int k = fr.nextInt();
			
			// What is the task?
			// A: It is to represent n as exactly k powers of two.
			if (k > n) {
				out.println("NO");
				continue OUTER;
			}
			
			// How will we solve the problem?
			// A: Represent n as powers of two greedily and then 
			// we can split the powers to get to 'k' powers.
			
			PriorityQueue<Long> powers = new PriorityQueue<>(Collections.reverseOrder());
			while (ncpy != 0) {
				long power = largestFittingPowerOf2(ncpy);
				ncpy -= power;
				powers.add(power);
			}
			
			if (k < powers.size()) {
				out.println("NO");
				continue OUTER;
			}
			
			int numSplits = k - powers.size();
			
			for (int i = 0; i < numSplits; i++) {
				long maxPow = powers.remove();
				
				if (maxPow == 1) {
					out.println("NO");
					continue OUTER;
				}
				
				powers.add(maxPow / 2);
				powers.add(maxPow / 2);
			}
			
			out.println("YES");
			for (long pow : powers)
				out.print(pow + " ");
			out.println();
		}
		out.close();
	}
	
	// Returns the binary string of length at least bits.
	static String toBinaryString(int num, int bits) {
		StringBuilder sb = new StringBuilder(Long.toBinaryString(num));
		sb.reverse();
		for (int i = sb.length(); i < bits; i++)
			sb.append(0);
		return sb.reverse().toString();
	}
	
	static class CountMap extends HashMap<Long, Integer>{

		public Integer put(Long key) {
			if (super.containsKey(key)) {
				return super.put(key, super.get(key) + 1);
			} else {
				return super.put(key, 1);
			}
		}
		
		public Integer put(int key) {
			if (super.containsKey((long) key)) {
				return super.put((long) key, super.get((long) key) + 1);
			} else {
				return super.put((long) key, 1);
			}
		}
		
		public Integer remove(Long key) {
			Integer count = super.get(key);
			if (count == null) return -1;
			if (count == 1)
				return super.remove(key);
			else
				return super.put(key, super.get(key) - 1);
		}
		
		public Integer remove(int key) {
			Integer count = super.get((long) key);
			if (count == null) return -1;
			if (count == 1)
				return super.remove((long) key);
			else
				return super.put((long) key, super.get((long) key) - 1);
		}
	}
	
	// Checks if s has subsequence t.
	static boolean hasSubsequence(String s, String t) {
		
		char[] schars = s.toCharArray();
		char[] tchars = t.toCharArray();
		int slen = schars.length, tlen = tchars.length;
		int tctr = 0;
		
		if (slen < tlen) return false;
		
		for (int i = 0; i < slen || i < tlen; i++) {
			if (tctr == tlen) break;
			if (schars[i] == tchars[tctr]) {
				tctr++;
			}
		}
		
		if (tctr == tlen) return true;
		
		return false;
	}
	
	static class Point implements Comparable<Point> {
		long x;
		long y;
		int id;
		
		Point() {
			x = y = id = 0;
		}
		
		Point(Point p) {
			this.x = p.x;
			this.y = p.y;
			this.id = p.id;
		}
		
		Point(long a, long b, int id) {
			this.x = a;
			this.y = b;
			this.id = id;
		}
		
		@Override
		public int compareTo(Point o) {
			if (this.x > o.x)
				return 1;
			if (this.x < o.x)
				return -1;
			if (this.y > o.y)
				return 1;
			if (this.y < o.y)
				return -1;
			return 0;
		}
	}
	
	static class PointComparator implements Comparator<Point> {
		@Override
		public int compare(Point o1, Point o2) {
			if (o1.y > o2.y)
				return 1;
			if (o1.y < o2.y)
				return -1;
			if (o1.x > o2.x)
				return 1;
			if (o1.x < o2.x)
				return -1;
			return 0;
		}
	}
	
	// Returns the largest power of k that fits into n.
	static int largestFittingPower(long n, long k) {
		int lo = 0, hi = log2(Long.MAX_VALUE);
		int largestPower = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			
			long val = (long) Math.pow(k, mid);
			if (val <= n) {
				largestPower = mid;
				lo = mid + 1;
			} else {
				hi = mid - 1;
			}
		}
		return largestPower;
	}
	
	static long largestFittingPowerOf2(long n) {
		return Long.highestOneBit(n);
	}
	
	static String bitSetToString(int set) {
		// We have to print all the elements that are present
		// in the set.
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < 30; i++) {
			if (((set >> i) & 1) == 1) {
				// The 'i'th bit is on meaning that the element 'i' is 
				// present in the set.
				sb.append((i + 1) + " ");
			}
		}
		sb.append("\n");
		return sb.toString();
	}
	
	static String displayBitSet(long set) {
		// We have to print all the elements that are present
		// in the set.
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < 60; i++) {
			if (((set >> i) & 1) == 1) {
				// The 'i'th bit is on meaning that the element 'i' is 
				// present in the set.
				sb.append((i + 1) + " ");
			}
		}
		sb.append("\n");
		return sb.toString();
	}
	
	static int addToBitSet(int set, int element) {
		set = (set) | (1 << (element - 1));
		return set;
	}
	
	static int removeFromBitSet(int set, int element) {
		// Checking whether the bit is present.
		if ((set & (1 << (element - 1))) == 0) return set;
		set = set ^ (1 << (element - 1));
		return set;
	}
	
	// Returns map of factor and its power in the number.
	static TreeMap<Long, Integer> primeFactorization(long num) {
		TreeMap<Long, Integer> map = new TreeMap<>();
		while (num % 2 == 0) {
			num /= 2;
			Integer pwrCnt = map.get(2L);
			map.put(2L, pwrCnt != null ? pwrCnt + 1 : 1);
		}
		
		for (long i = 3; i * i <= num; i += 2) {
			while (num % i == 0) {
				num /= i;
				Integer pwrCnt = map.get(i);
				map.put(i, pwrCnt != null ? pwrCnt + 1 : 1);
			}
		}
		
		// If the number is prime, we have to add it to the 
		// map.
		if (num != 1)
			map.put(num, 1);
		
		return map;
	}
	
	// Returns map of factor and its power in the number.
	static TreeMap<Integer, Integer> primeFactorization(int num) {
		TreeMap<Integer, Integer> map = new TreeMap<>();
		while (num % 2 == 0) {
			num /= 2;
			Integer pwrCnt = map.get(2);
			map.put(2, pwrCnt != null ? pwrCnt + 1 : 1);
		}
		
		for (int i = 3; i * i <= num; i += 2) {
			while (num % i == 0) {
				num /= i;
				Integer pwrCnt = map.get(i);
				map.put(i, pwrCnt != null ? pwrCnt + 1 : 1);
			}
		}
		
		// If the number is prime, we have to add it to the 
		// map.
		if (num != 1)
			map.put(num, 1);
		
		return map;
	}
	
	static Set<Long> divisors(long num) {
		Set<Long> divisors = new TreeSet<Long>();
		divisors.add(1L);
		divisors.add(num);
		
		for (long i = 2; i * i <= num; i++) {
			if (num % i == 0) {
				divisors.add(num/i);
				divisors.add(i);
			}
		}
		return divisors;
	}
	
	static void dfs(int node, boolean[] marked, ArrayList<Integer>[] adj) {
		if (marked[node]) return;

		marked[node] = true;
		for (int adjc : adj[node])
			dfs(adjc, marked, adj);
	}
	
	// Returns the index of the first element
	// larger than or equal to val.
	static int bsearch(int[] arr, int val, int lo, int hi) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] >= val) {
				idx = mid;
				hi = mid - 1;
			} else
				lo = mid + 1;
		}
		return idx;
	}
	
	static int bsearch(long[] arr, long val, int lo, int hi) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] >= val) {
				idx = mid;
				hi = mid - 1;
			} else
				lo = mid + 1;
		}
		return idx;
	}
	
	// Returns the index of the last element 
	// smaller than or equal to val.
	static int bsearch(long[] arr, long val, int lo, int hi, boolean sMode) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] > val) {
				hi = mid - 1;
			} else {
				idx = mid;
				lo = mid + 1;
			}
		}
		return idx;
	}
	
	static int bsearch(int[] arr, long val, int lo, int hi, boolean sMode) {
		int idx = -1;
		while (lo <= hi) {
			int mid = lo + (hi - lo)/2;
			if (arr[mid] > val) {
				hi = mid - 1;
			} else {
				idx = mid;
				lo = mid + 1;
			}
		}
		return idx;
	}
	
	static int factorial(int n) {
		if (n <= 1)
			return 1;
		return n * factorial(n - 1);
	}
	
	static long factorial(long n) {
		if (n <= 1)
			return 1;
		long factorial = 1;
		for (int i = 1; i <= n; i++)
			factorial = mod(factorial * i);
		return factorial;
	}
	
	static long factorialInDivision(long a, long b) {
		if (a == b)
			return 1;
		if (b < a) {
			long temp = a;
			a = b;
			b = temp;
		}
		
		long factorial = 1;
		for (long i = a + 1; i <= b; i++)
			factorial = mod(factorial * i);
		return factorial;
	}
	
	static BigInteger factorialInDivision(BigInteger a, BigInteger b) {
		if (a.equals(b))
			return BigInteger.ONE;
		return a.multiply(factorialInDivision(a.subtract(BigInteger.ONE), b));
	}
	
	static long nCr(long n, long r)
	{
		long p = gigamod;
	    // Base case
	    if (r == 0)
	        return 1;
	 
	    // Fill factorial array so that we
	    // can find all factorial of r, n
	    // and n-r
	    long fac[] = new long[(int)n + 1];
	    fac[0] = 1;
	    for (int i = 1; i <= n; i++)
	        fac[i] = fac[i - 1] * i % p;
	 
	    return (fac[(int)n] * modInverse(fac[(int)r], p) % p
	                    * modInverse(fac[(int)n - (int)r], p) % p) % p;
	}
	
	static long modInverse(long n, long p)
	{
	    return power(n, p - 2, p);
	}
	
	static long power(long x, long y, long p)
	{
	    long res = 1; // Initialize result
	 
	    x = x % p; // Update x if it is more than or
	    // equal to p
	 
	    while (y > 0)
	    {
	        // If y is odd, multiply x with result
	        if ((y & 1)==1)
	            res = (res * x) % p;
	 
	        // y must be even now
	        y = y >> 1; // y = y/2
	        x = (x * x) % p;
	    }
	    return res;
	}
	
	static long nPr(long n, long r) {
		return factorialInDivision(n, n - r);
	}
	
	static int log2(long n) {
		return (int)(Math.log(n) / Math.log(2));
	}
	
	static int logk(long n, long k) {
		return (int)(Math.log(n) / Math.log(k));
	}
	
	// Sieve of Eratosthenes: 
	static boolean[] primeGenerator(int upto) {
		boolean[] isPrime = new boolean[upto + 1];
		Arrays.fill(isPrime, true);
		isPrime[1] = isPrime[0] = false;

		for (long i = 2; i * i < upto + 1; i++)
			if (isPrime[(int) i])
				// Mark all the multiples greater than or equal
				// to the square of i to be false.
				for (long j = i; j * i < upto + 1; j++)
					isPrime[(int) j * (int) i] = false;
		return isPrime;
	}

	static int gcd(int a, int b) {
		if (b == 0) {
			return a;
		} else {
			return gcd(b, a % b);
		}
	}
	
	static long gcd(long a, long b) {
		if (b == 0) {
			return a;
		} else {
			return gcd(b, a % b);
		}
	}
	
	static int gcd(int[] arr) {
		int n = arr.length;
		int gcd = arr[0];
		for (int i = 1; i < n; i++) {
			gcd = gcd(gcd, arr[i]);
		}
		return gcd;
	}
	
	static long gcd(long[] arr) {
		int n = arr.length;
		long gcd = arr[0];
		for (int i = 1; i < n; i++) {
			gcd = gcd(gcd, arr[i]);
		}
		return gcd;
	}
	
	static long lcm(int[] arr) {
		int lcm = arr[0];
		int n = arr.length;
		for (int i = 1; i < n; i++) {
			lcm = (lcm * arr[i]) / gcd(lcm, arr[i]);
		}
		return lcm;
	}
	
	static long lcm(long[] arr) {
		long lcm = arr[0];
		int n = arr.length;
		for (int i = 1; i < n; i++) {
			lcm = (lcm * arr[i]) / gcd(lcm, arr[i]);
		}
		return lcm;
	}
	
	static long lcm(int a, int b) {
		return (a * b)/gcd(a, b);
	}
	
	static long lcm(long a, long b) {
		return (a * b)/gcd(a, b);
	}
	
	static boolean less(int a, int b) {
		return a < b ? true : false;
	}
	
	static boolean isSorted(int[] a) {
		for (int i = 1; i < a.length; i++) {
			if (less(a[i], a[i - 1]))
				return false;
		}
		return true;
	}
	
	static boolean isSorted(long[] a) {
		for (int i = 1; i < a.length; i++) {
			if (a[i] < a[i - 1])
				return false;
		}
		return true;
	}

	static void swap(int a, int b) {
		int temp = a;
		a = b;
		b = temp;
	}
	
	static void swap(long a, long b) {
		long temp = a;
		a = b;
		b = temp;
	}
	
	static void swap(double a, double b) {
		double temp = a;
		a = b;
		b = temp;
	}
	
	static void swap(int[] a, int i, int j) {
		int temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(long[] a, int i, int j) {
		long temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(double[] a, int i, int j) {
		double temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void swap(char[] a, int i, int j) {
		char temp = a[i];
		a[i] = a[j];
		a[j] = temp;
	}
	
	static void sort(int[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(char[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(long[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void sort(double[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
	}
	
	static void reverseSort(int[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(char[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(long[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void reverseSort(double[] arr) {
		shuffleArray(arr, 0, arr.length - 1);
		Arrays.sort(arr);
		int n = arr.length;
		for (int i = 0; i < n/2; i++)
			swap(arr, i, n - 1 - i);
	}
	
	static void shuffleArray(long[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			long tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static void shuffleArray(int[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			int tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static void shuffleArray(double[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			double tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	private static void shuffleArray(char[] arr, int startPos, int endPos) {
		Random rnd = new Random();
		for (int i = startPos; i < endPos; ++i) {
			char tmp = arr[i];
			int randomPos = i + rnd.nextInt(endPos - i);
			arr[i] = arr[randomPos];
			arr[randomPos] = tmp;
		}
	}
	
	static boolean isPrime(int n) {
		if (n <= 1)
			return false;
		if (n <= 3)
			return true;

		if (n % 2 == 0 || n % 3 == 0)
			return false;

		for (int i = 5; i * i <= n; i = i + 6)
			if (n % i == 0 || n % (i + 2) == 0)
				return false;

		return true;
	}

	static String toString(int[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(boolean[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(long[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + " ");
		return sb.toString();
	}
	
	static String toString(char[] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++)
			sb.append(dp[i] + "");
		return sb.toString();
	}
	
	static String toString(int[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(long[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(double[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static String toString(char[][] dp) {
		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < dp.length; i++) {
			for (int j = 0; j < dp[i].length; j++) {
				sb.append(dp[i][j] + " ");
			}
			sb.append('\n');
		}
		return sb.toString();
	}
	
	static char toChar(int i) {
		return (char) (i + 48);
	}
	
	static long mod(long a, long m) {
	    return (a%m + m) % m;
	}
	
	static long mod(long num) {
		return (num % gigamod + gigamod) % gigamod;
	}
	
	// Uses weighted quick-union with path compression.
	static class UnionFind {
	    private static int[] parent;  // parent[i] = parent of i
	    private static int[] size;    // size[i] = number of sites in tree rooted at i
	                           // Note: not necessarily correct if i is not a root node
	    private int count;     // number of components

	    public UnionFind(int n) {
	        count = n;
	        parent = new int[n];
	        size = new int[n];
	        for (int i = 0; i < n; i++) {
	            parent[i] = i;
	            size[i] = 1;
	        }
	    }

	    // Number of connected components.
	    public int count() {
	        return count;
	    }
	  
	    // Find the root of p.
	    public int find(int p) {
	        int root = p;
	        while (root != parent[root])
	            root = parent[root];
	        while (p != root) {
	            int newp = parent[p];
	            parent[p] = root;
	            p = newp;
	        }
	        return root;
	    }

	    public boolean connected(int p, int q) {
	        return find(p) == find(q);
	    }
	    
	    public int numConnectedTo(int node) {
	    	return size[find(node)];
	    }
	    
	    // Weighted union.
	    public void union(int p, int q) {
	        int rootP = find(p);
	        int rootQ = find(q);
	        
	        if (rootP == rootQ) return;

	        // make smaller root point to larger one
	        if (size[rootP] < size[rootQ]) {
	            parent[rootP] = rootQ;
	            size[rootQ] += size[rootP];
	        }
	        else {
	            parent[rootQ] = rootP;
	            size[rootP] += size[rootQ];
	        }
	        count--;
	    }
	}
	
	static class UGraph {
		// Adjacency list.
		private HashSet<Integer>[] adj;
		private static final String NEWLINE = "\n";
		private int E;
		
		public UGraph(int V) {
			adj = (HashSet<Integer>[]) new HashSet[V];
			E = 0;
			for (int i = 0; i < V; i++)
				adj[i] = new HashSet<Integer>();
		}
		
		public void addEdge(int from, int to) {
			if (adj[from].contains(to)) return;
			E++;
			adj[from].add(to);
			adj[to].add(from);
		}
		
		public HashSet<Integer> adj(int from) {
			return adj[from];
		}
		
		public int V() {
			return adj.length;
		}
		
		public int E() {
			return E;
		}
		
		public String toString() {
			StringBuilder s = new StringBuilder();
			s.append(V() + " vertices, " + E() + " edges " + NEWLINE);
			for (int v = 0; v < V(); v++) {
				s.append(v + ": ");
				for (int w : adj[v]) {
					s.append(w + " ");
				}
				s.append(NEWLINE);
			}
			return s.toString();
		}
		
		public static void dfsMark(int source, boolean[] marked, UGraph g) {
			if (marked[source]) return;
			
			marked[source] = true;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, marked, g);
		}
		
		public static void bfsOrder(int source, UGraph g) {
		}
		
		private static void dfsMark(int source, int[] colorIds, int color, UGraph g) {
			if (colorIds[source] != -1) return;
			
			colorIds[source] = color;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, colorIds, color, g);
		}
		
		public static int[] connectedComponents(UGraph g) {
			int n = g.V();
			int[] componentId = new int[n];
			Arrays.fill(componentId, -1);
			int colorCtr = 0;
			for (int i = 0; i < n; i++) {
				if (componentId[i] != -1) continue;
				dfsMark(i, componentId, colorCtr, g);
				colorCtr++;
			}
			
			return componentId;
		}
	}
	
	static class Digraph {
		// Adjacency list.
		private HashSet<Integer>[] adj;
		private static final String NEWLINE = "\n";
		private int E;
		
		public Digraph(int V) {
			adj = (HashSet<Integer>[]) new HashSet[V];
			E = 0;
			for (int i = 0; i < V; i++)
				adj[i] = new HashSet<Integer>();
		}
		
		public void addEdge(int from, int to) {
			if (adj[from].contains(to)) return;
			E++;
			adj[from].add(to);
		}
		
		public Iterable<Integer> adj(int from) {
			return adj[from];
		}
		
		public int V() {
			return adj.length;
		}
		
		public int E() {
			return E;
		}
		
		public String toString() {
			StringBuilder s = new StringBuilder();
			s.append(V() + " vertices, " + E() + " edges " + NEWLINE);
			for (int v = 0; v < V(); v++) {
				s.append(v + ": ");
				for (int w : adj[v]) {
					s.append(w + " ");
				}
				s.append(NEWLINE);
			}
			return s.toString();
		}
		
		public static void dfsMark(int source, boolean[] marked, Digraph g) {
			if (marked[source]) return;
			
			marked[source] = true;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, marked, g);
		}
		
		public static void bfsOrder(int source, Digraph g) {
		}
		
		private static void dfsMark(int source, int[] colorIds, int color, Digraph g) {
			if (colorIds[source] != -1) return;
			
			colorIds[source] = color;
			Iterable<Integer> adj = g.adj(source);
			for (int adjc : adj)
				dfsMark(adjc, colorIds, color, g);
		}
		
		public static int[] connectedComponents(Digraph g) {
			int n = g.V();
			int[] componentId = new int[n];
			Arrays.fill(componentId, -1);
			int colorCtr = 0;
			for (int i = 0; i < n; i++) {
				if (componentId[i] != -1) continue;
				dfsMark(i, componentId, colorCtr, g);
				colorCtr++;
			}
			
			return componentId;
		}
		
		public static Stack<Integer> topologicalSort(Digraph dg)
		{
			// dg has to be a directed acyclic graph.
			// We'll have to run dfs on the digraph and push the deepest nodes on stack first.
			// We'll need a Stack<Integer> and a int[] marked.
			Stack<Integer> topologicalStack = new Stack<Integer>();
			boolean[] marked = new boolean[dg.V()];
			
			// Calling dfs
			for (int i = 0; i < dg.V(); i++)
			{
				if (!marked[i]) runDfs(dg, topologicalStack, marked, i);
			}
			
			return topologicalStack;
		}
		
		static void runDfs(Digraph dg, Stack<Integer> topologicalStack, boolean[] marked, int source)
		{
			marked[source] = true;
			for (Integer adjVertex : dg.adj(source))
			{
				if (!marked[adjVertex]) runDfs(dg, topologicalStack, marked, adjVertex);
			}
			topologicalStack.add(source);
		}
	}
	
	static class FastReader {
		private BufferedReader bfr;
		private StringTokenizer st;

		public FastReader() {
			bfr = new BufferedReader(new InputStreamReader(System.in));
		}

		String next() {
			if (st == null || !st.hasMoreElements()) {
				try {
					st = new StringTokenizer(bfr.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());
		}

		char nextChar() {
			return next().toCharArray()[0];
		}

		String nextString() {
			return next();
		}

		int[] nextIntArray(int n) {
			int[] arr = new int[n];
			for (int i = 0; i < n; i++)
				arr[i] = nextInt();
			return arr;
		}
		
		int[] nextOneIntArray(int n) {
			int[] arr = new int[n + 1];
			for (int i = 1; i < n; i++)
				arr[i] = nextInt();
			return arr;
		}

		double[] nextDoubleArray(int n) {
			double[] arr = new double[n];
			for (int i = 0; i < arr.length; i++)
				arr[i] = nextDouble();
			return arr;
		}

		long[] nextLongArray(int n) {
			long[] arr = new long[n];
			for (int i = 0; i < n; i++)
				arr[i] = nextLong();
			return arr;
		}

		public char[] nextCharArray(int n) {
			char[] chars = new char[n];
			for (int i = 0; i < n; i++)
				chars[i] = fr.nextChar();
			return chars;
		}
	}
	
	private static class IndexMaxPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
	    private int maxN;        // maximum number of elements on PQ
	    private int n;           // number of elements on PQ
	    private int[] pq;        // binary heap using 1-based indexing
	    private int[] qp;        // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
	    private Key[] keys;      // keys[i] = priority of i

	    /**
	     * Initializes an empty indexed priority queue with indices between {@code 0}
	     * and {@code maxN - 1}.
	     *
	     * @param  maxN the keys on this priority queue are index from {@code 0} to {@code maxN - 1}
	     * @throws IllegalArgumentException if {@code maxN < 0}
	     */
	    public IndexMaxPQ(int maxN) {
	        if (maxN < 0) throw new IllegalArgumentException();
	        this.maxN = maxN;
	        n = 0;
	        keys = (Key[]) new Comparable[maxN + 1];    // make this of length maxN??
	        pq   = new int[maxN + 1];
	        qp   = new int[maxN + 1];                   // make this of length maxN??
	        for (int i = 0; i <= maxN; i++)
	            qp[i] = -1;
	    }

	    /**
	     * Returns true if this priority queue is empty.
	     *
	     * @return {@code true} if this priority queue is empty;
	     *         {@code false} otherwise
	     */
	    public boolean isEmpty() {
	        return n == 0;
	    }

	    /**
	     * Is {@code i} an index on this priority queue?
	     *
	     * @param  i an index
	     * @return {@code true} if {@code i} is an index on this priority queue;
	     *         {@code false} otherwise
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     */
	    public boolean contains(int i) {
	        validateIndex(i);
	        return qp[i] != -1;
	    }

	    /**
	     * Returns the number of keys on this priority queue.
	     *
	     * @return the number of keys on this priority queue 
	     */
	    public int size() {
	        return n;
	    }

	   /**
	     * Associate key with index i.
	     *
	     * @param  i an index
	     * @param  key the key to associate with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if there already is an item
	     *         associated with index {@code i}
	     */
	    public void insert(int i, Key key) {
	        validateIndex(i);
	        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
	        n++;
	        qp[i] = n;
	        pq[n] = i;
	        keys[i] = key;
	        swim(n);
	    }

	    /**
	     * Returns an index associated with a maximum key.
	     *
	     * @return an index associated with a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int maxIndex() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return pq[1];
	    }

	    /**
	     * Returns a maximum key.
	     *
	     * @return a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public Key maxKey() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return keys[pq[1]];
	    }

	    /**
	     * Removes a maximum key and returns its associated index.
	     *
	     * @return an index associated with a maximum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int delMax() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        int max = pq[1];
	        exch(1, n--);
	        sink(1);

	        assert pq[n+1] == max;
	        qp[max] = -1;        // delete
	        keys[max] = null;    // to help with garbage collection
	        pq[n+1] = -1;        // not needed
	        return max;
	    }

	    /**
	     * Returns the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to return
	     * @return the key associated with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public Key keyOf(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        else return keys[i];
	    }

	    /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     */
	    public void changeKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	        sink(qp[i]);
	    }

	   /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @deprecated Replaced by {@code changeKey(int, Key)}.
	     */
	    @Deprecated
	    public void change(int i, Key key) {
	        validateIndex(i);
	        changeKey(i, key);
	    }

	    /**
	     * Increase the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to increase
	     * @param  key increase the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key <= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void increaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) > 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key that is strictly less than the key in the priority queue");

	        keys[i] = key;
	        swim(qp[i]);
	    }

	    /**
	     * Decrease the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to decrease
	     * @param  key decrease the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key >= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void decreaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) < 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key that is strictly greater than the key in the priority queue");
	        keys[i] = key;
	        sink(qp[i]);
	    }

	    /**
	     * Remove the key on the priority queue associated with index {@code i}.
	     *
	     * @param  i the index of the key to remove
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void delete(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        int index = qp[i];
	        exch(index, n--);
	        swim(index);
	        sink(index);
	        keys[i] = null;
	        qp[i] = -1;
	    }

	    // throw an IllegalArgumentException if i is an invalid index
	    private void validateIndex(int i) {
	        if (i < 0) throw new IllegalArgumentException("index is negative: " + i);
	        if (i >= maxN) throw new IllegalArgumentException("index >= capacity: " + i);
	    }

	   /***************************************************************************
	    * General helper functions.
	    ***************************************************************************/
	    private boolean less(int i, int j) {
	        return keys[pq[i]].compareTo(keys[pq[j]]) < 0;
	    }

	    private void exch(int i, int j) {
	        int swap = pq[i];
	        pq[i] = pq[j];
	        pq[j] = swap;
	        qp[pq[i]] = i;
	        qp[pq[j]] = j;
	    }


	   /***************************************************************************
	    * Heap helper functions.
	    ***************************************************************************/
	    private void swim(int k) {
	        while (k > 1 && less(k/2, k)) {
	            exch(k, k/2);
	            k = k/2;
	        }
	    }

	    private void sink(int k) {
	        while (2*k <= n) {
	            int j = 2*k;
	            if (j < n && less(j, j+1)) j++;
	            if (!less(k, j)) break;
	            exch(k, j);
	            k = j;
	        }
	    }


	    /**
	     * Returns an iterator that iterates over the keys on the
	     * priority queue in descending order.
	     * The iterator doesn't implement {@code remove()} since it's optional.
	     *
	     * @return an iterator that iterates over the keys in descending order
	     */
	    public Iterator<Integer> iterator() {
	        return new HeapIterator();
	    }

	    private class HeapIterator implements Iterator<Integer> {
	        // create a new pq
	        private IndexMaxPQ<Key> copy;

	        // add all elements to copy of heap
	        // takes linear time since already in heap order so no keys move
	        public HeapIterator() {
	            copy = new IndexMaxPQ<Key>(pq.length - 1);
	            for (int i = 1; i <= n; i++)
	                copy.insert(pq[i], keys[pq[i]]);
	        }

	        public boolean hasNext()  { return !copy.isEmpty();                     }
	        public void remove()      { throw new UnsupportedOperationException();  }

	        public Integer next() {
	            if (!hasNext()) throw new NoSuchElementException();
	            return copy.delMax();
	        }
	    }

	    /*public static void main(String[] args) {
	        // insert a bunch of strings
	        String[] strings = { "it", "was", "the", "best", "of", "times", "it", "was", "the", "worst" };

	        IndexMaxPQ<String> pq = new IndexMaxPQ<String>(strings.length);
	        for (int i = 0; i < strings.length; i++) {
	            pq.insert(i, strings[i]);
	        }

	        // print each key using the iterator
	        for (int i : pq) {
	            StdOut.println(i + " " + strings[i]);
	        }

	        StdOut.println();

	        // increase or decrease the key
	        for (int i = 0; i < strings.length; i++) {
	            if (StdRandom.uniform() < 0.5)
	                pq.increaseKey(i, strings[i] + strings[i]);
	            else
	                pq.decreaseKey(i, strings[i].substring(0, 1));
	        }

	        // delete and print each key
	        while (!pq.isEmpty()) {
	            String key = pq.maxKey();
	            int i = pq.delMax();
	            StdOut.println(i + " " + key);
	        }
	        StdOut.println();

	        // reinsert the same strings
	        for (int i = 0; i < strings.length; i++) {
	            pq.insert(i, strings[i]);
	        }

	        // delete them in random order
	        int[] perm = new int[strings.length];
	        for (int i = 0; i < strings.length; i++)
	            perm[i] = i;
	        StdRandom.shuffle(perm);
	        for (int i = 0; i < perm.length; i++) {
	            String key = pq.keyOf(perm[i]);
	            pq.delete(perm[i]);
	            StdOut.println(perm[i] + " " + key);
	        }

	    }*/
	}
	
	public static class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
	    private int maxN;        // maximum number of elements on PQ
	    private int n;           // number of elements on PQ
	    private int[] pq;        // binary heap using 1-based indexing
	    private int[] qp;        // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
	    private Key[] keys;      // keys[i] = priority of i

	    /**
	     * Initializes an empty indexed priority queue with indices between {@code 0}
	     * and {@code maxN - 1}.
	     * @param  maxN the keys on this priority queue are index from {@code 0}
	     *         {@code maxN - 1}
	     * @throws IllegalArgumentException if {@code maxN < 0}
	     */
	    public IndexMinPQ(int maxN) {
	        if (maxN < 0) throw new IllegalArgumentException();
	        this.maxN = maxN;
	        n = 0;
	        keys = (Key[]) new Comparable[maxN + 1];    // make this of length maxN??
	        pq   = new int[maxN + 1];
	        qp   = new int[maxN + 1];                   // make this of length maxN??
	        for (int i = 0; i <= maxN; i++)
	            qp[i] = -1;
	    }

	    /**
	     * Returns true if this priority queue is empty.
	     *
	     * @return {@code true} if this priority queue is empty;
	     *         {@code false} otherwise
	     */
	    public boolean isEmpty() {
	        return n == 0;
	    }

	    /**
	     * Is {@code i} an index on this priority queue?
	     *
	     * @param  i an index
	     * @return {@code true} if {@code i} is an index on this priority queue;
	     *         {@code false} otherwise
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     */
	    public boolean contains(int i) {
	        validateIndex(i);
	        return qp[i] != -1;
	    }

	    /**
	     * Returns the number of keys on this priority queue.
	     *
	     * @return the number of keys on this priority queue
	     */
	    public int size() {
	        return n;
	    }

	    /**
	     * Associates key with index {@code i}.
	     *
	     * @param  i an index
	     * @param  key the key to associate with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if there already is an item associated
	     *         with index {@code i}
	     */
	    public void insert(int i, Key key) {
	        validateIndex(i);
	        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
	        n++;
	        qp[i] = n;
	        pq[n] = i;
	        keys[i] = key;
	        swim(n);
	    }

	    /**
	     * Returns an index associated with a minimum key.
	     *
	     * @return an index associated with a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int minIndex() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return pq[1];
	    }

	    /**
	     * Returns a minimum key.
	     *
	     * @return a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public Key minKey() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return keys[pq[1]];
	    }

	    /**
	     * Removes a minimum key and returns its associated index.
	     * @return an index associated with a minimum key
	     * @throws NoSuchElementException if this priority queue is empty
	     */
	    public int delMin() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        int min = pq[1];
	        exch(1, n--);
	        sink(1);
	        assert min == pq[n+1];
	        qp[min] = -1;        // delete
	        keys[min] = null;    // to help with garbage collection
	        pq[n+1] = -1;        // not needed
	        return min;
	    }

	    /**
	     * Returns the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to return
	     * @return the key associated with index {@code i}
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public Key keyOf(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        else return keys[i];
	    }

	    /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void changeKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	        sink(qp[i]);
	    }

	    /**
	     * Change the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to change
	     * @param  key change the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @deprecated Replaced by {@code changeKey(int, Key)}.
	     */
	    @Deprecated
	    public void change(int i, Key key) {
	        changeKey(i, key);
	    }

	    /**
	     * Decrease the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to decrease
	     * @param  key decrease the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key >= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void decreaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) < 0)
	            throw new IllegalArgumentException("Calling decreaseKey() with a key strictly greater than the key in the priority queue");
	        keys[i] = key;
	        swim(qp[i]);
	    }

	    /**
	     * Increase the key associated with index {@code i} to the specified value.
	     *
	     * @param  i the index of the key to increase
	     * @param  key increase the key associated with index {@code i} to this key
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws IllegalArgumentException if {@code key <= keyOf(i)}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void increaseKey(int i, Key key) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        if (keys[i].compareTo(key) == 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key equal to the key in the priority queue");
	        if (keys[i].compareTo(key) > 0)
	            throw new IllegalArgumentException("Calling increaseKey() with a key strictly less than the key in the priority queue");
	        keys[i] = key;
	        sink(qp[i]);
	    }

	    /**
	     * Remove the key associated with index {@code i}.
	     *
	     * @param  i the index of the key to remove
	     * @throws IllegalArgumentException unless {@code 0 <= i < maxN}
	     * @throws NoSuchElementException no key is associated with index {@code i}
	     */
	    public void delete(int i) {
	        validateIndex(i);
	        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
	        int index = qp[i];
	        exch(index, n--);
	        swim(index);
	        sink(index);
	        keys[i] = null;
	        qp[i] = -1;
	    }

	    // throw an IllegalArgumentException if i is an invalid index
	    private void validateIndex(int i) {
	        if (i < 0) throw new IllegalArgumentException("index is negative: " + i);
	        if (i >= maxN) throw new IllegalArgumentException("index >= capacity: " + i);
	    }

	   /***************************************************************************
	    * General helper functions.
	    ***************************************************************************/
	    private boolean greater(int i, int j) {
	        return keys[pq[i]].compareTo(keys[pq[j]]) > 0;
	    }

	    private void exch(int i, int j) {
	        int swap = pq[i];
	        pq[i] = pq[j];
	        pq[j] = swap;
	        qp[pq[i]] = i;
	        qp[pq[j]] = j;
	    }


	   /***************************************************************************
	    * Heap helper functions.
	    ***************************************************************************/
	    private void swim(int k) {
	        while (k > 1 && greater(k/2, k)) {
	            exch(k, k/2);
	            k = k/2;
	        }
	    }

	    private void sink(int k) {
	        while (2*k <= n) {
	            int j = 2*k;
	            if (j < n && greater(j, j+1)) j++;
	            if (!greater(k, j)) break;
	            exch(k, j);
	            k = j;
	        }
	    }


	   /***************************************************************************
	    * Iterators.
	    ***************************************************************************/

	    /**
	     * Returns an iterator that iterates over the keys on the
	     * priority queue in ascending order.
	     * The iterator doesn't implement {@code remove()} since it's optional.
	     *
	     * @return an iterator that iterates over the keys in ascending order
	     */
	    public Iterator<Integer> iterator() { return new HeapIterator(); }

	    private class HeapIterator implements Iterator<Integer> {
	        // create a new pq
	        private IndexMinPQ<Key> copy;

	        // add all elements to copy of heap
	        // takes linear time since already in heap order so no keys move
	        public HeapIterator() {
	            copy = new IndexMinPQ<Key>(pq.length - 1);
	            for (int i = 1; i <= n; i++)
	                copy.insert(pq[i], keys[pq[i]]);
	        }

	        public boolean hasNext()  { return !copy.isEmpty();                     }
	        public void remove()      { throw new UnsupportedOperationException();  }

	        public Integer next() {
	            if (!hasNext()) throw new NoSuchElementException();
	            return copy.delMin();
	        }
	    }


	    /**
	     * Unit tests the {@code IndexMinPQ} data type.
	     *
	     * @param args the command-line arguments
	     */
	   /* public static void main(String[] args) {
	        // insert a bunch of strings
	        String[] strings = { "it", "was", "the", "best", "of", "times", "it", "was", "the", "worst" };

	        IndexMinPQ<String> pq = new IndexMinPQ<String>(strings.length);
	        for (int i = 0; i < strings.length; i++) {
	            pq.insert(i, strings[i]);
	        }

	        // delete and print each key
	        while (!pq.isEmpty()) {
	            int i = pq.delMin();
	            StdOut.println(i + " " + strings[i]);
	        }
	        StdOut.println();

	        // reinsert the same strings
	        for (int i = 0; i < strings.length; i++) {
	            pq.insert(i, strings[i]);
	        }

	        // print each key using the iterator
	        for (int i : pq) {
	            StdOut.println(i + " " + strings[i]);
	        }
	        while (!pq.isEmpty()) {
	            pq.delMin();
	        }

	    }*/
	}
}

// NOTES: 
// ASCII VALUE OF 'A': 65
// ASCII VALUE OF 'a': 97
// Range of long: 9 * 10^18
// ASCII VALUE OF '0': 48