Concatenation of Consecutive Binary Numbers

class Solution {
public:
    int combinedBinary(int maximum) {
        const int MODULO = 1000000007;
        int bitSize = 0;
        long total = 0;
        for (int j = 1; j <= maximum; j++) {
            if ((j & (j - 1)) == 0) {
                bitSize++;
            }
            total = ((total << bitSize) | j) % MODULO;
        }
        return total;
    }
};

This solution addresses the problem of concatenating consecutive binary numbers into a single decimal value modulo 1000000007. Using C++, the approach involves:

• Tracking the bit size required to represent each number. This is necessary since each new number might require an additional bit as numbers increase.
• Employing bit manipulation to handle the concatenation and modulo operation.

Here's how the solution is strategically implemented:

• Define a function combinedBinary that takes an integer maximum as input and returns an integer.
• Initialize bitSize to 0 and total to 0. bitSize will keep track of the cumulative bit length required to represent all numbers up to the current number directly.
• Iterate through each number from 1 to maximum. For each iteration:
  • Check if the current number is a power of two by employing a bitwise AND between the number and its predecessor (j & (j - 1)). If the result is 0, increment bitSize.
  • Update total by left shifting its current value by bitSize and then performing a bitwise OR with the current number (j). Immediately apply the modulo operation to keep the result manageable and within the limits of the data type.
• At the end of the loop, return total as the result.

The solution efficiently handles both the binary concatenation and the modulo constraint by leveraging bitwise operations, ensuring performance is optimized even for larger values of maximum.

class Solution {
    public int combineBinaries(int numbers) {
        final int MODULO = 1000000007;
        int bitLength = 0;
        long combinedResult = 0;
        for (int index = 1; index <= numbers; index++) {
            // Check the power of two to increment bit length
            if ((index & (index - 1)) == 0) {
                bitLength++;
            }
            combinedResult = ((combinedResult << bitLength) | index) % MODULO;
        }
        return (int) combinedResult;
    }
}

The solution provided tackles the problem of concatenating the binary representations of numbers from 1 up to a given integer and then computing the result modulo (10^9+7). The approach involves using bitwise operations efficiently within Java.

• First, define the modulo constant as (\text{MODULO} = 1000000007) to ensure the results fit within typical integer limits after concatenation.
• Initialize bitLength to keep track of the number of bits required to represent the current number in binary form.
• Initialize combinedResult as a long to accommodate the potentially large number resulting from the concatenations.
• Loop through each number from 1 to the given parameter numbers:
  • Check if the current number is a power of two using the condition (index & (index - 1)) == 0. If true, increment bitLength because an additional bit is needed whenever a power of two is reached.
  • Perform the concatenation in binary by left-shifting combinedResult by bitLength and then applying a bitwise OR operation with index. Post this operation, apply the modulo to the result to keep it within bounds.

Return the final combinedResult cast back to an int as the solution requires. Implement careful handling of binary arithmetic and mod operations to ensure efficiency and correctness throughout the operation.

class Solution:
    def concatenatedBinaryInteger(self, num: int) -> int:
        MODULO = 10**9 + 7
        bit_length = 0
        concatenated_value = 0
        for value in range(1, num + 1):
            if value & (value - 1) == 0:
                bit_length += 1
            concatenated_value = ((concatenated_value << bit_length) | value) % MODULO
        return concatenated_value

The task involves concatenating the binary representations of all integers from 1 up to a given number, num, and then returning the result as an integer modulo (10^9 + 7). The implemented solution in Python handles this by leveraging bitwise operations and modular arithmetic.

The algorithm operates as follows:

1. Initialize MODULO to (10^9 + 7), which is used to ensure the result remains manageable in size and avoids overflow.
2. Initialize bit_length to 0, which keeps track of the current number of bits required to represent the numbers encountered so far.
3. Initialize concatenated_value to 0, which will store the final concatenated binary value as an integer.
4. Use a loop to iterate through each integer from 1 to num.
5. For each integer, check if it's a power of two by using the condition (value & (value - 1) == 0). If true, increment bit_length because a new power of two introduces an additional bit in binary representation.
6. Compute the new concatenated value:
   • Left-shift concatenated_value by bit_length to make space for the bits of the current value.
   • Use the bitwise OR operation to append the bits of value to concatenated_value.
   • Apply modulo MODULO to the result to avoid overflow.
7. Finally, return concatenated_value after the loop completes.

By carefully managing the bit shifts and modular operations, the code efficiently concatenates binary numbers and ensures the resulting value is within the specified limits using modulo arithmetic. This approach addresses the potential exponential growth of the concatenated binary number by using properties of modulo operations to keep computations feasible and correct.