Triangle - Practice Coding Problems

Triangle - Problem

Triangle Path Sum Challenge

Imagine you're standing at the top of a mountain represented as a triangular array of numbers. Your goal is to find the minimum cost path from the summit to the base.

🎯 The Rules:
• Start at the top of the triangle (index 0)
• At each step, you can only move to adjacent positions in the row below
• From position i, you can move to either position i or i + 1 in the next row
• Find the path with the minimum sum of all numbers encountered

Example: Given triangle [[2],[3,4],[6,5,7],[4,1,8,3]], the minimum path is 2 → 3 → 5 → 1 = 11

Input & Output

example_1.py — Basic Triangle

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

› Output: 11

💡 Note: The minimum path is 2 → 3 → 5 → 1 with sum = 11. At each level, we choose the adjacent position that leads to the minimum total.

example_2.py — Single Element

$ Input: triangle = [[-10]]

› Output: -10

💡 Note: With only one element, the minimum (and only) path sum is the element itself: -10.

example_3.py — Two Rows

$ Input: triangle = [[1],[2,3]]

› Output: 3

💡 Note: Two possible paths: 1→2 (sum=3) and 1→3 (sum=4). The minimum is 3.

Constraints

1 ≤ triangle.length ≤ 200
-10⁴ ≤ triangle[i][j] ≤ 10⁴
Triangle property: triangle[i] has exactly i+1 elements

Visualization

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Understanding the Visualization

Survey the Base

Start at the bottom row - these are our base cases with known costs

Work Upward

For each position, choose the minimum cost path from the two adjacent positions below

Optimal Substructure

Each position now stores the minimum cost to reach the bottom from that point

Reach the Summit

The top position contains the minimum total path cost

Key Takeaway

🎯 Key Insight: Bottom-up DP ensures we always make optimal local decisions that lead to the global optimum!

Asked in

a Amazon 45 G Google 38 ⊞ Microsoft 32 f Meta 28

The optimal solution uses bottom-up dynamic programming with O(n²) time and O(1) space complexity. By starting from the bottom row and working upward, we make locally optimal choices that guarantee a globally optimal solution.

Common Approaches

Approach	Time	Space	Notes
✓ Dynamic Programming (Bottom-Up Optimal)	O(n²)	O(1)	Build solution from bottom row upward, making optimal choices at each level

Dynamic Programming (Bottom-Up Optimal) — Algorithm Steps

Start from the second-to-last row of the triangle
For each position, add the minimum of the two adjacent positions below
Move up row by row, updating each position with its optimal path sum
Continue until we reach the top of the triangle
The top element now contains the minimum path sum

Visualization

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Step-by-Step Walkthrough

Start at Bottom

Begin with the last row values

Move Up

For each position, add minimum of two positions below

Continue Upward

Repeat for each row moving toward the top

Reach Summit

Top element contains the minimum path sum

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <limits.h>

int minimumTotal(int** triangle, int triangleSize, int* triangleColSize) {
    // Start from second-to-last row and work upward
    for (int row = triangleSize - 2; row >= 0; row--) {
        for (int col = 0; col < triangleColSize[row]; col++) {
            int left = triangle[row + 1][col];
            int right = triangle[row + 1][col + 1];
            triangle[row][col] += (left < right) ? left : right;
        }
    }
    
    return triangle[0][0];
}

int main() {
    // Example usage
    int* triangle[4];
    int triangleColSize[4] = {1, 2, 3, 4};
    
    // Allocate and initialize triangle
    triangle[0] = (int*)malloc(1 * sizeof(int));
    triangle[0][0] = 2;
    
    printf("%d\n", minimumTotal(triangle, 4, triangleColSize));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n²)

Visit each element in the triangle exactly once, total elements = 1+2+...+n = n(n+1)/2

⚠ Quadratic Growth

Space Complexity

O(1)

In-place modification requires no extra space, or O(n) if we preserve input

✓ Linear Space

67.4K Views

High Frequency

~15 min Avg. Time

1.8K Likes

Ln 1, Col 1

Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Dynamic Programming (Bottom-Up Optimal) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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