Problem Statement
Capital Investment Optimizer
You're a financial advisor helping a startup maximize its capital before an Initial Public Offering (IPO).
The startup has a list of potential projects it can undertake. Each project requires a minimum amount of capital to start and will generate a certain profit upon completion.
Given the startup's initial capital and the fact that it can only complete a limited number of projects before the IPO, your task is to determine the maximum capital the startup can accumulate.
Specifically, you are given:
- An integer
krepresenting the maximum number of projects the startup can complete before the IPO. - An integer
wrepresenting the initial capital of the startup. - Two integer arrays
profitsandcapitalwhereprofits[i]is the profit of theith project andcapital[i]is the minimum capital required to start theith project.
Once a project is completed, its profit is added to the startup's total capital. The startup can only work on one project at a time.
Return the maximum total capital the startup can have after completing at most k projects.
Examples
Example 1:
Input: k = 2, w = 0, profits = [1,2,3], capital = [0,1,1]
Output: 4
Explanation: Since the initial capital is 0, you can only start the project indexed 0.
After finishing it, your capital becomes 1 and you can start either project indexed 1 or 2.
Since you can choose at most 2 projects, you need to finish the project indexed 2 to maximize capital.
Therefore, the final capital is 0 + 1 + 3 = 4.
Example 2:
Input: k = 3, w = 0, profits = [1,2,3], capital = [0,1,2]
Output: 6
Explanation: Since the initial capital is 0, you can only start the project indexed 0.
After finishing it, your capital becomes 1 and you can start the project indexed 1.
After finishing that, your capital becomes 3 and you can start the project indexed 2.
After finishing that, your capital becomes 6.
Therefore, the final capital is 0 + 1 + 2 + 3 = 6.
Constraints
- 1 <= k <= 10^5
- 0 <= w <= 10^9
- n == profits.length
- n == capital.length
- 1 <= n <= 10^5
- 0 <= profits[i] <= 10^4
- 0 <= capital[i] <= 10^9
Problem Breakdown
To solve this problem, we need to:
- The key insight is to always choose the project with the highest profit among all available projects
- A project is available if its required capital is less than or equal to the current capital
- We need to efficiently track available projects as our capital increases
- Using two priority queues (heaps) can help: one to sort projects by capital, and another to sort available projects by profit
- The greedy approach of always choosing the highest profit project is optimal for this problem
- We can process at most k projects, but might process fewer if we run out of available projects
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Capital Investment Optimizer
You're a financial advisor helping a startup maximize its capital before an Initial Public Offering (IPO).
The startup has a list of potential projects it can undertake. Each project requires a minimum amount of capital to start and will generate a certain profit upon completion.
Given the startup's initial capital and the fact that it can only complete a limited number of projects before the IPO, your task is to determine the maximum capital the startup can accumulate.
Specifically, you are given:
- An integer
krepresenting the maximum number of projects the startup can complete before the IPO. - An integer
wrepresenting the initial capital of the startup. - Two integer arrays
profitsandcapitalwhereprofits[i]is the profit of theith project andcapital[i]is the minimum capital required to start theith project.
Once a project is completed, its profit is added to the startup's total capital. The startup can only work on one project at a time.
Return the maximum total capital the startup can have after completing at most k projects.
Examples
Example 1:
Since the initial capital is 0, you can only start the project indexed 0. After finishing it, your capital becomes 1 and you can start either project indexed 1 or 2. Since you can choose at most 2 projects, you need to finish the project indexed 2 to maximize capital. Therefore, the final capital is 0 + 1 + 3 = 4.
Example 2:
Since the initial capital is 0, you can only start the project indexed 0. After finishing it, your capital becomes 1 and you can start the project indexed 1. After finishing that, your capital becomes 3 and you can start the project indexed 2. After finishing that, your capital becomes 6. Therefore, the final capital is 0 + 1 + 2 + 3 = 6.
Problem Breakdown
Constraints:
- 1 <= k <= 10^5
- 0 <= w <= 10^9
- n == profits.length
- n == capital.length
- 1 <= n <= 10^5
- 0 <= profits[i] <= 10^4
- 0 <= capital[i] <= 10^9
Key Insights:
The key insight is to always choose the project with the highest profit among all available projects
A project is available if its required capital is less than or equal to the current capital
We need to efficiently track available projects as our capital increases
Using two priority queues (heaps) can help: one to sort projects by capital, and another to sort available projects by profit
The greedy approach of always choosing the highest profit project is optimal for this problem
We can process at most k projects, but might process fewer if we run out of available projects
Real-World Application
This problem has several practical applications:
Investment Portfolio Management
Optimizing investment decisions to maximize returns with limited capital.
Startup Funding Allocation
Deciding which projects to fund with limited resources to maximize company valuation.
Resource Allocation
Allocating limited resources to projects with different requirements and returns.
Financial Planning
Planning investment strategies to maximize wealth growth over a fixed period.
Venture Capital Decisions
Making investment decisions across multiple startups with varying capital requirements and potential returns.