Sum of Two Integers

mediumFREEBit ManipulationBlind 75

Given two integers a and b, return their sum without using the + or - operators.

Example

Input

1
2

Output

Reasoning

Inputs: a and b are ordinary signed integers, each between $-1000$ and $1000$ . Here a $= 1$ and b $= 2$ .
Output: a single integer equal to $a + b$ , so the sample answer is $1 + 2 = 3$ . The challenge is producing it without the plus or minus operators.
The math in base 2: $1 = 01_2$ and $2 = 10_2$ . Adding column by column, the ones column holds $1 + 0 = 1$ and the twos column holds $0 + 1 = 1$ . No column ever sums to $2$ , so nothing spills into the next column, and the result is $11_2 = 2^1 + 2^0 = 3$ .
Notice: in a case like $2 + 3$ ( $10_2 + 11_2$ ) the twos column holds two 1s, which spills a 1 into the fours column and gives $101_2 = 5$ .
Signs matter: for $-1 + 1$ the answer must be exactly 0, and a negative pair such as $-5 + 3$ must come back as $-2$ , a genuine negative integer.

Constraints

-1000 <= a, b <= 1000

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Concept primer

Background Knowledge

The problem "Sum of Two Integers" falls under the topic of Bit Manipulation, which involves manipulating the binary representation of numbers to perform operations. To tackle this problem, it's essential to understand how numbers are represented in binary and how bitwise operations work. In binary, each digit (or bit) can be either 0 or 1, and numbers are represented using a series of bits. For example, the decimal number 5 is represented as 101 in binary.

Bitwise operations are used to manipulate the bits of a number. The three primary bitwise operations are AND (&), OR (|), and XOR (^). The XOR operation is particularly useful for this problem, as it returns 1 if the two bits are different and 0 if they are the same. This operation can be used to add two numbers without using the + operator. Additionally, understanding how to use bitwise shifts (<< and >>) to multiply or divide numbers by powers of 2 is crucial.

To solve this problem, you'll also need to understand the concept of carry in addition. When adding two numbers, a carry is generated when the sum of two bits is greater than 1. This carry is then added to the next bit position. In binary, a carry is equivalent to a 1 being carried over to the next position. By using bitwise operations to handle the carry, you can effectively add two numbers without using the + operator.

Algorithm/Approach

The general approach to solving this problem involves using bitwise operations to add the two numbers. The algorithm will iterate through the bits of the two numbers, performing XOR and AND operations to calculate the sum and carry. The XOR operation will be used to calculate the sum of the bits without considering the carry, while the AND operation will be used to calculate the carry.

Step-by-Step Strategy

To implement the solution:

Initialize a variable to store the sum and another to store the carry.
Use a loop to iterate through the bits of the two numbers.
Inside the loop, use the XOR operation to calculate the sum of the current bits without considering the carry.
Use the AND operation to calculate the carry.
Use bitwise shifts to move the carry to the next bit position.
Repeat steps 3-5 until there is no carry left.

Common Pitfalls

When implementing the solution, watch out for:

Not handling the carry correctly
Not using bitwise operations correctly
Not iterating through all the bits of the two numbers

Time & Space Complexity

The expected time complexity for this problem is O(log(max(a, b))), where a and b are the input integers. This is because the algorithm needs to iterate through the bits of the two numbers, and the number of bits is proportional to the logarithm of the maximum value. The space complexity is O(1), as the algorithm only uses a constant amount of space to store the sum and carry.

solution.py

Python 3.13

Test results