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

DIRECTORY.md

@@ -66,6 +66,8 @@
       * [Test Longest Common Subsequence](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/dynamic_programming/longest_common_subsequence/test_longest_common_subsequence.py)
     * Min Distance
       * [Test Min Distance](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/dynamic_programming/min_distance/test_min_distance.py)
+    * Min Path Sum
+      * [Test Min Path Sum](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py)
     * Unique Paths
       * [Test Unique Paths](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/dynamic_programming/unique_paths/test_unique_paths.py)
     * Word Break
@@ -250,6 +252,7 @@
       * [Test Doubly Linked List](https://github.com/BrianLusina/PythonSnips/blob/master/datastructures/linked_lists/doubly_linked_list/test_doubly_linked_list.py)
       * [Test Doubly Linked List Move Tail To Head](https://github.com/BrianLusina/PythonSnips/blob/master/datastructures/linked_lists/doubly_linked_list/test_doubly_linked_list_move_tail_to_head.py)
       * [Test Doubly Linked List Palindrome](https://github.com/BrianLusina/PythonSnips/blob/master/datastructures/linked_lists/doubly_linked_list/test_doubly_linked_list_palindrome.py)
+    * [Linked List Utils](https://github.com/BrianLusina/PythonSnips/blob/master/datastructures/linked_lists/linked_list_utils.py)
     * Mergeklinkedlists
       * [Test Merge](https://github.com/BrianLusina/PythonSnips/blob/master/datastructures/linked_lists/mergeklinkedlists/test_merge.py)
     * Singly Linked List

algorithms/max_path_sum/README.md → algorithms/dynamic_programming/max_path_sum/README.md

@@ -1,3 +1,5 @@
+# Max Path Sum in Triangle
+
 By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top
 to bottom is 23.
 

algorithms/max_path_sum/__init__.py → algorithms/dynamic_programming/max_path_sum/__init__.py

@@ -2,8 +2,10 @@
 Finds the maximum path sum in a given triangle of numbers
 """
 
+from typing import List
 
-def max_path_sum_in_triangle(triangle):
+
+def max_path_sum_in_triangle(triangle: List[List[int]]) -> int:
     """
     Finds the maximum sum path in a triangle(tree) and returns it
     :param triangle:

algorithms/dynamic_programming/min_path_sum/README.md

@@ -0,0 +1,55 @@
+# Min Path Sum in Triangle
+
+Given an array, triangle, return the minimum path sum from top to bottom. You may move to an adjacent number in the row
+below at each step. More formally, if you are at index i in the current row, you may move to either index i or index 
+i+1 in the next row.
+
+## Constraints
+
+- 1 <= triangle.length <= 200
+- triangle[0].length == 1
+- triangle[i].length == triangle[i - 1].length + 1
+- -10^4 <= triangle[i][j] <= 10^4
+
+## Solution
+
+The goal is to find the minimum path sum from the top of a triangular array to its base, moving at each step to one of 
+the two adjacent numbers in the next row. A greedy choice at the top may fail because a small value early on can lead to
+a costly region later, and plain recursion is inefficient since it revisits the same subproblems many times. This 
+problem is a natural fit for dynamic programming:
+the best path through a cell depends only on the best paths beneath it (optimal substructure), and many different routes
+share the same suffixes (overlapping subproblems).
+
+To solve it efficiently, we take a bottom-up approach. We begin with the last row, whose values represent the known 
+final costs of any path ending there. From there, we move upward one row at a time. For each number, we determine its 
+minimum path cost by adding its own value to the smaller of the two pre-calculated path costs in the row directly below 
+it. This process effectively “folds” the triangle’s path information upward, continuously updating each row with the 
+optimal costs from the level below. By this process’s peak, the single top number has been transformed to hold the total
+of the most efficient path through the entire structure.
+
+The following steps can be performed to implement the algorithm above:
+
+1. First, we create a one-dimensional list, `dp`, to store our minimum path sums. This list is initialized as a copy of 
+   the last row of the triangle, i.e., triangle[-1]. This serves as our base case, because the minimum path cost from 
+   any number in the last row to the bottom is simply its own value.
+2. Next, we iterate from the second-to-last row (rowIdx = len(triangle) - 2) of the triangle and move upward, one row at
+   a time, until we reach the top (rowIdx = 0). This bottom-up order ensures that when we process a row, the optimal 
+   path costs for the row below it are already calculated and stored in the dp list.
+   - We create a nested loop inside the main loop that iterates through each number in the current row. 
+     - For the current number, triangle[rowIdx][colIdx], we calculate its minimum path sum using:
+       - min(dp[colIdx], dp[colIdx + 1])
+       - After finding the minimum of the two, we add it to triangle[rowIdx][colIdx]. 
+       - Finally, we update the dp list at the current position with this new, smaller total.
+
+3. After the loops complete, the dp list contains the fully collapsed path information. The final answer for the entire 
+   journey, from the top to the bottom, is now the list’s first element, dp[0].
+
+### Time Complexity
+
+The time complexity is O(n^2) where n is the number of rows in the triangle. This is because the algorithm processes
+each element exactly once, and the total number of elements in a triangle with n rows is about n * (n + 1)/2, which 
+grows quadratically.
+
+### Space Complexity
+
+The space complexity is O(n) since we only maintain a single working array `dp` with one entry per row.

algorithms/dynamic_programming/min_path_sum/__init__.py

@@ -0,0 +1,62 @@
+from typing import List
+
+
+def min_path_sum(triangle: List[List[int]]) -> int:
+    """
+    Finds the minimum path sum in a given triangle of numbers
+    This uses bottom up dynamic programming by starting at the bottom, we ensure that every decision made at a higher row
+    is based on the perfect knowledge of the best possible paths below it
+
+    Complexity:
+    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.
+    Space Complexity is O(1) since the triangle input array is updated in place
+
+    Args:
+        triangle(list): A list of lists of integers representing the triangle
+    Returns:
+        The minimum path sum in the triangle
+    """
+    row_count = len(triangle)
+
+    # start from the second to last row, since the bottom row has no children to begin with
+    for row in range(row_count - 2, -1, -1):
+        # ensures that we visit every element for the current row
+        for col in range(row + 1):
+            # Each cell is updated to include the minimum path sum from the row below
+            triangle[row][col] += min(
+                triangle[row + 1][col], triangle[row + 1][col + 1]
+            )
+
+    # the result trickles to the apex
+    return triangle[0][0]
+
+
+def min_path_sum_2(triangle: List[List[int]]) -> int:
+    """
+    Finds the minimum path sum in a given triangle of numbers
+    This uses bottom up dynamic programming by starting at the bottom, we ensure that every decision made at a higher row
+    is based on the perfect knowledge of the best possible paths below it
+
+    Complexity:
+    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.
+    Space Complexity is O(n) since the triangle input array's last row is copied over
+
+    Args:
+        triangle(list): A list of lists of integers representing the triangle
+    Returns:
+        The minimum path sum in the triangle
+    """
+    dp = triangle[-1][:]
+    row_count = len(triangle)
+
+    # start from the second to last row, since the bottom row has no children to begin with
+    for row_idx in range(row_count - 2, -1, -1):
+        # ensures that we visit every element for the current row
+        for col_idx in range(row_idx + 1):
+            # Each cell is updated to include the minimum path sum from the row below
+            dp[col_idx] = triangle[row_idx][col_idx] + min(dp[col_idx], dp[col_idx + 1])
+
+    # the result trickles to the apex
+    return dp[0]

algorithms/dynamic_programming/min_path_sum/test_min_path_sum.py

@@ -0,0 +1,31 @@
+import unittest
+import copy
+from typing import List
+from parameterized import parameterized
+from algorithms.dynamic_programming.min_path_sum import min_path_sum, min_path_sum_2
+
+TEST_CASES = [
+    ([[5]], 5),
+    ([[2], [3, 4]], 5),
+    ([[1000], [2000, 3000]], 3000),
+    ([[2], [3, 4], [6, 5, 7], [4, 1, 8, 3]], 11),
+    ([[7], [3, 8], [8, 1, 0], [2, 7, 4, 4], [4, 5, 2, 6, 5]], 17),
+]
+
+
+class MinPathSumInTriangleTestCase(unittest.TestCase):
+    @parameterized.expand(TEST_CASES)
+    def test_min_path_sum_in_triangle(self, triangle: List[List[int]], expected: int):
+        input_triangle = copy.deepcopy(triangle)
+        actual = min_path_sum(input_triangle)
+        self.assertEqual(expected, actual)
+
+    @parameterized.expand(TEST_CASES)
+    def test_min_path_sum_2_in_triangle(self, triangle: List[List[int]], expected: int):
+        input_triangle = copy.deepcopy(triangle)
+        actual = min_path_sum_2(input_triangle)
+        self.assertEqual(expected, actual)
+
+
+if __name__ == "__main__":
+    unittest.main()

datastructures/linked_lists/linked_list_utils.py

@@ -0,0 +1,21 @@
+from typing import Optional
+from datastructures.linked_lists import Node
+
+
+def find_middle_node(head: Optional[Node]) -> Optional[Node]:
+    """
+    Traverse the linked list to find the middle node
+    Time Complexity: O(n) where n is the number of nodes in the linked list
+    Space Complexity: O(1) as constant extra space is needed
+    @return: Middle Node or None
+    """
+    if not head:
+        return None
+
+    fast_pointer, slow_pointer = head, head
+
+    while fast_pointer and fast_pointer.next:
+        slow_pointer = slow_pointer.next
+        fast_pointer = fast_pointer.next.next
+
+    return slow_pointer