What is the time complexity of Triangle?

The optimal solution has a time complexity of O(n^2).

What data structures are used to solve Triangle?

This problem uses Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Triangle?

Insight 1: Dynamic Programming is well-suited for this problem because we can break it down into overlapping subproblems (finding the minimum path from any given point to the bottom). Insight 2: Bottom-up DP is more efficient in terms of space since we only need to keep track of the minimum sums of the previous row to compute the current row's minimum sums.

Which companies ask LeetCode 120. Triangle in interviews?

This problem is frequently asked at Agoda, DE Shaw, Salesforce.

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Triangle - LeetCode 120 Solution

Q: How to solve Triangle?

The provided solution uses a bottom-up dynamic programming approach with O(n) space complexity. It starts from the bottom row of the triangle and iterates upwards. For each row, it calculates the minimum path sum to reach each element in that row from the bottom row. The `dp` array stores the minimum path sums from each element in the current row to the bottom. By iterating upwards, the final `dp[0]` will contain the minimum path sum from the top of the triangle to the bottom.

Triangle - Complete Solution Guide

Triangle is LeetCode problem 120, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

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. More formally, if you are on index i on the current row, you may move to either index i or index i + 1 on the next row. Example 1: Input: triangle = [[2],[3,4],[6,5,7],[4,1,8,3]] Output: 11 Example 2: Input: triangle = [[-10]] Output: -10 Constraints: 1 <= triangle.length <= 200 triangle[0].length == 1 triangle[i].length == triangle[i - 1].length + 1 -10^4 <

Detailed Explanation

The problem asks us to find the minimum path sum from the top to the bottom of a triangle-shaped array of numbers. From any number in the triangle, you can only move to the number directly below it or the number to the right of the number directly below it. The goal is to find the path from the top that results in the smallest sum of numbers visited.

Solution Approach

The provided solution uses a bottom-up dynamic programming approach with O(n) space complexity. It starts from the bottom row of the triangle and iterates upwards. For each row, it calculates the minimum path sum to reach each element in that row from the bottom row. The `dp` array stores the minimum path sums from each element in the current row to the bottom. By iterating upwards, the final `dp[0]` will contain the minimum path sum from the top of the triangle to the bottom.

Step-by-Step Algorithm

Step 1: Initialize a `dp` array with the values from the last row of the triangle. This represents the minimum path sum from each element in the last row to itself.
Step 2: Iterate upwards from the second-to-last row to the first row.
Step 3: For each element in the current row, calculate the minimum path sum by adding the element's value to the minimum of the two elements directly below it in the `dp` array (i.e., `dp[j]` and `dp[j+1]`). Update `dp[j]` with this new minimum path sum.
Step 4: After iterating through all rows, the value at `dp[0]` will contain the minimum path sum from the top of the triangle to the bottom. Return this value.

Key Insights

Insight 1: Dynamic Programming is well-suited for this problem because we can break it down into overlapping subproblems (finding the minimum path from any given point to the bottom).
Insight 2: Bottom-up DP is more efficient in terms of space since we only need to keep track of the minimum sums of the previous row to compute the current row's minimum sums.
Insight 3: We can achieve O(n) space complexity by using a single array to store the minimum path sums from the bottom row up to the current row, overwriting the previous row's values as we go.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(n)

Topics

This problem involves: Dynamic Programming.

Companies

Asked at: Agoda, DE Shaw, Salesforce.

Triangle | LeetCode 120

Medium

Dynamic Programming

Time: O(n^2)Space: O(n)

Solution Code

Available in:

import java.util.*;
class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        int n = triangle.size();
        int[] dp = new int[n];
        for (int i = 0; i < n; i++) dp[i] = triangle.get(n-1).get(i);
        for (int i = n - 2; i >= 0; i--) {
            for (int j = 0; j <= i; j++) {
                dp[j] = triangle.get(i).get(j) + Math.min(dp[j], dp[j+1]);
            }
        }
        return dp[0];
    }
}

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Problem Info

Description

Problem 120: Triangle - Detailed explanation and solution approaches.

Topics

Dynamic Programming

Companies

AgodaDE ShawSalesforce

Stats

Difficulty:Medium

Languages:4

Companies:3

Problem #:120

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Step 1: Initialize a `dp` array with the values from the last row of the triangle. This represents the minimum path sum from each element in the last row to itself.
2
Step 2: Iterate upwards from the second-to-last row to the first row.
3
Step 3: For each element in the current row, calculate the minimum path sum by adding the element's value to the minimum of the two elements directly below it in the `dp` array (i.e., `dp[j]` and `dp[j+1]`). Update `dp[j]` with this new minimum path sum.
4
Step 4: After iterating through all rows, the value at `dp[0]` will contain the minimum path sum from the top of the triangle to the bottom. Return this value.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is derived from the nested loops iterating through the triangle. The space complexity is due to the `dp` array used to store intermediate results, which has a size equal to the number of rows in the triangle, fulfilling the follow up requirement in the description.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Top-Down Dynamic Programming (Memoization)

A recursive approach with memoization. Start from the top and recursively explore the two possible paths to the next row. Store the minimum path sums for each subproblem to avoid redundant calculations.

Time: O(n^2)Space: O(n^2) - O(n) for the recursion stack and O(n^2) for the memoization table. Can also be O(n) if space optimization is applied.

Trade-offs: Top-down DP can be more intuitive for some, but it generally has a higher space complexity due to the recursion stack and memoization table. However, the given solution has O(n) space which is better.

✨

Key Insights

1Insight 1: Dynamic Programming is well-suited for this problem because we can break it down into overlapping subproblems (finding the minimum path from any given point to the bottom).
2Insight 2: Bottom-up DP is more efficient in terms of space since we only need to keep track of the minimum sums of the previous row to compute the current row's minimum sums.
3Insight 3: We can achieve O(n) space complexity by using a single array to store the minimum path sums from the bottom row up to the current row, overwriting the previous row's values as we go.

📚

Key Terms Explained

For Students

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Triangle with only one row: The solution correctly returns the value in the only row because the dp array is initialized with this value and directly returned.
Edge case 2: Large values in the triangle: The data type used (int) should be large enough to hold the sum of values, considering the constraint -10^4 <= triangle[i][j] <= 10^4 and triangle.length <= 200. A long could prevent integer overflow in some situations.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Using a top-down approach without memoization. This leads to exponential time complexity due to repeated calculations of the same subproblems.

✗Common mistake 2: Not initializing the `dp` array correctly. The initial values in the `dp` array should represent the minimum path sums from the bottom row to itself, not arbitrary values.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Pathfinding in a cost grid: Imagine a grid where each cell has a cost associated with traversing it. Finding the least costly path from a starting point to a destination can be solved with similar dynamic programming techniques.

🏦Real-world scenario 2: Resource allocation: Optimizing the allocation of resources over multiple stages, where each stage has a cost associated with each resource choice, is a dynamic programming problem in disguise.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Start with recursion, then memoize. The hardest part is identifying the state. Ask yourself: "What information do I need to make the current decision?"

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Forgetting base cases or initializing the DP array incorrectly. Start by solving the smallest subproblems.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 29, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 6

❓

Frequently Asked Questions

How do I identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Triangle"?

For this medium-level Dynamic Programming problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether standard techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?