Python string 'in' operator implementation algorithm and time complexity

I am thinking of how the in operator implement, for instance

>>> s1 = 'abcdef'
>>> s2 = 'bcd'
>>> s2 in s1
True

In CPython, which algorithm is used to implement the string match, and what is the time complexity? Is there any official document or wiki about this?

It's a combination of Boyer-Moore and Horspool.

You can view the C code here:

Fast search/count implementation, based on a mix between Boyer-Moore and Horspool, with a few more bells and whistles on the top. For some more background, see: https://web.archive.org/web/20201107074620/http://effbot.org/zone/stringlib.htm.

From the link above:

When designing the new algorithm, I used the following constraints:

• should be faster than the current brute-force algorithm for all test cases (based on real-life code), including Jim Hugunin’s worst-case test
• small setup overhead; no dynamic allocation in the fast path (O(m) for speed, O(1) for storage)
• sublinear search behaviour in good cases (O(n/m))
• no worse than the current algorithm in worst case (O(nm))
• should work well for both 8-bit strings and 16-bit or 32-bit Unicode strings (no O(σ) dependencies)
• many real-life searches should be good, very few should be worst case
• reasonably simple implementation