Question

1680. Concatenation of Consecutive Binary Numbers

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Description

Given an integer n, return _the decimal value of the binary string formed by concatenating the binary representations of _1_ to _n_ in order, modulo _109 + 7.

 

Example 1:

Input: n = 1
Output: 1
Explanation: "1" in binary corresponds to the decimal value 1. 

Example 2:

Input: n = 3
Output: 27
Explanation: In binary, 1, 2, and 3 corresponds to "1", "10", and "11".
After concatenating them, we have "11011", which corresponds to the decimal value 27.

Example 3:

Input: n = 12
Output: 505379714
Explanation: The concatenation results in "1101110010111011110001001101010111100".
The decimal value of that is 118505380540.
After modulo 109 + 7, the result is 505379714.

 

Constraints:

• 1 <= n <= 105

Solutions

Solution 1: Bit Manipulation

By observing the pattern of number concatenation, we can find that when concatenating to the $i$-th number, the result $ans$ formed by concatenating the previous $i-1$ numbers is actually shifted to the left by a certain number of bits, and then $i$ is added. The number of bits shifted, $shift$, is the number of binary digits in $i$. Since $i$ is continuously incremented by $1$, the number of bits shifted either remains the same as the last shift or increases by one. When $i$ is a power of $2$, that is, when there is only one bit in the binary number of $i$ that is $1$, the number of bits shifted increases by $1$ compared to the last time.

The time complexity is $O(n)$, where $n$ is the given integer. The space complexity is $O(1)$.

class Solution:
  def concatenatedBinary(self, n: int) -> int:
    mod = 10**9 + 7
    ans = 0
    for i in range(1, n + 1):
      ans = (ans << i.bit_length() | i) % mod
    return ans

class Solution {
  public int concatenatedBinary(int n) {
    final int mod = (int) 1e9 + 7;
    long ans = 0;
    for (int i = 1; i <= n; ++i) {
      ans = (ans << (32 - Integer.numberOfLeadingZeros(i)) | i) % mod;
    }
    return (int) ans;
  }
}

class Solution {
public:
  int concatenatedBinary(int n) {
    const int mod = 1e9 + 7;
    long ans = 0;
    for (int i = 1; i <= n; ++i) {
      ans = (ans << (32 - __builtin_clz(i)) | i) % mod;
    }
    return ans;
  }
};

func concatenatedBinary(n int) (ans int) {
  const mod = 1e9 + 7
  for i := 1; i <= n; i++ {
    ans = (ans<<bits.Len(uint(i)) | i) % mod
  }
  return
}

function concatenatedBinary(n: number): number {
  const mod = BigInt(10 ** 9 + 7);
  let ans = 0n;
  let shift = 0n;
  for (let i = 1n; i <= n; ++i) {
    if ((i & (i - 1n)) == 0n) {
      ++shift;
    }
    ans = ((ans << shift) | i) % mod;
  }
  return Number(ans);
}

Solution 2

class Solution:
  def concatenatedBinary(self, n: int) -> int:
    mod = 10**9 + 7
    ans = shift = 0
    for i in range(1, n + 1):
      if (i & (i - 1)) == 0:
        shift += 1
      ans = (ans << shift | i) % mod
    return ans

class Solution {
  public int concatenatedBinary(int n) {
    final int mod = (int) 1e9 + 7;
    long ans = 0;
    int shift = 0;
    for (int i = 1; i <= n; ++i) {
      if ((i & (i - 1)) == 0) {
        ++shift;
      }
      ans = (ans << shift | i) % mod;
    }
    return (int) ans;
  }
}

class Solution {
public:
  int concatenatedBinary(int n) {
    const int mod = 1e9 + 7;
    long ans = 0;
    int shift = 0;
    for (int i = 1; i <= n; ++i) {
      if ((i & (i - 1)) == 0) {
        ++shift;
      }
      ans = (ans << shift | i) % mod;
    }
    return ans;
  }
};

func concatenatedBinary(n int) (ans int) {
  const mod = 1e9 + 7
  shift := 0
  for i := 1; i <= n; i++ {
    if i&(i-1) == 0 {
      shift++
    }
    ans = (ans<<shift | i) % mod
  }
  return
}