Given a positive integer n, print all n-bit binary numbers having more 1’s than 0’s for any prefix of the number.
Examples:
Input : n = 2 Output : 11 10 Input : n = 4 Output : 1111 1110 1101 1100 1011 1010
A simple but not efficient solution will be to generate all N-bit binary numbers and print those numbers that satisfy the conditions. The time complexity of this solution is exponential.
An efficient solution is to generate only those N-bit numbers that satisfy the given conditions. We use recursion, At each point in the recursion, we append 0 and 1 to the partially formed number and recur with one less digit.
Implementation:
Try It Yourself
// C++ program to print all N-bit binary
#include <bits/stdc++.h>
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 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
# 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# 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.
<?php
// PHP program to print all N-bit binary
// function to generate n digit numbers
function printRec($number, $extraOnes,
$remainingPlaces)
{
// if number generated
if (0 == $remainingPlaces)
{
echo( $number . " ");
return;
}
// Append 1 at the current number and
// reduce the remaining places by one
printRec($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,
$remainingPlaces - 1);
}
function printNums($n)
{
$str = "";
printRec($str, 0, $n);
}
// Driver Code
$n = 4;
// Function call
printNums($n);
// This code is contributed by Mukul Singh.
<script>
// Javascript program to print all N-bit binary
// function to generate n digit numbers
function printRec(number, extraOnes, remainingPlaces)
{
// if number generated
if (0 == remainingPlaces) {
document.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);
}
function printNums(n)
{
let str = "";
printRec(str, 0, n);
}
// driver function
let n = 4;
// Function call
printNums(n);
</script>
Output
1111 1110 1101 1100 1011 1010
Time Complexity: O(n)
Auxiliary Space: O(n)
A non-recursive solution also exists, the idea is to directly generate the numbers in the range of 2N to 2(N-1) then require, only these which satisfies the condition:
Implementation:
// C++ program to print all N-bit binary
#include <bits/stdc++.h>
#include <iostream>
using namespace std;
// Function to get the binary 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<string> NBitBinary(int N)
{
vector<string> 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<string> results = NBitBinary(n);
for (int i = 0; i < results.size(); ++i)
cout << results[i] << " ";
cout << endl;
return 0;
}
// Java program to print all N-bit binary
import java.io.*;
import java.util.*;
class GFG
{
// Function to get the binary 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<String> NBitBinary(int N)
{
ArrayList<String> r = new ArrayList<String>();
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<String> 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 program to print
# all N-bit binary
# Function to get the binary
# 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# program to print all N-bit binary
using System;
using System.Collections.Generic;
class GFG{
// Function to get the binary 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<string> NBitBinary(int N)
{
List<string> r = new List<string>();
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<string> results = NBitBinary(n);
for (int i = 0; i < results.Count; ++i)
Console.Write(results[i] + " ");
Console.WriteLine();
}
}
// This code is contributed by rag2127
<script>
// Javascript program to print all N-bit binary
// Function to get the binary representation
// of the number N
function getBinaryRep(N, num_of_bits)
{
let 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;
}
function NBitBinary(N)
{
let r = [];
let first = 1 << (N - 1);
let last = first * 2;
// generate numbers in the range of (2^N)-1 to 2^(N-1)
// inclusive
for (let i = last - 1; i >= first; --i)
{
let zero_cnt = 0;
let one_cnt = 0;
let t = i;
let 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
let all_prefix_match = true;
let msk = (1 << num_of_bits) - 2;
let prefix_shift = 1;
while (msk > 0)
{
let prefix = (msk & i) >> prefix_shift;
let prefix_one_cnt = 0;
let 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.push(getBinaryRep(i, num_of_bits));
}
}
}
return r;
}
let n = 4;
// Function call
let results = NBitBinary(n);
for (let i = 0; i < results.length; ++i)
document.write(results[i]+" ");
document.write("</br>");
// This code is contributed by mukesh07.
</script>
Output
1111 1110 1101 1100 1011 1010
Time Complexity: O(m*n)
Auxiliary Space: O(n)