371. Sum of Two Integers - Explanation

Description

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

Example 1:

Input: a = 1, b = 1

Output: 2

Example 2:

Input: a = 4, b = 7

Output: 11

Constraints:

-1000 <= a, b <= 1000

Topics

Math Bit Manipulation

Recommended Time & Space Complexity

You should aim for a solution with O(1) time and O(1) space.

Hint 1

A brute force approach would use the addition operator. Can you think of a way to perform addition without using it? Maybe you should consider solving this using bit manipulation.

Hint 2

We can use the bitwise XOR operator to perform addition. If both a and b have 1 at the same bit position, the sum at that position is 0, and a carry of 1 is generated. If the bits are different, the sum at that position is 1. Additionally, we account for the carry from the previous step in the next iteration.

Hint 3

We iterate bit by bit from 0 to 31 since the given integers are 32-bit. We track the carry, initially set to 0, and initialize the result as res. During iteration, the XOR of the bits at the i-th position of both integers and the carry determines the current bit of res. How can you handle negative numbers?

Hint 4

To handle negative numbers, if the final result exceeds the maximum positive 32-bit integer, it means the number should be negative. We adjust it using bitwise operations: flipping the bits with res ^ ((2 ^ 32) - 1) and applying ~ to restore the correct two’s complement representation. This ensures the result correctly represents signed 32-bit integers.

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Prerequisites

Before attempting this problem, you should be comfortable with:

Bit Manipulation Basics - Understanding bitwise operators (AND, OR, XOR, NOT) and how they work at the binary level
Two's Complement Representation - How negative numbers are represented in binary using 32-bit signed integers
Binary Addition - Understanding how carry propagation works when adding binary numbers bit by bit

1. Brute Force

Intuition

The goal is to compute the sum of two integers.

In the brute force approach, we rely directly on the language’s built-in arithmetic addition operator. This is the most straightforward and intuitive solution because:

Addition is a fundamental operation supported natively by all programming languages
The language runtime already handles all edge cases such as:
- negative numbers
- carry propagation
- integer representation

This approach focuses purely on correctness and simplicity, without worrying about implementation details.

Algorithm

Take the two input integers a and b.
Use the built-in addition operation to compute a + b.
Return the result.

class Solution:
    def getSum(self, a: int, b: int) -> int:
        return a + b

Time & Space Complexity

Time complexity: $O(1)$
Space complexity: $O(1)$

2. Bit Manipulation

Intuition

The problem asks us to compute the sum of two integers without using the + or - operators.

At the bit level, addition works using two simple ideas:

XOR (^) gives the sum of two bits without considering carry
AND (&) + left shift determines where a carry is generated

For example (single bit):

0 + 0 → sum = 0, carry = 0
1 + 0 → sum = 1, carry = 0
1 + 1 → sum = 0, carry = 1

By repeating this logic for all bit positions, we can simulate normal addition exactly as it happens in hardware.

Because integers are stored in fixed-width (32-bit) two's complement form, we also need to:

limit results to 32 bits
correctly convert the result back if it represents a negative number

Algorithm

Initialize:
- result to store the final sum
- carry = 0
- a mask to keep numbers within 32 bits
For each bit position from 0 to 31:
- Extract the i-th bit from both numbers
- Compute the current sum bit using XOR:
```
sum_bit = a_bit XOR b_bit XOR carry
```
- Update the carry:
```
carry = (a_bit + b_bit + carry) ≥ 2
```
- Set the i-th bit in the result if sum_bit is 1
After processing all bits:
- If the result represents a negative number in 32-bit two's complement:
  - Convert it back to a signed integer
Return the result

class Solution:
    def getSum(self, a: int, b: int) -> int:
        carry = 0
        res = 0
        mask = 0xFFFFFFFF

        for i in range(32):
            a_bit = (a >> i) & 1
            b_bit = (b >> i) & 1
            cur_bit = a_bit ^ b_bit ^ carry
            carry = (a_bit + b_bit + carry) >= 2
            if cur_bit:
                res |= (1 << i)

        if res > 0x7FFFFFFF:
            res = ~(res ^ mask)

        return res

public class Solution {
    public int getSum(int a, int b) {
        int carry = 0, res = 0, mask = 0xFFFFFFFF;

        for (int i = 0; i < 32; i++) {
            int a_bit = (a >> i) & 1;
            int b_bit = (b >> i) & 1;
            int cur_bit = a_bit ^ b_bit ^ carry;
            carry = (a_bit + b_bit + carry) >= 2 ? 1 : 0;
            if (cur_bit != 0) {
                res |= (1 << i);
            }
        }

        if (res > 0x7FFFFFFF) {
            res = ~(res ^ mask);
        }

        return res;
    }
}

class Solution {
public:
    int getSum(int a, int b) {
        int carry = 0, res = 0, mask = 0xFFFFFFFF;

        for (int i = 0; i < 32; i++) {
            int a_bit = (a >> i) & 1;
            int b_bit = (b >> i) & 1;
            int cur_bit = a_bit ^ b_bit ^ carry;
            carry = (a_bit + b_bit + carry) >= 2 ? 1 : 0;
            if (cur_bit) {
                res |= (1 << i);
            }
        }

        if (res > 0x7FFFFFFF) {
            res = ~(res ^ mask);
        }

        return res;
    }
};

class Solution {
    /**
     * @param {number} a
     * @param {number} b
     * @return {number}
     */
    getSum(a, b) {
        let carry = 0,
            res = 0,
            mask = 0xffffffff;

        for (let i = 0; i < 32; i++) {
            let a_bit = (a >> i) & 1;
            let b_bit = (b >> i) & 1;
            let cur_bit = a_bit ^ b_bit ^ carry;
            carry = a_bit + b_bit + carry >= 2 ? 1 : 0;
            if (cur_bit) {
                res |= 1 << i;
            }
        }

        if (res > 0x7fffffff) {
            res = ~(res ^ mask);
        }

        return res;
    }
}

public class Solution {
    public int GetSum(int a, int b) {
        int carry = 0, res = 0, mask = unchecked((int)0xFFFFFFFF);

        for (int i = 0; i < 32; i++) {
            int a_bit = (a >> i) & 1;
            int b_bit = (b >> i) & 1;
            int cur_bit = a_bit ^ b_bit ^ carry;
            carry = (a_bit + b_bit + carry) >= 2 ? 1 : 0;
            if (cur_bit != 0) {
                res |= (1 << i);
            }
        }

        if (res > Int32.MaxValue) {
            res = ~(res ^ mask);
        }

        return res;
    }
}

func getSum(a int, b int) int {
    carry := 0
    res := 0
    mask := 0xFFFFFFFF

    for i := 0; i < 32; i++ {
        aBit := (a >> i) & 1
        bBit := (b >> i) & 1
        curBit := aBit ^ bBit ^ carry
        if (aBit + bBit + carry) >= 2 {
            carry = 1
        } else {
            carry = 0
        }
        if curBit == 1 {
            res |= (1 << i)
        }
    }

    if res > 0x7FFFFFFF {
        res = ^(res ^ mask)
    }

    return res
}

class Solution {
    fun getSum(a: Int, b: Int): Int {
        var carry = 0
        var res = 0
        val mask = 0xFFFFFFFF.toInt()

        for (i in 0 until 32) {
            val aBit = (a shr i) and 1
            val bBit = (b shr i) and 1
            val curBit = aBit xor bBit xor carry
            carry = if (aBit + bBit + carry >= 2) 1 else 0
            if (curBit == 1) {
                res = res or (1 shl i)
            }
        }

        if (res > 0x7FFFFFFF) {
            res = res.inv() xor mask
        }

        return res
    }
}

class Solution {
    func getSum(_ a: Int, _ b: Int) -> Int {
        var carry = 0
        var res = 0
        let mask = 0xFFFFFFFF

        for i in 0..<32 {
            let aBit = (a >> i) & 1
            let bBit = (b >> i) & 1
            let curBit = aBit ^ bBit ^ carry
            carry = (aBit + bBit + carry) >= 2 ? 1 : 0
            if curBit == 1 {
                res |= (1 << i)
            }
        }

        if res > 0x7FFFFFFF {
            res = ~(res ^ mask)
        }

        return res
    }
}

impl Solution {
    pub fn get_sum(a: i32, b: i32) -> i32 {
        let mut carry = 0i32;
        let mut res = 0i32;

        for i in 0..32 {
            let a_bit = (a >> i) & 1;
            let b_bit = (b >> i) & 1;
            let cur_bit = a_bit ^ b_bit ^ carry;
            carry = if a_bit + b_bit + carry >= 2 { 1 } else { 0 };
            if cur_bit == 1 {
                res |= 1 << i;
            }
        }

        res
    }
}

Time & Space Complexity

Time complexity: $O(1)$
Space complexity: $O(1)$

3. Bit Manipulation (Optimal)

Intuition

We need to add two integers without using + or -.
Binary addition can be built from two operations:

Sum without carry
- a XOR b gives the bit-by-bit sum ignoring carry
  (because 1 XOR 1 = 0, which matches sum without carry)
Carry information
- a AND b tells us where both bits are 1, which creates a carry
- shifting left by 1 (<< 1) moves that carry to the next higher bit

So we can repeatedly:

compute the carry
update the partial sum using XOR
add the carry again (by setting b = carry)

We keep doing this until there is no carry left (b == 0).

Because many languages use fixed-width integers (like 32-bit signed integers), we use a mask to keep only the lower 32 bits at each step. Finally, if the result represents a negative number in 32-bit two's complement form, we convert it back to a signed integer.

Algorithm

Define constants for 32-bit handling:
- mask to keep only 32 bits
- max_int as the largest 32-bit signed integer
While b is not zero:
- Compute carry:
  - carry = (a AND b) << 1
- Compute sum without carry:
  - a = (a XOR b), then apply the 32-bit mask
- Move carry into b (also masked to 32 bits)
After the loop, a holds the 32-bit result.
If a is within signed range, return it directly.
Otherwise, convert from unsigned 32-bit to a negative signed value and return it.

class Solution:
    def getSum(self, a: int, b: int) -> int:
        mask = 0xFFFFFFFF
        max_int = 0x7FFFFFFF

        while b != 0:
            carry = (a & b) << 1
            a = (a ^ b) & mask
            b = carry & mask

        return a if a <= max_int else ~(a ^ mask)

public class Solution {
    public int getSum(int a, int b) {
        while (b != 0) {
            int carry = (a & b) << 1;
            a ^= b;
            b = carry;
        }
        return a;
    }
}

class Solution {
    /**
     * @param {number} a
     * @param {number} b
     * @return {number}
     */
    getSum(a, b) {
        while (b !== 0) {
            let carry = (a & b) << 1;
            a ^= b;
            b = carry;
        }
        return a;
    }
}

public class Solution {
    public int GetSum(int a, int b) {
        while (b != 0) {
            int carry = (a & b) << 1;
            a ^= b;
            b = carry;
        }
        return a;
    }
}

func getSum(a int, b int) int {
    mask := 0xFFFFFFFF
    maxInt := 0x7FFFFFFF

    for b != 0 {
        carry := (a & b) << 1
        a = (a ^ b) & mask
        b = carry & mask
    }

    if a <= maxInt {
        return a
    }
    return ^(a ^ mask)
}

class Solution {
    fun getSum(a: Int, b: Int): Int {
        var carry = 0
        var res = 0
        val mask = 0xFFFFFFFF.toInt()

        for (i in 0 until 32) {
            val aBit = (a shr i) and 1
            val bBit = (b shr i) and 1
            val curBit = aBit xor bBit xor carry
            carry = if (aBit + bBit + carry >= 2) 1 else 0
            if (curBit == 1) {
                res = res or (1 shl i)
            }
        }

        if (res > 0x7FFFFFFF) {
            res = res.inv() xor mask
        }

        return res
    }
}

class Solution {
    func getSum(_ a: Int, _ b: Int) -> Int {
        let mask = 0xFFFFFFFF
        let maxInt = 0x7FFFFFFF
        var a = a
        var b = b

        while b != 0 {
            let carry = (a & b) << 1
            a = (a ^ b) & mask
            b = carry & mask
        }

        return a <= maxInt ? a : ~(a ^ mask)
    }
}

impl Solution {
    pub fn get_sum(a: i32, b: i32) -> i32 {
        let mut a = a;
        let mut b = b;
        while b != 0 {
            let carry = (a & b) << 1;
            a ^= b;
            b = carry;
        }
        a
    }
}

Time & Space Complexity

Time complexity: $O(1)$
Space complexity: $O(1)$

Common Pitfalls

Forgetting to Handle Negative Numbers

When using bit manipulation to add integers, negative numbers are represented in two's complement form. In languages like Python that have arbitrary-precision integers, you must explicitly mask results to 32 bits and convert back to signed representation when the result exceeds 0x7FFFFFFF. Failing to do this will produce incorrect results for negative inputs or negative sums.

Infinite Loop with Negative Carry

In languages with fixed-width integers (like Java, C++), left-shifting a negative carry can cause issues. When b becomes negative and you compute (a & b) << 1, the carry propagates correctly due to two's complement representation. However, in Python without masking, the carry can grow indefinitely, causing an infinite loop. Always apply a 32-bit mask to both a and b in each iteration.

Confusing XOR and AND Operations

A common mistake is mixing up the roles of XOR and AND in the addition algorithm. XOR (^) computes the sum without carry, while AND (&) followed by left shift computes the carry. Swapping these operations or forgetting the left shift on the carry will produce wrong results. Remember: XOR gives you what stays, AND shifted gives you what carries over.