Triangle

Problem Statement

Find the minimum path sum from top to bottom of a triangle.

Key Insight

This problem requires dynamic programming or hash-based techniques to solve efficiently. Start by understanding the recurrence relation or the hash structure needed.

Examples

Input	Output	Explanation
Example 1	Result 1	Explanation here
Example 2	Result 2	Explanation here

Constraints

• Constraints go here

Real-World Applications

Where this pattern appears

• Database indexing and query optimization
• Buffer management in system memory
• Time-series data analysis and sliding windows
• Image pixel manipulation and batch processing

Complexity Comparison

Approach	Time	Space	Best For
Top-Down DP	Bottom-Up DP	Optimized DP	TBD

Approach 1: Top-Down DP

★ Recommended

Detailed explanation of approach 1.

⏱ Time O(n) 💾 Space O(n)

Pseudocode

function solution():
    // Pseudocode here

Solution Code

Python

triangle_approach1.py

"""
LeetCode #120: Triangle
Approach: {approach}
"""
from typing import List, Dict, Set


def solution({param}) -> {return_type}:
    """
    {description}

    Time Complexity: O(n)
    Space Complexity: O(n)
    """
    pass


if __name__ == "__main__":
    # Test cases
    print(solution({test_input}))

C++

triangle_approach1.cpp

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
#include <vector>
#include <unordered_map>
#include <string>
#include <iostream>

using namespace std;

// Time Complexity: O(n)
// Space Complexity: O(n)
class Solution {
public:
    // Solution here
private:
    // Helper functions
};

int main() {
    Solution sol;
    // Test cases
    return 0;
}

Java

triangle_approach1.java

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
import java.util.*;

class Solution {
    /**
     * Time Complexity: O(n)
     * Space Complexity: O(n)
     */
    public Object solution(Object input) {
        // Solution here
        return null;
    }
}

public class Main {
    public static void main(String[] args) {
        Solution sol = new Solution();
        // Test cases
    }
}

JavaScript

triangle_approach1.js

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */

/**
 * @param {type} input
 * @return {type}
 * Time Complexity: O(n)
 * Space Complexity: O(n)
 */
function solution(input) {
    // Solution here
    return null;
}

// Test cases
console.log(solution(input));

Rust

triangle_approach1.rs

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
use std::collections::{HashMap, HashSet};

// Time Complexity: O(n)
// Space Complexity: O(n)
pub fn solution(input: &[i32]) -> i32 {
    // Solution here
    0
}

fn main() {
    // Test cases
    println!("{:?}", solution(&[]));
}

Go

triangle_approach1.go

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
package main

import (
    "fmt"
)

// Time Complexity: O(n)
// Space Complexity: O(n)
func solution(input []int) int {
    // Solution here
    return 0
}

func main() {
    // Test cases
    fmt.Println(solution([]int{}))
}

Approach 2: Bottom-Up DP

🎯 Interview Favourite

Detailed explanation of approach 2.

⏱ Time O(n) 💾 Space O(n)

Solution Code

Python

triangle_approach2.py

"""
LeetCode #120: Triangle
Approach: {approach}
"""
from typing import List, Dict, Set


def solution({param}) -> {return_type}:
    """
    {description}

    Time Complexity: O(n)
    Space Complexity: O(n)
    """
    pass


if __name__ == "__main__":
    # Test cases
    print(solution({test_input}))

C++

triangle_approach2.cpp

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
#include <vector>
#include <unordered_map>
#include <string>
#include <iostream>

using namespace std;

// Time Complexity: O(n)
// Space Complexity: O(n)
class Solution {
public:
    // Solution here
private:
    // Helper functions
};

int main() {
    Solution sol;
    // Test cases
    return 0;
}

Java

triangle_approach2.java

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
import java.util.*;

class Solution {
    /**
     * Time Complexity: O(n)
     * Space Complexity: O(n)
     */
    public Object solution(Object input) {
        // Solution here
        return null;
    }
}

public class Main {
    public static void main(String[] args) {
        Solution sol = new Solution();
        // Test cases
    }
}

JavaScript

triangle_approach2.js

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */

/**
 * @param {type} input
 * @return {type}
 * Time Complexity: O(n)
 * Space Complexity: O(n)
 */
function solution(input) {
    // Solution here
    return null;
}

// Test cases
console.log(solution(input));

Rust

triangle_approach2.rs

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
use std::collections::{HashMap, HashSet};

// Time Complexity: O(n)
// Space Complexity: O(n)
pub fn solution(input: &[i32]) -> i32 {
    // Solution here
    0
}

fn main() {
    // Test cases
    println!("{:?}", solution(&[]));
}

Go

triangle_approach2.go

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
package main

import (
    "fmt"
)

// Time Complexity: O(n)
// Space Complexity: O(n)
func solution(input []int) int {
    // Solution here
    return 0
}

func main() {
    // Test cases
    fmt.Println(solution([]int{}))
}

Approach 3: Optimized DP

✓ Simple

Detailed explanation of approach 3.

⏱ Time O(n) 💾 Space O(n)

Solution Code

Python

triangle_approach3.py

"""
LeetCode #120: Triangle
Approach: {approach}
"""
from typing import List, Dict, Set


def solution({param}) -> {return_type}:
    """
    {description}

    Time Complexity: O(n)
    Space Complexity: O(n)
    """
    pass


if __name__ == "__main__":
    # Test cases
    print(solution({test_input}))

C++

triangle_approach3.cpp

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
#include <vector>
#include <unordered_map>
#include <string>
#include <iostream>

using namespace std;

// Time Complexity: O(n)
// Space Complexity: O(n)
class Solution {
public:
    // Solution here
private:
    // Helper functions
};

int main() {
    Solution sol;
    // Test cases
    return 0;
}

Java

triangle_approach3.java

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
import java.util.*;

class Solution {
    /**
     * Time Complexity: O(n)
     * Space Complexity: O(n)
     */
    public Object solution(Object input) {
        // Solution here
        return null;
    }
}

public class Main {
    public static void main(String[] args) {
        Solution sol = new Solution();
        // Test cases
    }
}

JavaScript

triangle_approach3.js

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */

/**
 * @param {type} input
 * @return {type}
 * Time Complexity: O(n)
 * Space Complexity: O(n)
 */
function solution(input) {
    // Solution here
    return null;
}

// Test cases
console.log(solution(input));

Rust

triangle_approach3.rs

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
use std::collections::{HashMap, HashSet};

// Time Complexity: O(n)
// Space Complexity: O(n)
pub fn solution(input: &[i32]) -> i32 {
    // Solution here
    0
}

fn main() {
    // Test cases
    println!("{:?}", solution(&[]));
}

Go

triangle_approach3.go

/*
 * LeetCode #120: Triangle
 * Approach: {approach}
 */
package main

import (
    "fmt"
)

// Time Complexity: O(n)
// Space Complexity: O(n)
func solution(input []int) int {
    // Solution here
    return 0
}

func main() {
    // Test cases
    fmt.Println(solution([]int{}))
}

Interview Tips

Key trade-offs

• Discuss time/space trade-offs
• Mention edge cases
• Explain optimization steps