Reversing XOR and Bitwise operation in Python

Question

I tried searching a lot but I am not able to find a solution to reversing XOR and Bitwise operation combined.

num[i] = num[i]^( num[i] >> 1 );

How can I reverse this operation using Python. I tried the XOR concept explained here: What is inverse function to XOR?

Still unable to solve the math.

Answer 1

That's Gray code. There's also a chapter about it in Hackers' Delight. There's some code in that wikipedia article, but to avoid a link only answer, here's how you construct the inverse:
do x ^= x >> (1 << i) for i = 0 .. ceil(log_2(bits)) - 1.

So for 32 bits integers,

x ^= x >> 1;
x ^= x >> 2;
x ^= x >> 4;
x ^= x >> 8;
x ^= x >> 16;

For n-bit integers: (not fully tested, but seems to work so far)

def gray2binary(x):
    shiftamount = 1;
    while x >> shiftamount:
        x ^= x >> shiftamount
        shiftamount <<= 1
    return x

Answer 2

For a much faster version of the conversion see @harold's answer.

---

Let's consider 2-bit numbers:

00 = 00 ^ 00 (0 -> 0)
01 = 01 ^ 00 (1 -> 1)
11 = 10 ^ 01 (2 -> 3)
10 = 11 ^ 01 (3 -> 2)

If y[i] is i-th bit (little-endian) then from y = x ^ (x >> 1) follows:

y[1]y[0] = x[1]x[0] ^ 0x[1] # note: y[1]y[0] means `(y[1] << 1) | y[0]` here

It means that:

y[1] = x[1] ^ 0
y[0] = x[0] ^ x[1]

If we know y then to get x:

 y[i] = (y & ( 1 << i )) >> i
 x[1] = y[1] ^ 0
 x[0] = y[0] ^ x[1] = y[0] ^ (y[1] ^ 0)
 x = (x[1] << 1) | x[0]

You can generalize it for n-bit number:

def getbit(x, i):
    return (x >> i) & 1

def y2x(y):
    assert y >= 0    
    xbits = [0] * (y.bit_length() + 1)
    for i in range(len(xbits) - 2, -1, -1):
        xbits[i] = getbit(y, i) ^ xbits[i + 1] 

    x = 0
    for i, bit in enumerate(xbits):
        x |= (bit << i) 
    return x

y2x() could be simplified to work with numbers without the bit array:

def y2x(y):
    assert y >= 0    
    x = 0
    for i in range(y.bit_length() - 1, -1, -1):
        if getbit(y, i) ^ getbit(x, i + 1):
            x |= (1 << i) # set i-th bit
    return x

Example

print("Dec Gray Binary")
for x in range(8):
    y = x ^ (x >> 1)
    print("{x: ^3} {y:03b}  {x:03b}".format(x=x, y=y))
    assert x == y2x(y)

Output

Dec Gray Binary
 0  000  000
 1  001  001
 2  011  010
 3  010  011
 4  110  100
 5  111  101
 6  101  110
 7  100  111