打印所有前缀中的1大于0的N位二进制数

给定一个正整数n，请为该数字的任何前缀打印所有n位二进制数字，这些数字的1大于0。

例子：

Input : n = 2
Output : 11 10

Input : n = 4
Output : 1111 1110 1101 1100 1011 1010

一个简单但效率不高的解决方案是生成所有N位二进制数并打印满足条件的数字。该解决方案的时间复杂度是指数级的。

一种有效的解决方案是仅生成满足给定条件的N位数字。我们使用递归。在递归的每个点上，我们将0和1附加到部分形成的数字上，然后再递减一位。

C++

// C++ program to print all N-bit binary
#include 
using namespace std;
 
/* function to generate n  digit numbers*/
void printRec(string number, int extraOnes,
              int remainingPlaces)
{
    /* if number generated */
    if (0 == remainingPlaces) {
        cout << number << " ";
        return;
    }
 
    /* Append 1 at the current number and reduce
       the remaining places by one */
    printRec(number + "1", extraOnes + 1,
             remainingPlaces - 1);
 
    /* If more ones than zeros, append 0 to the
       current number and reduce the remaining
       places by one*/
    if (0 < extraOnes)
        printRec(number + "0", extraOnes - 1,
                 remainingPlaces - 1);
}
 
void printNums(int n)
{
    string str = "";
    printRec(str, 0, n);
}
 
// Driver code
int main()
{
    int n = 4;
   
    // Function call
    printNums(n);
    return 0;
}

Java

// Java program to print all N-bit binary
import java.io.*;
 
class GFG {
    // function to generate n digit numbers
    static void printRec(String number,
                         int extraOnes,
                         int remainingPlaces)
    {
        // if number generated
        if (0 == remainingPlaces) {
            System.out.print(number + " ");
            return;
        }
 
        // Append 1 at the current number and
        // reduce the remaining places by one
        printRec(number + "1", extraOnes + 1,
                 remainingPlaces - 1);
 
        // If more ones than zeros, append 0 to the
        // current number and reduce the remaining
        // places by one
        if (0 < extraOnes)
            printRec(number + "0", extraOnes - 1,
                     remainingPlaces - 1);
    }
 
    static void printNums(int n)
    {
        String str = "";
        printRec(str, 0, n);
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 4;
       
        // Function call
        printNums(n);
    }
}
 
// This code is contributed by vt_m

Python3

# Python 3 program to print all N-bit binary
 
# function to generate n digit numbers
 
 
def printRec(number, extraOnes, remainingPlaces):
 
    # if number generated
    if (0 == remainingPlaces):
        print(number, end=" ")
        return
 
    # Append 1 at the current number and
    # reduce the remaining places by one
    printRec(number + "1", extraOnes + 1,
             remainingPlaces - 1)
 
    # If more ones than zeros, append 0 to
    # the current number and reduce the
    # remaining places by one
    if (0 < extraOnes):
        printRec(number + "0", extraOnes - 1,
                 remainingPlaces - 1)
 
 
def printNums(n):
    str = ""
    printRec(str, 0, n)
 
 
# Driver Code
if __name__ == '__main__':
    n = 4
      
    # Function call
    printNums(n)
 
# This code is contributed by
# Surendra_Gangwar

C#

// C# program to print all N-bit binary
using System;
 
class GFG {
 
    // function to generate n digit numbers
    static void printRec(String number,
                         int extraOnes,
                         int remainingPlaces)
    {
 
        // if number generated
        if (0 == remainingPlaces)
        {
            Console.Write(number + " ");
            return;
        }
 
        // Append 1 at the current number and
        // reduce the remaining places by one
        printRec(number + "1", extraOnes + 1,
                 remainingPlaces - 1);
 
        // If more ones than zeros, append
        // 0 to the current number and
        // reduce the remaining places
        // by one
        if (0 < extraOnes)
            printRec(number + "0", extraOnes - 1,
                     remainingPlaces - 1);
    }
    static void printNums(int n)
    {
        String str = "";
        printRec(str, 0, n);
    }
 
    // Driver code
    public static void Main()
    {
        int n = 4;
       
        // Function call
        printNums(n);
    }
}
 
// This code is contributed by Nitin Mittal.

---

输出

1111 1110 1101 1100 1011 1010 

还存在非递归解决方案，其思想是直接生成2 ^ N到2 ^(N-1)范围内的数字，然后仅查询满足条件的数字：

C++

// C++ program to print all N-bit binary
#include 
#include 
using namespace std;
 
 
// Function to get the biary representation
// of the number N
string getBinaryRep(int N, int num_of_bits)
{
    string r = "";
    num_of_bits--;
   
    // loop for each bit
    while (num_of_bits >= 0)
    {
        if (N & (1 << num_of_bits))
            r.append("1");
        else
            r.append("0");
        num_of_bits--;
    }
    return r;
}
 
vector NBitBinary(int N)
{
    vector r;
    int first = 1 << (N - 1);
    int last = first * 2;
     
    // generate numbers in the range of (2^N)-1 to 2^(N-1)
    // inclusive
    for (int i = last - 1; i >= first; --i)
    {
        int zero_cnt = 0;
        int one_cnt = 0;
        int t = i;
        int num_of_bits = 0;
         
        // longest prefix check
        while (t)
        {
            if (t & 1)
                one_cnt++;
            else
                zero_cnt++;
            num_of_bits++;
            t = t >> 1;
        }
       
        // if counts of 1 is greater than
        // counts of zero
        if (one_cnt >= zero_cnt)
        {
            // do sub-prefixes check
            bool all_prefix_match = true;
            int msk = (1 << num_of_bits) - 2;
            int prefix_shift = 1;
            while (msk)
            {
 
                int prefix = (msk & i) >> prefix_shift;
                int prefix_one_cnt = 0;
                int prefix_zero_cnt = 0;
                while (prefix)
                {
                    if (prefix & 1)
                        prefix_one_cnt++;
                    else
                        prefix_zero_cnt++;
                    prefix = prefix >> 1;
                }
                if (prefix_zero_cnt > prefix_one_cnt)
                {
                    all_prefix_match = false;
                    break;
                }
                prefix_shift++;
                msk = msk & (msk << 1);
            }
            if (all_prefix_match)
            {
                r.push_back(getBinaryRep(i, num_of_bits));
            }
        }
    }
    return r;
}
 
// Driver code
int main()
{
    int n = 4;
   
    // Function call
    vector results = NBitBinary(n);
    for (int i = 0; i < results.size(); ++i)
        cout << results[i] << " ";
    cout << endl;
    return 0;
}

Java

// Java program to print all N-bit binary
import java.io.*;
import java.util.*;
class GFG
{
 
  // Function to get the biary representation
  // of the number N
  static String getBinaryRep(int N, int num_of_bits)
  {
    String r = "";
    num_of_bits--;
 
    // loop for each bit
    while (num_of_bits >= 0)
    {
      if ((N & (1 << num_of_bits))!=0)
        r += "1";
      else
        r += "0";
      num_of_bits--;
    }
    return r;
  }
 
  static ArrayList NBitBinary(int N)
  {
    ArrayList r = new ArrayList();
    int first = 1 << (N - 1);
    int last = first * 2;
 
    // generate numbers in the range of (2^N)-1 to 2^(N-1)
    // inclusive
    for (int i = last - 1; i >= first; --i)
    {
      int zero_cnt = 0;
      int one_cnt = 0;
      int t = i;
      int num_of_bits = 0;
 
      // longest prefix check
      while (t > 0)
      {
        if ((t & 1) != 0)
          one_cnt++;
        else
          zero_cnt++;
        num_of_bits++;
        t = t >> 1;
      }
 
      // if counts of 1 is greater than
      // counts of zero
      if (one_cnt >= zero_cnt)
      {
        // do sub-prefixes check
        boolean all_prefix_match = true;
        int msk = (1 << num_of_bits) - 2;
        int prefix_shift = 1;
        while (msk > 0)
        {
 
          int prefix = (msk & i) >> prefix_shift;
          int prefix_one_cnt = 0;
          int prefix_zero_cnt = 0;
          while (prefix > 0)
          {
            if ((prefix & 1)!=0)
              prefix_one_cnt++;
            else
              prefix_zero_cnt++;
            prefix = prefix >> 1;
          }
          if (prefix_zero_cnt > prefix_one_cnt)
          {
            all_prefix_match = false;
            break;
          }
          prefix_shift++;
          msk = msk & (msk << 1);
        }
        if (all_prefix_match)
        {
          r.add(getBinaryRep(i, num_of_bits));
        }
      }
    }
    return r;
  }
 
  // Driver code
  public static void main (String[] args)
  {
 
    int n = 4;
 
    // Function call
    ArrayList results = NBitBinary(n);
    for (int i = 0; i < results.size(); ++i)
      System.out.print(results.get(i)+" ");
    System.out.println();
  }
}
 
// This code is contributed by avanitrachhadiya2155

Python3

# Python3 program to print
# all N-bit binary
 
# Function to get the biary
# representation of the number N
def getBinaryRep(N, num_of_bits):
 
    r = "";
    num_of_bits -= 1
   
    # loop for each bit
    while (num_of_bits >= 0):   
        if (N & (1 << num_of_bits)):
            r += ("1");
        else:
            r += ("0");
        num_of_bits -= 1
     
    return r;
 
def NBitBinary(N):
 
    r = []
    first = 1 << (N - 1);
    last = first * 2;
     
    # generate numbers in the range
    # of (2^N)-1 to 2^(N-1) inclusive
    for i in range (last - 1,
                    first - 1, -1):   
        zero_cnt = 0;
        one_cnt = 0;
        t = i;
        num_of_bits = 0;
         
        # longest prefix check
        while (t):       
            if (t & 1):
                one_cnt += 1
            else:
                zero_cnt += 1
            num_of_bits += 1
            t = t >> 1;       
       
        # if counts of 1 is greater
        # than counts of zero
        if (one_cnt >= zero_cnt):
         
            # do sub-prefixes check
            all_prefix_match = True;
            msk = (1 << num_of_bits) - 2;
            prefix_shift = 1;
             
            while (msk):           
                prefix = ((msk & i) >>
                           prefix_shift);
                prefix_one_cnt = 0;
                prefix_zero_cnt = 0;
                 
                while (prefix):               
                    if (prefix & 1):
                        prefix_one_cnt += 1
                    else:
                        prefix_zero_cnt += 1
                    prefix = prefix >> 1;
                 
                if (prefix_zero_cnt >
                    prefix_one_cnt):               
                    all_prefix_match = False;
                    break;
                 
                prefix_shift += 1
                msk = msk & (msk << 1);
             
            if (all_prefix_match):           
                r.append(getBinaryRep(i,
                                      num_of_bits));         
    return r
 
# Driver code
if __name__ == "__main__":
  
    n = 4;
   
    # Function call
    results = NBitBinary(n);
    for i in range (len(results)):
        print (results[i],
               end = " ")
    print ()
     
# This code is contributed by Chitranayal

C＃

// C# program to print all N-bit binary
using System;
using System.Collections.Generic;
 
class GFG{
     
// Function to get the biary representation
// of the number N
static string getBinaryRep(int N, int num_of_bits)
{
    string r = "";
    num_of_bits--;
     
    // loop for each bit
    while (num_of_bits >= 0)
    {
        if ((N & (1 << num_of_bits)) != 0)
            r += "1";
        else
            r += "0";
             
        num_of_bits--;
    }
    return r;
}
     
static List NBitBinary(int N)
{
    List r = new List();
    int first = 1 << (N - 1);
    int last = first * 2;
     
    // Generate numbers in the range of (2^N)-1 to 2^(N-1)
    // inclusive
    for(int i = last - 1; i >= first; --i)
    {
        int zero_cnt = 0;
        int one_cnt = 0;
        int t = i;
        int num_of_bits = 0;
         
        // longest prefix check
        while (t > 0)
        {
            if ((t & 1) != 0)
                one_cnt++;
            else
                zero_cnt++;
                 
            num_of_bits++;
            t = t >> 1;
        }
     
        // If counts of 1 is greater than
        // counts of zero
        if (one_cnt >= zero_cnt)
        {
             
            // Do sub-prefixes check
            bool all_prefix_match = true;
            int msk = (1 << num_of_bits) - 2;
            int prefix_shift = 1;
             
            while (msk > 0)
            {
                int prefix = (msk & i) >> prefix_shift;
                int prefix_one_cnt = 0;
                int prefix_zero_cnt = 0;
                 
                while (prefix > 0)
                {
                    if ((prefix & 1)!=0)
                        prefix_one_cnt++;
                    else
                        prefix_zero_cnt++;
                         
                    prefix = prefix >> 1;
                }
                if (prefix_zero_cnt > prefix_one_cnt)
                {
                    all_prefix_match = false;
                    break;
                }
                prefix_shift++;
                msk = msk & (msk << 1);
            }
            if (all_prefix_match)
            {
                r.Add(getBinaryRep(i, num_of_bits));
            }
        }
    }
    return r;
}
 
// Driver code
static public void Main()
{
    int n = 4;
     
    // Function call
    List results = NBitBinary(n);
    for (int i = 0; i < results.Count; ++i)
        Console.Write(results[i] + " ");
         
    Console.WriteLine();
}
}
 
// This code is contributed by rag2127

输出

1111 1110 1101 1100 1011 1010