计算从1到n的所有数字中的总置位位数

给定一个正整数n，计算从1到n的所有数字的二进制表示形式的置位总数。

例子：

Input: n = 3
Output:  4

Input: n = 6
Output: 9

Input: n = 7
Output: 12

Input: n = 8
Output: 13

资料来源：亚马逊访谈问题

方法1(简单)
一个简单的解决方案是运行一个从1到n的循环，并对从1到n的所有数字中的设置位的计数求和。

C++

// A simple program to count set bits
// in all numbers from 1 to n.
#include 
 
// A utility function to count set bits
// in a number x
unsigned int countSetBitsUtil(unsigned int x);
 
// Returns count of set bits present in all
// numbers from 1 to n
unsigned int countSetBits(unsigned int n)
{
    int bitCount = 0; // initialize the result
 
    for (int i = 1; i <= n; i++)
        bitCount += countSetBitsUtil(i);
 
    return bitCount;
}
 
// A utility function to count set bits
// in a number x
unsigned int countSetBitsUtil(unsigned int x)
{
    if (x <= 0)
        return 0;
    return (x % 2 == 0 ? 0 : 1) + countSetBitsUtil(x / 2);
}
 
// Driver program to test above functions
int main()
{
    int n = 4;
    printf("Total set bit count is %d", countSetBits(n));
    return 0;
}

Java

// A simple program to count set bits
// in all numbers from 1 to n.
 
class GFG{
 
    // Returns count of set bits present
    //  in all numbers from 1 to n
    static int countSetBits( int n)
    {
        // initialize the result
        int bitCount = 0;
     
        for (int i = 1; i <= n; i++)
            bitCount += countSetBitsUtil(i);
     
        return bitCount;
    }
     
    // A utility function to count set bits
    // in a number x
    static int countSetBitsUtil( int x)
    {
        if (x <= 0)
            return 0;
        return (x % 2 == 0 ? 0 : 1) +
               countSetBitsUtil(x / 2);
    }
     
    // Driver program
    public static void main(String[] args)
    {
        int n = 4;
        System.out.print("Total set bit count is ");
        System.out.print(countSetBits(n));
    }
}
 
// This code is contributed by
// Smitha Dinesh Semwal

Python3

# A simple program to count set bits
# in all numbers from 1 to n.
 
# Returns count of set bits present in all
# numbers from 1 to n
def countSetBits(n):
     
    # initialize the result
    bitCount = 0
 
    for i in range(1, n + 1):
        bitCount += countSetBitsUtil(i)
 
    return bitCount
 
 
# A utility function to count set bits
# in a number x
def countSetBitsUtil(x):
 
    if (x <= 0):
        return 0
    return (0 if int(x % 2) == 0 else 1) +  countSetBitsUtil(int(x / 2))
 
 
# Driver program
if __name__=='__main__':
    n = 4
    print("Total set bit count is", countSetBits(n))
      
# This code is contributed by
# Smitha Dinesh Semwal

C#

// A simple C# program to count set bits
// in all numbers from 1 to n.
using System;
 
class GFG
{
    // Returns count of set bits present
    // in all numbers from 1 to n
    static int countSetBits( int n)
    {
        // initialize the result
        int bitCount = 0;
     
        for (int i = 1; i <= n; i++)
            bitCount += countSetBitsUtil(i);
     
        return bitCount;
    }
     
    // A utility function to count set bits
    // in a number x
    static int countSetBitsUtil( int x)
    {
        if (x <= 0)
            return 0;
        return (x % 2 == 0 ? 0 : 1) +
            countSetBitsUtil(x / 2);
    }
     
    // Driver program
    public static void Main()
    {
        int n = 4;
        Console.Write("Total set bit count is ");
        Console.Write(countSetBits(n));
    }
}
 
// This code is contributed by Sam007

Javascript

// A simple program to count set bits
// in all numbers from 1 to n.
 
     
    // Returns count of set bits present
    //  in all numbers from 1 to n
    function countSetBits(n)
    {
        // initialize the result
        let bitCount = 0;
        for (let i = 1; i <= n; i++)
        {
            bitCount += countSetBitsUtil(i);
        }
        return bitCount;
    }
     
    // A utility function to count set bits
    // in a number x
    function countSetBitsUtil(x)
    {
        if (x <= 0)
        {
            return 0;
        }
        return (x % 2 == 0 ? 0 : 1) + countSetBitsUtil(Math.floor(x/2));
    }
     
    // Driver program
    let n = 4;
    document.write("Total set bit count is ");
    document.write(countSetBits(n));
     
    // This code is contributed by rag2127

输出：

Total set bit count is 5

时间复杂度：O(nLogn)

方法2(比方法1简单高效)
如果我们从距离i的最右边观察到位，则在垂直序列中2 ^ i位置后位会反转。
例如n = 5;
0 = 0000
1 = 0001
2 = 0010
3 = 0011
4 = 0100
5 = 0101
观察最右边的位(i = 0)，这些位在(2 ^ 0 = 1)之后翻转
观察最右边的第3位(i = 2)，这些位在(2 ^ 2 = 4)之后被翻转
因此，我们可以以垂直方式对位进行计数，以使在第i个最右边的位置，在2 ^ i次迭代后，位将被翻转；

C++

#include 
using namespace std;
 
// Function which counts set bits from 0 to n
int countSetBits(int n)
{
    int i = 0;
 
    // ans store sum of set bits from 0 to n 
    int ans = 0;
 
    // while n greater than equal to 2^i
    while ((1 << i) <= n) {
 
        // This k will get flipped after
        // 2^i iterations
        bool k = 0;
 
        // change is iterator from 2^i to 1
        int change = 1 << i;
 
        // This will loop from 0 to n for
        // every bit position
        for (int j = 0; j <= n; j++) {
 
            ans += k;
 
            if (change == 1) {
                k = !k; // When change = 1 flip the bit
                change = 1 << i; // again set change to 2^i
            }
            else {
                change--;
            }
        }
 
        // increment the position
        i++;
    }
 
    return ans;
}
 
// Main Function
int main()
{
    int n = 17;
    cout << countSetBits(n) << endl;
    return 0;
}

Java

public class GFG {
     
    // Function which counts set bits from 0 to n
    static int countSetBits(int n)
    {
        int i = 0;
 
        // ans store sum of set bits from 0 to n
        int ans = 0;
 
        // while n greater than equal to 2^i
        while ((1 << i) <= n) {
 
            // This k will get flipped after
            // 2^i iterations
            boolean k = false;
 
            // change is iterator from 2^i to 1
            int change = 1 << i;
 
            // This will loop from 0 to n for
            // every bit position
            for (int j = 0; j <= n; j++) {
 
                if (k == true)
                    ans += 1;
                else
                    ans += 0;
 
                if (change == 1) {
                     
                    // When change = 1 flip the bit
                    k = !k;
                     
                    // again set change to 2^i
                    change = 1 << i;
                }
                else {
                    change--;
                }
            }
 
            // increment the position
            i++;
        }
 
        return ans;
    }
 
    // Driver program
    public static void main(String[] args)
    {
        int n = 17;
         
        System.out.println(countSetBits(n));
    }
}
 
// This code is contributed by Sam007.

Python3

# Function which counts set bits from 0 to n
def countSetBits(n) :
    i = 0
 
    # ans store sum of set bits from 0 to n 
    ans = 0
 
    # while n greater than equal to 2^i
    while ((1 << i) <= n) :
 
        # This k will get flipped after 
        # 2^i iterations
        k = 0
 
        # change is iterator from 2^i to 1
        change = 1 << i
 
        # This will loop from 0 to n for
        # every bit position
        for j in range(0, n+1) :
            ans += k
 
            if change == 1 :
                 
                #  When change = 1 flip the bit
                k = not k
 
                # again set change to 2^i
                change = 1 << i
 
            else :
                change -= 1
 
        # increment the position
        i += 1
 
    return ans
 
 
 
# Driver code
if __name__ == "__main__" :
 
    n = 17
    print(countSetBits(n))
  
# This code is contributed by ANKITRAI1

C＃

using System;
 
class GFG
{
    static int countSetBits(int n)
    {
        int i = 0;
 
        // ans store sum of set bits from 0 to n
        int ans = 0;
 
        // while n greater than equal to 2^i
        while ((1 << i) <= n) {
 
            // This k will get flipped after
            // 2^i iterations
            bool k = false;
 
            // change is iterator from 2^i to 1
            int change = 1 << i;
 
            // This will loop from 0 to n for
            // every bit position
            for (int j = 0; j <= n; j++) {
 
                if (k == true)
                    ans += 1;
                else
                    ans += 0;
 
                if (change == 1) {
                     
                    // When change = 1 flip the bit
                    k = !k;
                     
                    // again set change to 2^i
                    change = 1 << i;
                }
                else {
                    change--;
                }
            }
 
            // increment the position
            i++;
        }
 
        return ans;
    }
 
    // Driver program
    static void Main()
    {
        int n = 17;
        Console.Write(countSetBits(n));
    }
}
 
// This code is contributed by Sam007

Java脚本

 
// Javascript program for the above approach
 
// Function which counts set bits from 0 to n
function countSetBits(n)
{
    let i = 0;
 
    // ans store sum of set bits from 0 to n 
    let ans = 0;
 
    // while n greater than equal to 2^i
    while ((1 << i) <= n) {
 
        // This k will get flipped after
        // 2^i iterations
        let k = 0;
 
        // change is iterator from 2^i to 1
        let change = 1 << i;
 
        // This will loop from 0 to n for
        // every bit position
        for (let j = 0; j <= n; j++) {
 
            ans += k;
 
            if (change == 1) {
                // When change = 1 flip the bit
                k = !k;
                // again set change to 2^i
                change = 1 << i;
            }
            else {
                change--;
            }
        }
 
        // increment the position
        i++;
    }
 
    return ans;
}
     
 
// Driver Code
 
    let n = 17;
    document.write(countSetBits(n));
 

输出：

35

时间复杂度：O(k * n)
其中k =表示数字n的位数
k <= 64

方法3(整rick)
如果输入数字的形式为2 ^ b -1，例如1、3、7、15等，则设置位数为b * 2 ^(b-1)。这是因为对于从0到(2 ^ b)-1的所有数字，如果对列表进行补充和翻转，最终将得到相同的列表(一半的位打开，一半关闭)。
如果数字没有全部设置位，则某个位置m是最左边的设置位的位置。该位置的设置位数为n –(1 << m)+1。其余的设置位数分为两部分：
1)(m-1)中的位向下定位到最左边的位变为0的点，并且
2)该点以下的2 ^(m-1)数是上面的封闭形式。
一个简单的方法是考虑数字6：

0|0 0
0|0 1
0|1 0
0|1 1
-|--
1|0 0
1|0 1
1|1 0

最左边的置1位在位置2(位置被认为从0开始)。如果我们掩盖掉剩下的是2(最后一行右侧的“ 1 0”。)那么第二个位置(左下方的框)的位数是3(即2 + 1) 。 0-3(上面右上方的框)中的设置位是2 * 2 ^(2-1)=4。右下角的框是我们尚未计数的剩余位，是设置的数量可以递归计算最多2个数字(右下方框中最后一个条目的值)的位数。

C++

#include 
 
// A O(Logn) complexity program to count
// set bits in all numbers from 1 to n
using namespace std;
 
/* Returns position of leftmost set bit.
The rightmost position is considered
as 0 */
unsigned int getLeftmostBit(int n)
{
    int m = 0;
    while (n > 1)
    {
        n = n >> 1;
        m++;
    }
    return m;
}
 
/* Given the position of previous leftmost
set bit in n (or an upper bound on
leftmost position) returns the new
position of leftmost set bit in n */
unsigned int getNextLeftmostBit(int n, int m)
{
    unsigned int temp = 1 << m;
    while (n < temp) {
        temp = temp >> 1;
        m--;
    }
    return m;
}
 
// The main recursive function used by countSetBits()
unsigned int _countSetBits(unsigned int n, int m);
 
// Returns count of set bits present in
// all numbers from 1 to n
unsigned int countSetBits(unsigned int n)
{
    // Get the position of leftmost set
    // bit in n. This will be used as an
    // upper bound for next set bit function
    int m = getLeftmostBit(n);
 
    // Use the position
    return _countSetBits(n, m);
}
 
unsigned int _countSetBits(unsigned int n, int m)
{
    // Base Case: if n is 0, then set bit
    // count is 0
    if (n == 0)
        return 0;
 
    /* get position of next leftmost set bit */
    m = getNextLeftmostBit(n, m);
 
    // If n is of the form 2^x-1, i.e., if n
    // is like 1, 3, 7, 15, 31, .. etc,
    // then we are done.
    // Since positions are considered starting
    // from 0, 1 is added to m
    if (n == ((unsigned int)1 << (m + 1)) - 1)
        return (unsigned int)(m + 1) * (1 << m);
 
    // update n for next recursive call
    n = n - (1 << m);
    return (n + 1) + countSetBits(n) + m * (1 << (m - 1));
}
 
// Driver code
int main()
{
    int n = 17;
    cout<<"Total set bit count is "<< countSetBits(n);
    return 0;
}
 
// This code is contributed by rathbhupendra

C

// A O(Logn) complexity program to count
// set bits in all numbers from 1 to n
#include 
 
/* Returns position of leftmost set bit.
   The rightmost position is considered
   as 0 */
unsigned int getLeftmostBit(int n)
{
    int m = 0;
    while (n > 1) {
        n = n >> 1;
        m++;
    }
    return m;
}
 
/* Given the position of previous leftmost
   set bit in n (or an upper bound on
   leftmost position) returns the new
   position of leftmost set bit in n  */
unsigned int getNextLeftmostBit(int n, int m)
{
    unsigned int temp = 1 << m;
    while (n < temp) {
        temp = temp >> 1;
        m--;
    }
    return m;
}
 
// The main recursive function used by countSetBits()
unsigned int _countSetBits(unsigned int n, int m);
 
// Returns count of set bits present in
// all numbers from 1 to n
unsigned int countSetBits(unsigned int n)
{
    // Get the position of leftmost set
    // bit in n. This will be used as an
    // upper bound for next set bit function
    int m = getLeftmostBit(n);
 
    // Use the position
    return _countSetBits(n, m);
}
 
unsigned int _countSetBits(unsigned int n, int m)
{
    // Base Case: if n is 0, then set bit
    // count is 0
    if (n == 0)
        return 0;
 
    /* get position of next leftmost set bit */
    m = getNextLeftmostBit(n, m);
 
    // If n is of the form 2^x-1, i.e., if n
    // is like 1, 3, 7, 15, 31, .. etc,
    // then we are done.
    // Since positions are considered starting
    // from 0, 1 is added to m
    if (n == ((unsigned int)1 << (m + 1)) - 1)
        return (unsigned int)(m + 1) * (1 << m);
 
    // update n for next recursive call
    n = n - (1 << m);
    return (n + 1) + countSetBits(n) + m * (1 << (m - 1));
}
 
// Driver program to test above functions
int main()
{
    int n = 17;
    printf("Total set bit count is %d", countSetBits(n));
    return 0;
}

Java

// Java A O(Logn) complexity program to count
// set bits in all numbers from 1 to n
import java.io.*;
 
class GFG
{
     
/* Returns position of leftmost set bit.
The rightmost position is considered
as 0 */
static int getLeftmostBit(int n)
{
    int m = 0;
    while (n > 1)
    {
        n = n >> 1;
        m++;
    }
    return m;
}
 
/* Given the position of previous leftmost
set bit in n (or an upper bound on
leftmost position) returns the new
position of leftmost set bit in n */
static int getNextLeftmostBit(int n, int m)
{
    int temp = 1 << m;
    while (n < temp)
    {
        temp = temp >> 1;
        m--;
    }
    return m;
}
 
// The main recursive function used by countSetBits()
// Returns count of set bits present in
// all numbers from 1 to n
 
static int countSetBits(int n)
{
    // Get the position of leftmost set
    // bit in n. This will be used as an
    // upper bound for next set bit function
    int m = getLeftmostBit(n);
 
    // Use the position
    return countSetBits(n, m);
}
 
static int countSetBits( int n, int m)
{
    // Base Case: if n is 0, then set bit
    // count is 0
    if (n == 0)
        return 0;
 
    /* get position of next leftmost set bit */
    m = getNextLeftmostBit(n, m);
 
    // If n is of the form 2^x-1, i.e., if n
    // is like 1, 3, 7, 15, 31, .. etc,
    // then we are done.
    // Since positions are considered starting
    // from 0, 1 is added to m
    if (n == ((int)1 << (m + 1)) - 1)
        return (int)(m + 1) * (1 << m);
 
    // update n for next recursive call
    n = n - (1 << m);
    return (n + 1) + countSetBits(n) + m * (1 << (m - 1));
}
 
// Driver code
public static void main (String[] args)
{
 
    int n = 17;
    System.out.println ("Total set bit count is " + countSetBits(n));
}
}
 
// This code is contributed by ajit..

Python3

# A O(Logn) complexity program to count
# set bits in all numbers from 1 to n
 
"""
/* Returns position of leftmost set bit.
The rightmost position is considered
as 0 */
"""
def getLeftmostBit(n):
 
    m = 0
    while (n > 1) :
 
        n = n >> 1
        m += 1
 
    return m
 
"""
/* Given the position of previous leftmost
set bit in n (or an upper bound on
leftmost position) returns the new
position of leftmost set bit in n */
"""
def getNextLeftmostBit(n, m) :
 
    temp = 1 << m
    while (n < temp) :
        temp = temp >> 1
        m -= 1
 
    return m
 
# The main recursive function used by countSetBits()
# def _countSetBits(n, m)
 
# Returns count of set bits present in
# all numbers from 1 to n
def countSetBits(n) :
 
    # Get the position of leftmost set
    # bit in n. This will be used as an
    # upper bound for next set bit function
    m = getLeftmostBit(n)
 
    # Use the position
    return _countSetBits(n, m)
 
def _countSetBits(n, m) :
 
    # Base Case: if n is 0, then set bit
    # count is 0
    if (n == 0) :
        return 0
 
    # /* get position of next leftmost set bit */
    m = getNextLeftmostBit(n, m)
 
    # If n is of the form 2^x-1, i.e., if n
    # is like 1, 3, 7, 15, 31, .. etc,
    # then we are done.
    # Since positions are considered starting
    # from 0, 1 is added to m
    if (n == (1 << (m + 1)) - 1) :
        return ((m + 1) * (1 << m))
 
    # update n for next recursive call
    n = n - (1 << m)
    return (n + 1) + countSetBits(n) + m * (1 << (m - 1))
 
# Driver code
n = 17
print("Total set bit count is", countSetBits(n))
 
# This code is contributed by rathbhupendra

C＃

// C# A O(Logn) complexity program to count
// set bits in all numbers from 1 to n
using System;
 
class GFG
{
     
/* Returns position of leftmost set bit.
The rightmost position is considered
as 0 */
static int getLeftmostBit(int n)
{
    int m = 0;
    while (n > 1)
    {
        n = n >> 1;
        m++;
    }
    return m;
}
 
/* Given the position of previous leftmost
set bit in n (or an upper bound on
leftmost position) returns the new
position of leftmost set bit in n */
static int getNextLeftmostBit(int n, int m)
{
    int temp = 1 << m;
    while (n < temp)
    {
        temp = temp >> 1;
        m--;
    }
    return m;
}
 
// The main recursive function used by countSetBits()
// Returns count of set bits present in
// all numbers from 1 to n
static int countSetBits(int n)
{
    // Get the position of leftmost set
    // bit in n. This will be used as an
    // upper bound for next set bit function
    int m = getLeftmostBit(n);
 
    // Use the position
    return countSetBits(n, m);
}
 
static int countSetBits( int n, int m)
{
    // Base Case: if n is 0,
    // then set bit count is 0
    if (n == 0)
        return 0;
 
    /* get position of next leftmost set bit */
    m = getNextLeftmostBit(n, m);
 
    // If n is of the form 2^x-1, i.e., if n
    // is like 1, 3, 7, 15, 31, .. etc,
    // then we are done.
    // Since positions are considered starting
    // from 0, 1 is added to m
    if (n == ((int)1 << (m + 1)) - 1)
        return (int)(m + 1) * (1 << m);
 
    // update n for next recursive call
    n = n - (1 << m);
    return (n + 1) + countSetBits(n) +
                  m * (1 << (m - 1));
}
 
// Driver code
static public void Main ()
{
    int n = 17;
    Console.Write("Total set bit count is " +
                            countSetBits(n));
}
}
 
// This code is contributed by Tushil.

输出：

Total set bit count is 35

时间复杂度：O(Logn)。从实现的第一眼看，时间复杂度看起来更多。但是，如果我们仔细研究一下，则将对n中的所有0位执行getNextLeftmostBit()的while循环内的语句。并且执行递归的次数小于或等于n中的设置位。换句话说，如果控件进入getNextLeftmostBit()的while循环内部，则它将跳过递归的许多位。
感谢agatsu和IC提出此解决方案。
这是Piyush Kapoor建议的另一种解决方案。

一个简单的解决方案，使用以下事实：对于第i个最低有效位，答案为

(N/2^i)*2^(i-1)+ X

在哪里

X = N%(2^i)-(2^(i-1)-1)

iff

N%(2^i)>=(2^(i-1)-1)

C++

int getSetBitsFromOneToN(int N){
    int two = 2,ans = 0;
    int n = N;
    while(n){
        ans += (N/two)*(two>>1);
        if((N&(two-1)) > (two>>1)-1) ans += (N&(two-1)) - (two>>1)+1;
        two <<= 1;
        n >>= 1;
    }
    return ans;
}

Java

static int getSetBitsFromOneToN(int N){
    int two = 2,ans = 0;
    int n = N;
    while(n != 0)
    {
        ans += (N / two)*(two >> 1);
        if((N&(two - 1)) > (two >> 1) - 1)
            ans += (N&(two - 1)) - (two >> 1) + 1;
        two <<= 1;
        n >>= 1;
    }
    return ans;
}
 
// This code is contributed by divyeshrabadiya07.

Python3

def getSetBitsFromOneToN(N):
    two = 2
    ans = 0
    n = N
    while(n != 0)
    {
        ans += int(N / two) * (two >> 1)
        if((N & (two - 1)) > (two >> 1) - 1):
            ans += (N&(two - 1)) - (two >> 1) + 1
        two <<= 1;
        n >>= 1;
    }
    return ans
 
# This code is contributed by avanitrachhadiya2155

C＃

static int getSetBitsFromOneToN(int N){
    int two = 2,ans = 0;
    int n = N;
    while(n != 0)
    {
        ans += (N / two)*(two >> 1);
        if((N&(two - 1)) > (two >> 1) - 1)
            ans += (N&(two - 1)) - (two >> 1) + 1;
        two <<= 1;
        n >>= 1;
    }
    return ans;
}
 
// This code is contributed by divyesh072019.

Java脚本

 
function getSetBitsFromOneToN(N)
{
    var two = 2
    var ans = 0
    var n = N
     
    while(n != 0)
    {
        ans += Math.floor(N / two) * (two >> 1)
         
        if ((N & (two - 1)) > (two >> 1) - 1)
            ans += (N&(two - 1)) - (two >> 1) + 1
             
        two <<= 1;
        n >>= 1;
    }
    return ans
}
 
// This code is contributed by AnkThon