Give a recursive definition of the function ones(s) which counts the number of ones in a bit string s.
To define the function ones(s) recursively, we can consider the following cases:
Case 1: Base Case
If the bit string s is empty (i.e., s = ""), then the number of ones in the string is 0. This is our base case.
Case 2: Recursive Cases
If the bit string s is not empty, we can define the function ones(s) as follows:
- If the first bit of the string s is 1, then ones(s) is equal to 1 + ones(rest), where rest is the remaining substring obtained by removing the first bit from s.
- If the first bit of the string s is 0, then ones(s) is equal to ones(rest), where rest is the remaining substring obtained by removing the first bit from s.
In both recursive cases, we are reducing the problem size by removing the first bit of the string and counting the number of ones in the remaining substring.
Therefore, the recursive definition for the function ones(s) can be written as:
ones("") = 0 (Base Case)
ones(s) = 1 + ones(rest) if s[0] = 1 (Recursive Case)
ones(s) = ones(rest) if s[0] = 0 (Recursive Case)
Note: The notation s[0] represents the first bit of the string s, and rest represents the substring obtained by removing the first bit from s.
Is it a pseudocode or algorithm that you are looking for?
Counting the number of ones in a string of size n is of order O(n), whether by a recursive function or by marching, since every bit has to be examined.