feat(algorithms, dynamic-programming): minimum path sum in a triangle

[BrianLusina]

Describe your change:

Minimum path sum in a triangle

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• 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. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms have a URL in its comments that points to Wikipedia or other similar explanation.
• If this pull request resolves one or more open issues then the commit message contains Fixes: #{$ISSUE_NO}.

Summary by CodeRabbit

• New Features

  • Added two minimum-path-sum algorithms for triangle inputs (in-place and DP-array variants)
  • Added a linked-list utility to find the middle node

• Documentation

  • Added guides for max and min path-sum problems with explanations and complexity notes

• Tests

  • Added unit tests validating both minimum path-sum implementations across multiple triangle cases

_{✏️ Tip: You can customize this high-level summary in your review settings.}

[coderabbitai]

Walkthrough

Adds a new min path sum triangle module with two implementations and tests, adds type hints and README text to the existing max path sum module, introduces a linked list utility to find the middle node, and updates directory documentation to reference the new items.

Changes

Cohort / File(s)	Summary
Dynamic Programming: Max Path Sum algorithms/dynamic_programming/max_path_sum/README.md, algorithms/dynamic_programming/max_path_sum/__init__.py	Added README content and added typing (List) with signature annotation for max_path_sum_in_triangle(triangle: List[List[int]]) -> int
Dynamic Programming: Min Path Sum algorithms/dynamic_programming/min_path_sum/README.md, algorithms/dynamic_programming/min_path_sum/__init__.py, algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py	New module providing min_path_sum (in-place bottom-up) and min_path_sum_2 (bottom-up with DP array), both O(n²); added README and parameterized tests covering multiple triangle cases
Linked Lists Utilities datastructures/linked_lists/linked_list_utils.py	New find_middle_node(head: Optional[Node]) -> Optional[Node] implementing slow/fast pointer to return the middle node (O(n) time, O(1) space)
Directory Documentation DIRECTORY.md	Registered new Min Path Sum item and added Linked List Utils entry to the directory registry

Estimated code review effort

🎯 3 (Moderate) | ⏱️ ~20 minutes

• Review correctness and edge-case handling in min_path_sum and min_path_sum_2 (empty triangle, single-row).
• Verify tests in test_min_path_sum.py cover intended cases and assert identical outputs for both implementations.
• Confirm type annotation import and usage in max_path_sum/__init__.py.
• Check find_middle_node for correct handling of even/odd-length lists and imports of Node/Optional.

Suggested labels

LinkedList

Poem

🐰 I hopped through triangles, counting every sum,
Two paths I learned — in-place and then some,
I paused at the center with a slow and fast beat,
Type hints and README — tidy and neat! ✨

Pre-merge checks and finishing touches

❌ Failed checks (1 warning, 1 inconclusive)

Check name	Status	Explanation	Resolution
Docstring Coverage	⚠️ Warning	Docstring coverage is 66.67% which is insufficient. The required threshold is 80.00%.	You can run @coderabbitai generate docstrings to improve docstring coverage.
Description check	❓ Inconclusive	The description provides change summary and includes completed checklist items. However, it marks some checklist items as completed that appear inconsistent with the actual changes (e.g., 'bug/typo fix' when primarily adding new algorithms).	Clarify whether this PR only adds a new algorithm or also includes bug fixes. Verify that all checked items accurately reflect the actual changes made to the codebase.

✅ Passed checks (1 passed)

Check name	Status	Explanation
Title check	✅ Passed	The title accurately describes the main change: adding a minimum path sum algorithm in the dynamic programming section.

✨ Finishing touches

• 📝 Generate docstrings

🧪 Generate unit tests (beta)

• Create PR with unit tests
• Post copyable unit tests in a comment
• Commit unit tests in branch feat/algorithms-triangle

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📜 Recent review details

Configuration used: defaults

Review profile: CHILL

Plan: Pro

📥 Commits

Reviewing files that changed from the base of the PR and between 6aab5b8 and 8ebedca.

📒 Files selected for processing (3)

• algorithms/dynamic_programming/min_path_sum/README.md (1 hunks)
• algorithms/dynamic_programming/min_path_sum/__init__.py (1 hunks)
• algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py (1 hunks)

🚧 Files skipped from review as they are similar to previous changes (2)

• algorithms/dynamic_programming/min_path_sum/init.py
• algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py

🧰 Additional context used

🪛 markdownlint-cli2 (0.18.1)

algorithms/dynamic_programming/min_path_sum/README.md

12-12: Reference links and images should use a label that is defined
Missing link or image reference definition: "j"

(MD052, reference-links-images)

---

39-39: Reference links and images should use a label that is defined
Missing link or image reference definition: "colidx"

(MD052, reference-links-images)

---

41-41: Reference links and images should use a label that is defined
Missing link or image reference definition: "colidx"

(MD052, reference-links-images)

🔇 Additional comments (1)

algorithms/dynamic_programming/min_path_sum/README.md (1)

1-55: Documentation is comprehensive and well-written.

The README clearly explains the Min Path Sum in Triangle problem, provides strong motivation for the DP approach, details the algorithm steps, and correctly analyzes time (O(n²)) and space (O(n)) complexity. The previously flagged typo on line 55 has been corrected to "one entry per row."

Note: Static analysis linter hints flagging "[j]" and "[colIdx]" as missing markdown references are false positives—these are array subscript notation in the explanation text, not markdown link syntax.

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[coderabbitai]

Actionable comments posted: 3

🧹 Nitpick comments (1)

algorithms/dynamic_programming/min_path_sum/__init__.py (1)

4-32: Implementation is correct.

The bottom-up DP approach is correctly implemented. Minor typo in docstring: "roughtl" should be "roughly" (line 12).

🔎 Typo fix

     Time Complexity results in O(n^2), since we visit every number in the triangle exactly once. For n rows, there are
-    roughtl n^2/2 elements.
+    roughly n^2/2 elements.

📜 Review details

Configuration used: defaults

Review profile: CHILL

Plan: Pro

📥 Commits

Reviewing files that changed from the base of the PR and between 4d9387b and 6aab5b8.

📒 Files selected for processing (7)

• DIRECTORY.md (2 hunks)
• algorithms/dynamic_programming/max_path_sum/README.md (1 hunks)
• algorithms/dynamic_programming/max_path_sum/__init__.py (1 hunks)
• algorithms/dynamic_programming/min_path_sum/README.md (1 hunks)
• algorithms/dynamic_programming/min_path_sum/__init__.py (1 hunks)
• algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py (1 hunks)
• datastructures/linked_lists/linked_list_utils.py (1 hunks)

🧰 Additional context used

🧬 Code graph analysis (1)

algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py (1)

algorithms/dynamic_programming/min_path_sum/__init__.py (2)

• min_path_sum (4-32)
• min_path_sum_2 (35-62)

🪛 markdownlint-cli2 (0.18.1)

DIRECTORY.md

69-69: Unordered list indentation
Expected: 2; Actual: 4

(MD007, ul-indent)

---

70-70: Unordered list indentation
Expected: 4; Actual: 6

(MD007, ul-indent)

---

255-255: Unordered list indentation
Expected: 2; Actual: 4

(MD007, ul-indent)

🔇 Additional comments (4)

datastructures/linked_lists/linked_list_utils.py (1)

5-21: LGTM! Clean implementation of the fast/slow pointer technique.

The algorithm correctly finds the middle node with O(n) time and O(1) space complexity. For even-length lists, this returns the second of the two middle nodes, which is the standard convention.

DIRECTORY.md (1)

69-70: Documentation entries added correctly.

The new entries for Min Path Sum and Linked List Utils follow the existing file structure and naming conventions.

Also applies to: 255-255

algorithms/dynamic_programming/max_path_sum/README.md (1)

1-2: Good documentation improvement.

Adding the markdown header improves consistency with other README files in the project.

algorithms/dynamic_programming/max_path_sum/__init__.py (1)

5-8: Good addition of type hints.

The type annotations improve code clarity and enable better static analysis support without changing behavior.