fix: sub-interval midpoint formula in ternary search

[Kanika0306]

Describe your change:

Fixes #15069

• Fixed the sub-interval split point calculation formulas in searches/ternary_search.py for both ite_ternary_search and rec_ternary_search.
• Corrected the midpoint calculations when the search interval starts at an index greater than 0.
• Adjusted the search boundary updates to correctly handle the exclusive right index.
• Added doctests covering non-zero left boundaries and larger input arrays to prevent regressions.

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

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file.
• All new Python files are placed inside an existing directory.
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• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".

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[priya-sundaram-dev]

Reviewed this carefully — it's a correct and well-scoped fix. Nice work @Kanika0306.

Root cause: the old split points (left + right) // 3 + 1 and 2 * (left + right) // 3 + 1 aren't the true one-third/two-third boundaries of [left, right) once left > 0, and the trailing + 1 can push two_third past the end of the array — I reproduced an IndexError on the current master for several inputs. The new left + (right - left) // 3 / right - (right - left) // 3 are the standard ternary split points and stay in range.

Convention is consistent: right stays exclusive (right = len(array)), so the new boundary updates (right = one_third, left = two_third + 1, right = two_third) and the exclusive lin_search(left, right, ...) line up correctly. Also good catch fixing the __main__ call to rec_ternary_search(0, len(collection), ...) to match.

Verification: I ran both ite_ternary_search and rec_ternary_search exhaustively against a linear-search oracle for every array length 0..400 and every target (including out-of-range and empty lists): 0 mismatches, 0 exceptions. The added doctests pass. The current master fails the same sweep.

One small optional suggestion for future-proofing (non-blocking): a hypothesis/property test asserting parity with bisect would guard this class of off-by-one regression, but that's out of scope here. LGTM. 🤖 (For transparency: I'm an AI agent — I write and test everything I post myself. #ABotWroteThis)

[cclauss]

Nice! Looks good to me.

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[priya-sundaram-dev]

Approving. The new one_third = left + (right - left)//3 / two_third = right - (right - left)//3 formulation fixes the old off-by-one that could index out of the current [left, right] window, and the boundary updates (right = one_third, left = two_third + 1, else left = one_third + 1; right = two_third) keep the interval strictly shrinking so it always terminates.

I verified both ite_ and rec_ variants exhaustively against a linear-search oracle for every target in a range spanning below/in/above arrays of sizes 0..400 — 0 mismatches, no IndexError. The added large-list doctests are a good regression guard. LGTM. 👍

[Kanika0306]

Approving. The new one_third = left + (right - left)//3 / two_third = right - (right - left)//3 formulation fixes the old off-by-one that could index out of the current [left, right] window, and the boundary updates (right = one_third, left = two_third + 1, else left = one_third + 1; right = two_third) keep the interval strictly shrinking so it always terminates.

I verified both ite_ and rec_ variants exhaustively against a linear-search oracle for every target in a range spanning below/in/above arrays of sizes 0..400 — 0 mismatches, no IndexError. The added large-list doctests are a good regression guard. LGTM. 👍

Thankyou!
Could you let me know if there is any community channel which I can join

[priya-sundaram-dev]

Welcome, and glad the review helped! 🙌 The main community hub is the TheAlgorithms Discord — the invite is linked from the "Chat" badge near the top of this repo's README, and from https://the-algorithms.com/. That's the best place to ask questions, find good-first-issues, and coordinate with reviewers. Enjoy contributing!