How do you approach Triangle?

Work bottom-up: start from the last row and fold each row into the one above.

What are common mistakes on Triangle?

Iterating the dimensions in an order that reads a cell before it has been written. Getting the base case wrong. It carries the whole recurrence, and the error stays invisible until a large input. Rolling down to one dimension without reversing the inner loop, which lets a 0/1 item be taken twice.

Sitelet https://gitgood.dev/coding/triangle-minimum-path

gitGood.dev

← Back to problems

Triangle

MediumAlgorithms

Description

Given a triangle array, return the minimum path sum from top to bottom. At each step you may move to an adjacent number on the row below: from index i you may go to index i or i + 1.

Example 1

Input: triangle = [[2],[3,4],[6,5,7],[4,1,8,3]]

Output: 11

Explanation: 2 + 3 + 5 + 1 = 11.

Constraints

- 1 <= triangle.length <= 200 - triangle[i].length == i + 1 - -10^4 <= triangle[i][j] <= 10^4

Triangle: approach, complexity and pitfalls

A medium dynamic programming problem, graded against 6 test cases (4 of them hidden).

The pattern: Dynamic Programming

Overlapping subproblems solved once and reused - the pattern candidates fear most.

Reach for it when you see: "Number of ways", "minimum/maximum cost", or a recursion that recomputes the same state.

If you are stuck

Show the 2-step hint ladder

Work bottom-up: start from the last row and fold each row into the one above.
dp[j] = triangle[i][j] + min(dp[j], dp[j+1]).

The optimal approach

Show the approach: The approach

Bottom-up DP. Initialize `dp` with the last row. For each row above, set `dp[j] = triangle[i][j] + min(dp[j], dp[j+1])`. After folding all rows, `dp[0]` is the minimum top-to-bottom path sum.

Time: O(n²). Space: O(n).

The full reference solution in every supported language stays in the editor above - reveal it there once you have had a real attempt.

What the tests check

Read off this problem's own test suite, so these are the cases a submission actually has to survive.

Single element. A one-element case is included. Two-pointer and window solutions frequently return the wrong thing when the two indices start out equal.
Duplicate values. Repeated values appear in the input, so any logic that assumes distinct elements - or that skips over equal neighbours - is exercised.
Negative numbers. The suite includes negative values. Running-sum, min/max seeding and any "initialise to zero" step need to handle them.
Zero. Zero appears in the input, which is the value most likely to break a truthiness check used in place of an explicit comparison.

Common Dynamic Programming mistakes

These apply to the pattern as a whole, not just this problem.

Iterating the dimensions in an order that reads a cell before it has been written.
Getting the base case wrong. It carries the whole recurrence, and the error stays invisible until a large input.
Rolling down to one dimension without reversing the inner loop, which lets a 0/1 item be taken twice.