Perform multiplication of any two integers positive or negative through bit manipulations (#201)

This commit is contained in:
ADITYA 2018-09-08 15:51:03 -04:00 committed by Oleksii Trekhleb
parent 1a62078f26
commit bc8943dee2
5 changed files with 109 additions and 14 deletions

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@ -2,10 +2,10 @@
#### Get Bit
This method shifts the relevant bit to the zeroth position.
Then we perform `AND` operation with one which has bit
This method shifts the relevant bit to the zeroth position.
Then we perform `AND` operation with one which has bit
pattern like `0001`. This clears all bits from the original
number except the relevant one. If the relevant bit is one,
number except the relevant one. If the relevant bit is one,
the result is `1`, otherwise the result is `0`.
> See [getBit.js](getBit.js) for further details.
@ -24,7 +24,7 @@ other bits of the number.
This method shifts `1` over by `bitPosition` bits, creating a
value that looks like `00100`. Than it inverts this mask to get
the number that looks like `11011`. Then `AND` operation is
being applied to both the number and the mask. That operation
being applied to both the number and the mask. That operation
unsets the bit.
> See [clearBit.js](clearBit.js) for further details.
@ -35,21 +35,35 @@ This method is a combination of "Clear Bit" and "Set Bit" methods.
> See [updateBit.js](updateBit.js) for further details.
#### isEven
This method determines if the number provided is even.
```
Number: 5
isEven: false
Number: 4
isEven: true
```
> See [isEven.js](isEven.js) for further details.
#### Multiply By Two
This method shifts original number by one bit to the left.
Thus all binary number components (powers of two) are being
multiplying by two and thus the number itself is being
multiplying by two and thus the number itself is being
multiplied by two.
```
Before the shift
Number: 0b0101 = 5
Powers of two: 0 + 2^2 + 0 + 2^0
Powers of two: 0 + 2^2 + 0 + 2^0
After the shift
Number: 0b1010 = 10
Powers of two: 2^3 + 0 + 2^1 + 0
Powers of two: 2^3 + 0 + 2^1 + 0
```
> See [multiplyByTwo.js](multiplyByTwo.js) for further details.
@ -58,17 +72,17 @@ Powers of two: 2^3 + 0 + 2^1 + 0
This method shifts original number by one bit to the right.
Thus all binary number components (powers of two) are being
divided by two and thus the number itself is being
divided by two and thus the number itself is being
divided by two without remainder.
```
Before the shift
Number: 0b0101 = 5
Powers of two: 0 + 2^2 + 0 + 2^0
Powers of two: 0 + 2^2 + 0 + 2^0
After the shift
Number: 0b0010 = 2
Powers of two: 0 + 0 + 2^1 + 0
Powers of two: 0 + 0 + 2^1 + 0
```
> See [divideByTwo.js](divideByTwo.js) for further details.
@ -87,11 +101,29 @@ inverting all of the bits of the number and adding 1 to it.
0001 1
0010 2
0011 3
```
```
> See [switchSign.js](switchSign.js) for further details.
#### Multiply Two Numbers
#### Multiply Two Signed Numbers
This method multiplies two signed integer numbers using bitwise operators.
This method is based on the following :
```text
a * b can be written in the below formats
0 if a is zero or b is zero or both a and b are zeroes
2a * (b/2) if b is even
2a * (b - 1)/2 + a if b is odd and positive
2a * (b + 1)/2 - a if b is odd and negative
```
The advantage of this approach is that in each recursive step one of the operands reduces to half its original value.
Hence, the run time complexity is O(log b) where b is the operand that reduces to half on each recursive step.
> See [multiply.js](multiply.js) for further details.
#### Multiply Two Unsigned Numbers
This method multiplies two integer numbers using bitwise operators.
This method is based on that "Every number can be denoted as the sum of powers of 2".
@ -111,7 +143,7 @@ Then multiplying number `x` by `19` is equivalent of:
x * 19 = x * 2^4 + x * 2^1 + x * 2^0
```
Now we need to remember that `x * 2^4` is equivalent of shifting `x` left
Now we need to remember that `x * 2^4` is equivalent of shifting `x` left
by `4` bits (`x << 4`).
> See [multiplyUnsigned.js](multiplyUnsigned.js) for further details.
@ -158,7 +190,7 @@ When we shift 1 four times it will become bigger than 5.
#### Is Power of Two
This method checks if a number provided is power of two. It uses the following
This method checks if a number provided is power of two. It uses the following
property. Let's say that `powerNumber` is a number that has been formed as a power
of two (i.e. 2, 4, 8, 16 etc.). Then if we'll do `&` operation between `powerNumber`
and `powerNumber - 1` it will return `0` (in case if number is power of two).

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import isEven from '../isEven';
describe('isEven', () => {
it('should detect if a number is even', () => {
expect(isEven(0)).toBe(true);
expect(isEven(2)).toBe(true);
expect(isEven(-2)).toBe(true);
expect(isEven(1)).toBe(false);
expect(isEven(-1)).toBe(false);
});
});

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import multiply from '../multiply';
describe('multiply', () => {
it('should multiply two numbers', () => {
expect(multiply(0, 0)).toBe(0);
expect(multiply(2, 0)).toBe(0);
expect(multiply(0, 2)).toBe(0);
expect(multiply(1, 2)).toBe(2);
expect(multiply(2, 1)).toBe(2);
expect(multiply(6, 6)).toBe(36);
expect(multiply(-2, 4)).toBe(-8);
expect(multiply(4, -2)).toBe(-8);
expect(multiply(-4, -4)).toBe(16);
expect(multiply(4, -5)).toBe(-20);
});
});

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/**
* @param {number} number
* @return bool
*/
export default function isEven(number) {
return (number & 1) === 0;
}

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import divideByTwo from './divideByTwo';
import isEven from './isEven';
import multiplyByTwo from './multiplyByTwo';
/**
* FUNCTION DEFINITION
* multiply(a, b) = 0 if a is zero or b is zero or if both a and b are zeros
* multiply(a, b) = multiply(2a, b/2) if b is even
* multiply(a, b) = multiply(2a, (b-1)/2) + a if b is odd and b is positive
* multiply(a, b) = multiply(2a, (b+1)/2) - a if b is odd and b is negative
*
* COMPLEXITY
* O(log b)
* @param {number} a
* @param {number} b
* @return {number} a * b
*/
export default function multiply(a, b) {
if (b === 0 || a === 0) {
return 0;
}
const multiplyByOddPositive = () => multiply(multiplyByTwo(a), divideByTwo(b - 1)) + a;
const multiplyByOddNegative = () => multiply(multiplyByTwo(a), divideByTwo(b + 1)) - a;
const multiplyByEven = () => multiply(multiplyByTwo(a), divideByTwo(b));
const multiplyByOdd = () => (b > 0 ? multiplyByOddPositive() : multiplyByOddNegative());
return isEven(b) ? multiplyByEven() : multiplyByOdd();
}