Triangle
A medium Dynamic Programming interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Dynamic Programming
- Sheets
- 2
- Core for
- 9 roles
The problem
A triangle of numbers has one value in its first row, two in the second, and so on. Walk from the top to the bottom row; from a position in one row you may step to the same position or the next position in the row below. Return the smallest total a walk can have.
Example 1
- Input
- triangle = [[2],[5,3],[8,1,4],[6,7,2,9]]
- Output
- 8
- Why
- Path: 2->3->1->2 = 8.
Example 2
- Input
- triangle = [[-10]]
- Output
- -10
- Why
- Single row triangle, the minimum is the only element.
Example 3
- Input
- triangle = [[2],[3,4],[6,5,7],[4,1,8,3],[1,2,3,4,5]]
- Output
- 11
- Why
- The minimum path is 2→3→5→1 = 11.
Constraints
- 1 <= triangle.length <= 200
- triangle[i].length == i + 1
- -10^4 <= triangle[i][j] <= 10^4
How to think about it
Updated 2026-09-29Moving top-to-bottom requires boundary checks on the edges and a final scan across the entire bottom row. Flipping the direction to bottom-up collapses pairs cleanly: every element (r, c) chooses the minimum of its two downward neighbors (r+1, c) and (r+1, c+1), converging naturally at the apex.
Approaches, worst first
Recursive path search
time O(2^n) · space O(n)
Recurse from (r, c) down to (r+1, c) and (r+1, c+1). Without memoization, creates 2^n paths with massive overlap on central nodes.
Top-down DP table
time O(n^2) · space O(n^2)
Propagate sums from triangle[0][0] downward. Cells along the left and right edges have only one parent; interior cells take min of two parents. Requires scanning the bottom row for the answer.
Bottom-up 1D reductionWrite this one
time O(n^2) · space O(n)
Copy the bottom row into an array dp. Walk rows from n - 2 up to 0: for each column c, set dp[c] = triangle[r][c] + min(dp[c], dp[c+1]). The answer settles at dp[0].
Where people lose marks · 3
- Negative numbers: values can be negative, so initializing DP with 0 produces incorrect paths.
- Edge handling in top-down approach: j = 0 has no top-left parent, and j = i has no top-right parent. Out-of-bounds reads occur without specific branching.
- Single row triangle: a triangle of depth 1 must immediately return triangle[0][0].
The theory behind it
Dynamic Programming — the ground this problem stands on. All Dynamic Programming problems
What Dynamic Programming is
Dynamic programming is a method for solving a complex problem by breaking it into overlapping subproblems, solving each subproblem only once, and remembering the answers in a lookup table. Instead of recalculating identical questions over and over, future steps look up previous answers directly. By assembling these saved pieces from the bottom up or storing them during recursion, a task that would take billions of steps finishes in a fraction of a second.
When to reach for it
Reach for dynamic programming when questions ask for the maximum profit, minimum cost, total number of distinct ways to achieve a goal, or whether a target can be formed. Signals include overlapping choices where making a choice now affects what choices remain later, but greedy picking fails to find the true global optimum. If drawing a recursive decision tree reveals the same subproblem states repeating across branches, dynamic programming is needed.
How the pattern works
Identify the state variables that uniquely describe a subproblem, such as an array index and remaining capacity. Write the base cases first, representing states whose answers are known without calculation. Next, write the recurrence relation that expresses the current state using previously solved states, taking the minimum, maximum, or sum among your options. Build the solution either top-down by caching recursive returns in a memo table, or bottom-up by filling an array in topological dependency order. When each state depends only on the previous row, compress storage down to a single array.
More Dynamic Programming problems
Target Roles
Core requirement for 9 roles:
Track in your role's order
Pick your target role and all 370 problems resequence to what that interview actually asks.
Start freeProblem set and role mapping as of .