Shifts and masks

Understand the idea

Left shift by k multiplies a nonnegative integer by 2^k; right shift divides and floors. A mask 1<<i isolates a bit. Define width and signed behavior before translating across languages.

Work through an example

13>>2=3; 3<<2=12. Shifting loses low bits when shifting right. (13>>2)<<2 is not necessarily 13.

Time and space complexity

Fixed-width shift O(1); unbounded result size affects cost/storage.

Why it works

Shifted bits move by the specified number of positions, with explicit width rules.

Common mistakes

• Negative shift counts are invalid.
• Right shift of negatives can be arithmetic rather than logical.

Chapter lab · Python implementation

Bit operations treat an integer as binary positions. Our lab clears the lowest set bit repeatedly and enumerates subsets with masks. Examples use nonnegative integers unless an explicit width is supplied. Python integers are unbounded, so signed NOT and shifting differ from unsigned fixed-width machine examples.

This shared lab connects the chapter's topics. Follow the current lesson's worked example first, then trace the complete implementation. Additional algorithms described in topic exercises are variations you can implement independently.

def count_bits(value):
    if value<0: raise ValueError('nonnegative value required')
    count=0
    while value:
        value &= value-1
        count+=1
    return count

def subsets_mask(values):
    output=[]
    for mask in range(1<<len(values)):
        output.append([x for i,x in enumerate(values) if mask & (1<<i)])
    return output

def complement(value,width):
    if width<1 or not 0<=value<(1<<width):
        raise ValueError('value must fit width')
    return (~value)&((1<<width)-1)

Run and test the lab

Copy the implementation and these assertions into a Python 3 file and run it. An assertion failure means the actual result differs from the expectation. Add the current topic's exercise and boundary tests to deepen your understanding.

assert count_bits(13)==3 and count_bits(0)==0
assert complement(5,4)==10
assert subsets_mask(['a','b'])==[[],['a'],['b'],['a','b']]
assert all(count_bits(x)==bin(x).count('1') for x in range(256))