Maximum Path in Triangle

Logical Breakdown

• [ ] Subproblem Identification:
• [ ] Optimal Substructure:
• [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
• [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

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Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

\[ dp[i] = ... \]

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Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

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Complexity Analysis

Approach	Time Complexity	Space Complexity
Iterative DP	\(O(N)\)	\(O(1)\)

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Code Reference

package com.dsa.dynamic_programming;

import java.util.*;

/ * Problem: Maximum Path in Triangle * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: Maximum Path in Triangle * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a triangle array, find the MAXIMUM path sum from top to bottom. * Movement: From (row, i) -> (row+1, i) or (row+1, i+1). * * --------------------- * 2. Logical Breakdown * --------------------- * Identical to Min Sum Path, but using MAX function. * Solved Bottom-Up for efficiency. * * Recurrence: * dp[row][i] = triangle[row][i] + max(dp[row+1][i], dp[row+1][i+1]) * * --------------------- * 3. Complexity * --------------------- * Time: O(N^2) * Space: O(N) * * ============================================================================================ */ public class MaximumPathInTriangle { public int solve(int A) { // TODO: Implement solution return 0; }

public int maximumTotal(ArrayList<ArrayList<Integer>> triangle) {
    return 1;
}

}