Maximizing Capital for LeetCode's IPO Through Optimal Project Selection
To solve the problem of maximizing LeetCode\'s capital before its IPO by choosing at most \( k \) distinct projects, we can employ a greedy approach. This metho
To solve the problem of maximizing LeetCode\'s capital before its IPO by choosing at most \\( k \\) distinct projects, we can employ a greedy approach. This method ensures that at each step, we choose the project that offers the maximum profit while meeting the current capital requirement.\n\n### Approach Breakdown\n\n1. **Sort Projects by Capital Requirement**: First, we sort the projects based on the minimum capital required to start them. This helps in quickly finding all projects that can be initiated with the current available capital.\n\n2. **Use a Max-Heap for Profits**: To always select the project with the highest profit that can be started with the current capital, we use a max-heap (priority queue). This allows us to efficiently retrieve the project with the maximum profit at any point.\n\n3. **Iterate up to \\( k \\) Times**: We iterate up to \\( k \\) times, at each iteration:\n   - Add all projects that can be started with the current capital to the max-heap.\n   - Select the project with the maximum profit from the max-heap.\n   - Update the current capital by adding the profit of the selected project.\n   - Repeat until no more projects can be started or we reach the \\( k \\) iterations limit.\n\n### Detailed Steps with Example\n\nConsider the example:\n\n- k = 2\n- w = 0 \n- profits = [1, 2, 3]\n- capital = [0, 1, 1]\n\n#### Step 1: Sort Projects by Capital\nFirst, we pair each project\'s capital requirement with its profit and sort them by capital.\n\n- Pair and sort: projects= [(0, 1), (1, 2), (1, 3)]\n\n#### Step 2: Initialize Data Structures\n- Create a max-heap to store profits of projects that can be started.\n- Initialize current capital \\( w = 0 \\).\n\n#### Step 3: Iterate and Select Projects\n- **Iteration 1**:\n  - Check which projects can be started with \\( w = 0 \\).\n  - Only project \\((0, 1)\\) can be started.\n  - Add its profit to the max-heap: maxHeap = [1].\n  - Select the project with the highest profit (1).\n  - Update \\( w \\): \\( w = 0 + 1 = 1 \\).\n  \n- **Iteration 2**:\n  - With \\( w = 1 \\), both remaining projects \\((1, 2)\\) and \\((1, 3)\\) can be started.\n  - Add their profits to the max-heap:textmaxHeap = [2, 3].\n  - Select the project with the highest profit (3).\n  - Update \\( w \\): \\( w = 1 + 3 = 4 \\).\n\nAfter these iterations, the maximum capital \\( w \\) is 4.\n\n### Code Implementation\n\nHere\'s the implementation of the above logic in C++:\n\n```cpp\n#include <bits/stdc++.h>\nusing namespace std;\n\nclass Solution {\npublic:\n    int findMaximizedCapital(int k, int w, vector<int>& profits, vector<int>& capital) {\n        ios_base::sync_with_stdio(false);\n        int n = profits.size();\n        vector<pair<int, int>> minCapital;\n\n        for (int i = 0; i < n; i++)\n            minCapital.push_back({capital[i], profits[i]});\n\n        sort(minCapital.begin(), minCapital.end());\n\n        priority_queue<int> maxProfit;\n        int j = 0;\n        for (int i = 0; i < k; ++i) {\n            while (j < n && minCapital[j].first <= w) {\n                maxProfit.push(minCapital[j].second);\n                j++;\n            }\n\n            if (maxProfit.empty())\n                break;\n\n            w += maxProfit.top();\n            maxProfit.pop();\n        }\n\n        return w;\n    }\n};\n```\n\n## Visual Explanation\n\n1. **Initialization**:\n   - Projects: [(0, 1), (1, 2), (1, 3)]\n   - Capital: \\( w = 0 \\)\n   - Max-Heap: **empty**\n\n2. **Iteration 1**:\n   - Add projects with capital \u2264 0: [(0, 1)]\n   - Max-Heap: [1]\n   - Select max profit 1: Update \\( w = 1 \\)\n\n3. **Iteration 2**:\n   - Add projects with capital \u2264 1: [(1, 2), (1, 3)]\n   - Max-Heap: [2, 3]\n   - Select max profit 3: Update  w = 4 \n\nBy following these steps, we ensure that at each step, we are maximizing the profit while considering the capital constraints, thereby arriving at the optimal solution.
['Greedy', 'Heap (Priority Queue)', 'C++']