feat(searches): Add Floyd's Cycle-Finding Algorithm

[AdityaK60]

Describe your change:

This pull request introduces an implementation of Floyd's Cycle-Finding Algorithm, also known as the "Tortoise and Hare" algorithm. This well-known algorithm is used to efficiently detect a cycle in a sequence of iterated values and also finds the cycle's starting point and length.

The file floyds_cycle_finding.py has been added to the searches/ directory.

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests?
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
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[algorithms-keeper]

Click here to look at the relevant links ⬇️

🔗 Relevant Links

Repository:

• Contributing guidelines
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Python:

• Formatted string literals (f-strings)
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• doctest
• unittest
• pytest

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[algorithms-keeper]

Please provide descriptive name for the parameter: f

[cclauss]

@priya-sundaram-dev Are these tests running in an infinite loop in CI?

[priya-sundaram-dev]

Yes — confirmed locally. Two of the three doctests spin forever, which is why the build job never finishes:

• floyds_cycle_finding(lambda x: (2*x+3)%17, 0) → hangs
• floyds_cycle_finding(lambda x: [1,2,3,4,5,3][x], 0) → hangs
• the no-cycle example returns fast, but only because it hits the hare is None early-return before it can loop.

Root cause is the asymmetric setup in Phase 1: hare is initialized one step ahead (hare = successor_function(start_value)) and equality is tested at the top of the loop before moving. That means the value they meet on isn't guaranteed to sit at an index that's a multiple of the cycle length, which is the invariant Phase 2 relies on. So the Phase-2 while tortoise != hare loop never converges and just walks around the cycle forever.

The classic tortoise-and-hare form fixes it — start both at the same point and check equality after advancing:

tortoise = successor_function(start_value)
hare = successor_function(successor_function(start_value))
while tortoise != hare:
    if tortoise is None or hare is None:
        return None
    tortoise = successor_function(tortoise)
    hare = successor_function(successor_function(hare))

With that Phase 1, both examples terminate. One heads-up though: the list case then returns (3, 3), not the (2, 3) in the doctest — and (3, 3) is the correct answer, since the sequence is 0,1,2,3,4,5,3,4,5,… and the cycle [3,4,5] first appears at index 3. So the expected value in that doctest is off by one on μ as well and should be updated once the loop is fixed.

Happy to open a small PR with the corrected Phase 1 + fixed doctests if that's useful.