Problem Description

The problem requires us to convert a given integer n into its equivalent representation in base -2. Base -2 numbering system, also known as negabinary, is similar to binary except that the base is -2 instead of 2. This means that each digit position is worth -2 raised to the power of the position index, starting from 0. The challenge is to represent any integer n with digits 0 and 1 in base -2 such that there are no leading zeros, except when the resulting string should be "0".

Intuition

The solution is derived using a similar approach to converting numbers from decimal to binary, with some modifications due to the negative base. The conventional method to convert a decimal number to binary involves repeatedly dividing the number by 2 and taking the remainder as the next digit. The process continues until the number is completely reduced to 0. In the case of the negabinary system, we continue to divide by -2, but we need to handle negative remainders differently.

When converting to negabinary, if the remainder when dividing by -2 is negative, we adjust the number by adding 1 to prevent the use of negative digits. Since we're working with a negative base, adding 1 effectively subtracts the base from the original number, which in turn cancels out the negative remainder and keeps the number non-negative.

Each division gives us one binary digit, but due to the nature of base -2, we sometimes need to increment the quotient to handle the cases where the remainder would otherwise be negative. The pattern of division continues until the dividend (our number n) becomes zero. The algorithm appends '1' if the expression n % 2 evaluates to true (i.e., remainder is 1 when dividing by 2), which indicates that our current digit in negabinary is 1. Otherwise, it appends '0'. After each step, n is divided by 2 and k, which toggles between 1 and -1, is used to adjust n accordingly. The sequence of 1s and 0s is reversed at the end to get the final negabinary representation.

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Solution Approach

The Solution class defines a single method baseNeg2, which takes an integer n as input and returns a string representing the negabinary representation of that integer.

Here's how the implementation works:

• We initialize a variable k with the value 1. This will be used to alternate the subtraction value between 1 and -1 according to the negabinary rules.
• We create an empty list ans that will eventually contain the digits of the negabinary number, albeit in reverse order.
• We enter a while loop that will keep running until n is reduced to zero. On each iteration, we perform the following steps:
  • Check if n % 2 is 1 or 0 by using the modulo operator. This is equivalent to finding out if there is a remainder when n is divided by 2:
    • If n % 2 evaluates to 1, it means the current binary digit should be '1', and we append '1' to the ans list.
    • Furthermore, we adjust n by subtracting k. Since the base is negative, when the remainder is 1, we need to compensate by decrementing n by 1 when k is 1 or incrementing by 1 when k is -1. This ensures we are correctly building the negabinary representation.
    • If n % 2 evaluates to 0, we simply append '0' to the ans list. There is no need to adjust n since the remainder is 0.
  • Update n to be half of its previous value, which is done by using integer division n //= 2. Integer division truncates towards zero, making sure that we are getting closer to zero on each division.
  • Multiply k by -1. This simple operation alternates the value of k between 1 and -1, ensuring that we properly adjust n in the next iteration to account for the negative base.
• Once the while loop exits, we reverse the ans list to represent the digits in the correct order since we've been appending them in reverse.
• We then join the list into a string to represent the negabinary number. If the list is empty, which would occur if n were 0, we return '0'.

By the end of the loop, the ans list will contain the negabinary digits in reverse order, so we need to reverse it to get the correct representation. We perform this reverse operation using ans[::-1]. If the number n is 0, then ans will be an empty list, in which case we return '0'.

The Python join function concatenates the elements of the list into a single string, which is the final negabinary representation we return from our method.

This approach cleverly utilizes modular arithmetic and the property of negabinary representation to efficiently convert an integer to its equivalent in base -2. No additional data structures besides the list ans are used, and the entire conversion happens in a single pass through the loop with an O(log n) time complexity.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach by converting the decimal number 6 to its equivalent in base -2.

1. Start with n = 6 and initialize k = 1.
2. Initialize an empty list ans which will hold the digits of the negabinary number in reverse order.

The while-loop begins:

• Iteration 1:

  • n % 2 is 0 (since 6 is even), so we append '0' to ans. (ans = ['0'])
  • We do not adjust n because there is no remainder.
  • Now, perform integer division n //= 2, which gives n = 3.
  • Multiply k by -1 to get k = -1.

• Iteration 2:

  • n % 2 is 1, so we append '1' to ans. (ans = ['0', '1'])
  • Adjust n by subtracting -1 (since k = -1), giving us n = 3 - (-1) = 4.
  • Perform integer division n //= 2, yielding n = 2.
  • Multiply k by -1 to get k = 1.

• Iteration 3:

  • n % 2 is 0, so we append '0' to ans. (ans = ['0', '1', '0'])
  • n does not need adjustment as the remainder is 0.
  • Perform integer division n //= 2, which results in n = 1.
  • Multiply k by -1 to get k = -1.

• Iteration 4:

  • n % 2 is 1, so we append '1' to ans. (ans = ['0', '1', '0', '1'])
  • Adjust n by subtracting -1 (since k = -1), which makes n = 1 - (-1) = 2.
  • Perform integer division n //= 2, yielding n = 1.
  • Multiply k by -1 to get k = 1.

• Iteration 5:

  • n % 2 is 1, so we append '1' to ans. (ans = ['0', '1', '0', '1', '1'])
  • Adjust n by subtracting 1 (since k = 1), which leaves n = 1 - 1 = 0.
  • The division step and multiplication of k by -1 are no longer necessary as n is now 0, so the while loop terminates.

Now, ans = ['0', '1', '0', '1', '1'] holds the negabinary digits in reverse order. To fix this, we reverse ans to get ['1', '1', '0', '1', '0'].

Finally, join all elements of the list ans to get the string '11010', which is the negabinary representation of the decimal number 6.

Thus, the baseNeg2 method would return '11010' when called with the input 6.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(log(n)). This is because the loop iterates once for every bit that n produces when converging to 0. Since we're halving n at every step, albeit with an additional operation to account for negative base representation, the number of iterations grows logarithmically with respect to the input n.

The space complexity is O(log(n)) as well, this is due to the ans list that is used to store the binary representation of the number in base -2. The number of digits needed to represent n in base -2 scales similarly to the number of iterations, hence the space complexity follows the same logarithmic pattern.

Learn more about how to find time and space complexity quickly using problem constraints.