The Ship Company

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.heaps;

import java.util.*;

/ * ============================================================================================ * PROBLEM: The Ship Company (Max/Min Profit with Dynamic Pricing) * ============================================================================================ * * --------------------- * 1. Logical Breakdown & Mathematical Formulation * --------------------- * We have A passengers and N ships with \(B[i]\) vacant seats. * Ticket cost = current number of vacant seats in the chosen ship. * * Strategy for Maximum Profit: * - To maximize profit, we must always sell the most expensive ticket available. * - This means picking the ship with the maximum number of vacant seats at every step. * - Data Structure: Max-Heap. * - Algorithm: Extract \(max\), add to total, decrement \(max\), and if \(max > 0\), * push it back. * * Strategy for Minimum Profit: * - To minimize profit, we must always sell the cheapest ticket available. * - This means picking the ship with the minimum (but > 0) number of vacant seats. * - Data Structure: Min-Heap*. * - Algorithm: Extract \(min\), add to total, decrement \(min\), and if \(min > 0\), * push it back. * * --------------------- * 2. Complexity Analysis * --------------------- * Time Complexity: * O((N + A) log N) * - Initializing both heaps takes O(N log N). * - Processing A passengers, each with a heap extraction and insertion: O(A log N). * * Space Complexity: * O(N) * - To store the capacities of N ships in the Priority Queues. * * --------------------- * 3. Base Case Verification * --------------------- * - A = 1: Both max and min profits will be the maximum value in array B. * - All ships have 1 seat: Max and min profits will both be A. * ============================================================================================ / public class TheShipCompany { public int solve(int A) { // TODO: Implement solution return 0; }

public int[] solve(int i, int i1, int[] c2) {
    return new int[0];
}

}