Sum of Two Integers without Arithmetic operators

#dsa #python

Instructions

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

Approach

We cannot use arithmetic operators, so we have to use bit manipulation to achieve addition.
The XOR operator is useful for bit manipulation where its output is shown below.

1^1=0
0^0=0
0^1=1

From the code above we observe that 1^1=0 which is not equal to 1+1. We can use the AND(&) operator to handle carry to calculate the correct answer. We shift the carry to the left and repeat the operations until we get a result of 0 meaning there are no more carries.

Here is an example:

Python Implementation

def getSum(self, a: int, b: int) -> int:
        while (b != 0):
            carry = a & b
            a = a ^ b
            b = carry << 1
        return a

The time complexity is 0(n).
The solution is sufficient for most scenarios but we can improve it by consider that the range of bits for representing a value is not 32 in Python.
Therefore, we have to use a 32 bit mask of 1's represented by 0xfffffff.
The mask expands a number into a 32 bit / unsigned integer.
Our solution becomes:

def getSum(self, a: int, b: int) -> int:    
        mask = 0xffffffff
        while (b & mask) > 0:
            carry = (a & b) << 1
            a = (a ^ b)
            b = carry
        return (a & mask) if b > 0 else a

Approach 2

We can use logarithms to achieve the same result.
From our knowledge of logarithms, we know that

2 ^a x 2 ^b = 2 ^a+b and
log _a (a ⁿ)= n therefore,
log ₂ (2 ^a+b)= a+b

Python Implementation

def getSum(self, a: int, b: int) -> int: 
    return int(math.log2(pow(2,a)*pow(2,b)))

Useful Links

Mask
Mask

Top comments (4)

matthew-salerno • Feb 18 '22

A friend of mine and I tried writing some simple functions in python without using any arithmetic operators or numeric literals. It was pretty fun, mostly boiled down to taking the length of lists of empty lists and other objectively cursed things. It was still a fun exercise though and definitely worth trying. I like your approach of using logic operators, although it still seems a little too close to normal addition to me. The log method was really neat and definitely not something I would have thought of.