Divide Two Integers

Problem

https://leetcode.com/problems/divide-two-integers/

Given two integers dividend and divisor, divide two integers without using multiplication, division, and mod operator.

The integer division should truncate toward zero, which means losing its fractional part. For example, 8.345 would be truncated to 8, and -2.7335 would be truncated to -2.

Return the quotient after dividing dividend by divisor.

Note: Assume we are dealing with an environment that could only store integers within the 32-bit signed integer range: [−2:sup:`31`, 2:sup:`31`− 1]. For this problem, if the quotient is strictly greater than 2:sup:`31`- 1, then return 2:sup:`31`- 1, and if the quotient is strictly less than -2:sup:`31`, then return -2:sup:`31`.

Example 1:

Input: dividend = 10, divisor = 3
Output: 3
Explanation: 10/3 = 3.33333.. which is truncated to 3.

Example 2:

Input: dividend = 7, divisor = -3
Output: -2
Explanation: 7/-3 = -2.33333.. which is truncated to -2.

Constraints:

-2:sup:`31`<= dividend, divisor <= 2:sup:`31`- 1
divisor != 0

Pattern

Math, Bit Manipulation

Approaches

Subtraction

Explanation

We can use repeated subtraction of the dividend by the divisor to find the quotient without using multiplication, division, or modulus. However, loops in Python are slow so we will time out for large dividends and small divisors. Instead we can use len(range(0, stop, step)) to find the quotient. However,

len(range(0, stop, step)) == math.ceil(stop / step)

while we want math.floor(stop / step). To do this, we pull stop down by step - 1 or, equivalently, start at step - 1. So

start = abs(divisor) - 1
stop = abs(dividend)
step = abs(divisor)

The quotient should be negative when one of the dividend or divisor is negative. Finally, clamp the quotient to be in \([-2^{31}, 2^{31} - 1]\).

Code

def divide(dividend: int, divisor: int) -> int:
    """Given two integers ``dividend`` and ``divisor``, divide the two integers
    without using the multiplication, division, and mod operator.
    """
    quotient = len(range(abs(divisor) - 1, abs(dividend), abs(divisor)))

    if (dividend < 0) ^ (divisor < 0):
        quotient = -quotient

    return max(min(quotient, 2**31 - 1), -(2**31))

Test

>>> from divide_two_integers__subtraction import divide
>>> divide(10, 3)
3
>>> divide(7, -3)
-2

Complexity

\(q\) is the quotient

Measure	Complexity	Notes
Time	\(O(\|q\|)\)	calculating the quotient by counting the number of steps in the range
Auxiliary Space	\(O(1)\)	only the quotient variable; the `range` object is lazy and never materializes its elements

divide_two_integers__subtraction.divide(dividend: int, divisor: int) → int: Given two integers dividend and divisor, divide the two integers without using the multiplication, division, and mod operator.