Math Without Operator To Do It

Division/Power Without Exact Operators

I see this sort of questions among algorithm problems. I wrote Addtion without +/- operators in the post about XOR related questions. “Divide without divison” and “power without its operator or function” are examples as well.

Not like the addition, a divison and power need to repeat. Intuitive implementations would be simply repeat subtraction or multiplication. Those calculates correctly, however, time complexity tends to be O(n). Better ways are out there.

I’m going to write a memo how to calculate effectively such Math-y stuff without exact operators to divide/power.

Problem Description - Divide without division and mod

“Given two integers, dividend and divisor, calculate division without divide and modulo operators.”

The division itself is nothing special. An answer will be integer when diviend is devidend by divisor.

For example, given 10 (dividend) and -3 (divisor), the answer will be -3.

The idea to divide without division/modulo operators

Simple solution is count up one by one starting from zero, while subtracting the divisor from dividend. A counter is the quotient, which is the answer.

int sign = ...
int a = Math.abs(dividend);
int b = Math.abs(divisor);
int count = 0;
while (a >= b) {
    a -= b;
    count++;
}
return sign * count;

This solution’s performance is considered O(n), where n is the quotient. This is simple and correct, but surely can be improved.

To improve above, I should see the number a bit different way. All numbers can be expressed as:


X = x_0 * (2 ^ 0) + x_1 * (2 ^ 1) + x_2 * (2 ^ 2) + ..... + x_n * (2 ^ n)

For example, 10 is 10 * (2 ^ 0) + 1 * (2 ^ 1) + 0 * (2 ^ 2) + 1 * (2 ^ 3), and 3 is 1 * (2 ^ 0) + 1 * (2 ^ 1).

If I write how to manually calculate the division:


         1 1      <- quotient
    ---------
1 1) 1 0 1 0
       1 1
    ---------
       1 0 0
         1 1
    ---------
           1      <-- modulo

The first step is to find where is the highest bit. While shifting divisor to the left and counting how many time to repeat, I will get the answer.


devidend: 10 = 1010
divisor: 3 = 11
count: 1

the first iteration:
divisor: 11 << 1 => 110
count:   1 << 1  => 10

the second iteration
divisor: 110 << 1 => 1100 (exceeds 1010)
count:    10 << 1 => 100

The second step is to find what is the quotient. I’ve already learned how many times to repeat. So, I want to know each bit is zero or one.

This step goes like in the manual calculation.

1010 - 110 = 100    --> the second bit is one
100 - 11 = 1        --> the first bit is one
1 - ... (no more)

This divison’s time complexity turns to O(log(n)) .

Java code for dividing number without division

The result is:

-3
228

Problem Description - Impelemnt Power

Given two integers, x and y, implmenet a function (method) to calculate x ^ y.

I believe every computer language provides a way to calculate a power out of the box. However, the problem asks the implementation without using such existing feature.

The idea to construct from/to a string with markers

Also, there’s a simple solution. Repeating multiplication y times gives me the answer. The perfomrance of this simple solution will be O(n) (n = y).

There’s a way to improve this.

The improvied version calculates x ^ (y / 2) recursively. While returning, calcuate (x ^ (y / 2)) * (x ^ (y / 2)). If y is odd, multiply x.


y is even:   y = 2n

x ^ y = x ^ (2n) = (x ^ n) * (x ^ n)

y is odd:    y = 2n + 1

x ^ y = x ^ (2n + 1) = x * x ^ (2n) = x * (x ^ n) * (x ^ n)

This way, the time complexity goes down to O(log(n)).

Java code for constructing a binary tree from a string with markers

The result is:

1024
-512

Resources


Yoko Harada

Mostly for myself, writes memos about what I learned.