GATE-CS-2009 Problem 4

Problem statement

You are given a partially written function to solve the following problem:

def maxZeros(n):
  '''Find the maximum number of consecutive zeros in the binary representation of given integer n.'''

  # Initialize count variable
  count = 0

  # Loop to check consecutive zeros
  while n!=0:
    # Case - 1
    if n%2==0:
      # TODO: Code to be written here
      pass

    # Case - 2
    elif n%2==1:
      # TODO: Code to be written here
      pass

  return count

Your task is to complete the above function so that it passes all the test cases.

Example

The function should return the expected output for the following test cases:

assert(maxZeros(45) == 4)
assert(maxZeros(550) == 4)
assert(maxZeros(1022) == 0)

Solution

To solve this problem, we need to understand that the maximum number of consecutive zeros in the binary representation of a number can be found by counting the number of trailing zeros in its binary representation.

For example, the binary representation of 45 is:

101101

The maximum number of consecutive zeros in this binary representation is 4, which occurs after the two ones in the middle. Therefore, we need to count the number of trailing zeros i.e. the number of zeros after the last one.

101101
     ^

To count the number of trailing zeros, we can repeatedly divide the given number by 2 as long as it is divisible by 2. Every time we divide it by 2, we shift its binary representation to the right by 1 bit. Eventually, this will leave us with the rightmost 1 in its binary representation. Then, the number of trailing zeros will be equal to the number of bits we shifted to the right.

101101 (45)
010110 (22)
001011 (11)
000101 (5)
000010 (2)
000001 (1)

In the above example, we divided 45 by 2 five times. Therefore, the maximum number of consecutive zeros in its binary representation is 5-1=4.

Now, let's complete the given function based on this logic:

def maxZeros(n):
  '''Find the maximum number of consecutive zeros in the binary representation of given integer n.'''

  # Initialize count variable
  count = 0

  # Loop to check consecutive zeros
  while n!=0:
    # Case - 1
    if n%2==0:
      count += 1 # Increment count of trailing zeros
      n //= 2    # Right shift binary representation

    # Case - 2
    elif n%2==1:
      n //= 2    # Right shift binary representation

  return count

In this function, we check whether the current bit is 0 or 1. If it is 0, we increment the count of trailing zeros and right shift the binary representation of the number. If it is 1, we just right shift the binary representation. We repeat this until the binary representation becomes zero.

Conclusion

In this problem, we learned how to find the maximum number of consecutive zeros in the binary representation of a number. This involved counting the number of trailing zeros in its binary representation. We achieved this by repeatedly dividing the number by 2 and counting the number of bits shifted to the right.