Leetcode #120: Triangle
In this guide, we solve Leetcode #120 Triangle in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Given a triangle array, return the minimum path sum from top to bottom. For each step, you may move to an adjacent number of the row below.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Array, Dynamic Programming
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: triangle = [[2],[3,4],[6,5,7],[4,1,8,3]]
Output: 11
Explanation: The triangle looks like:
2
3 4
6 5 7
4 1 8 3
The minimum path sum from top to bottom is 2 + 3 + 5 + 1 = 11 (underlined above).
Python Solution
class Solution:
def minimumTotal(self, triangle: List[List[int]]) -> int:
n = len(triangle)
f = [[0] * (n + 1) for _ in range(n + 1)]
for i in range(n - 1, -1, -1):
for j in range(i + 1):
f[i][j] = min(f[i + 1][j], f[i + 1][j + 1]) + triangle[i][j]
return f[0][0]
Complexity
The time complexity is , and the space complexity is , where is the number of rows in the triangle. The space complexity is , where is the number of rows in the triangle.
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.