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medium
DP

Triangle Minimum Path Sum

**Problem Statement:** Given the required input arguments, write an efficient algorithm to solve the **Triangle Minimum Path Sum** problem. Implement the required logic as specified by standard definitions for this classic algorithmic challenge. **Hint / Expected Approach:** Bottom-up in-place dp from last row **Edge Cases to Consider:** - (1) Single row - (2) All same values - (3) Negative values on path

Examples

Example 1
Input: [[2],[3,4],[6,5,7],[4,1,8,3]]
Output: 11
Explanation: 2+3+5+1
Example 2
Input: [[-10]]
Output: -10
Explanation: Single element
Example 3
Input: [[1],[2,3]]
Output: 3
Explanation: 1+2

Constraints

  • ▪Input arguments are within valid ranges
  • ▪Optimize for execution speed
  • ▪Handle null/empty inputs gracefully

Watch Out For Edge Cases

  • ▪Single row
  • ▪All same values
  • ▪Negative values on path
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