Maximize funds with capital-gated projects

You start with initial funds W and may undertake at most K projects. Each project i has a required capital C[i] and yields profit P[i]; you can only start project i if your current funds are at least C[i], and upon completion your funds increase by P[i]. Choose up to K projects to maximize your final funds. Design an efficient algorithm, specify the data structures, analyze time and space complexity, and justify correctness. Discuss how your solution scales when N is as large as 2e5.

What a Strong Answer Covers

• A clear algorithm with the right data structures and enough pseudocode or code-level detail to implement it.
• A correctness argument that explains why the algorithm covers all required cases.
• Time and space complexity, plus at least one alternative approach when relevant.
• Focused tests for normal cases, edge cases, and failure modes.

Follow-up Questions

• How would the approach change if the input were streaming or too large for memory?
• What invariants would you assert in production code?
• Which tests would catch off-by-one, duplicate, or tie-breaking bugs?

Maximize funds with capital-gated projects

You start with initial funds w and may undertake at most k projects. Each project i has a required capital capital[i] and yields profit profits[i]. You can only start project i if your current funds are at least capital[i], and upon completion your funds increase by profits[i] (profits are additive; capital is not consumed). Choose up to k projects to maximize your final funds and return that maximum.

This is a classic greedy + heap problem: at each of the k steps, among all projects affordable with the current funds, pick the one with the largest profit.

Function signature: solution(k, w, profits, capital) returns the maximum final funds.

Example: k=2, w=0, profits=[1,2,3], capital=[0,1,1] -> 4. First only project 0 (capital 0) is affordable; take it (funds 0 -> 1). Now projects 1 and 2 (capital 1) are affordable; take project 2 (profit 3) for funds 1 -> 4.

Constraints

• 0 <= k <= N
• 0 <= w (and intermediate funds) fit in 64-bit integers
• len(profits) == len(capital) == N, with N up to 2e5
• 0 <= profits[i], 0 <= capital[i]
• Each project may be taken at most once

Hints

1. Greedy choice: at each step you want the affordable project with the highest profit, because taking it never reduces (and may increase) which projects become affordable later.
2. Sort projects by required capital so you can sweep in a pointer and unlock projects as your funds grow.
3. Use a max-heap keyed on profit to hold all currently affordable, not-yet-taken projects; pop the top each round.
4. Stop early if you run out of affordable projects (heap empty) before reaching k picks.

Solution

Solution Alignment

The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests.

Problem Restatement

You start with initial funds W and may undertake at most K projects. Each project i has a required capital C[i] and yields profit P[i]; you can only start project i if your current funds are at least C[i], and upon completion your funds increase by P[i]. Choose up to K projects to maximize your final funds. Design an efficient algorithm, specify the data structures, analyze time and space complexity, and justify correctness. Discuss how your solution scales when N is as large as 2e5.

Recommended Approach

Start with a brute-force baseline to confirm correctness, then identify the repeated work or ordering property that enables a better data structure such as a hash map, heap, stack, queue, two pointers, prefix sums, BFS/DFS, or dynamic programming. Write the implementation around a small invariant and test that invariant directly.

Correctness

The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call.

Complexity

State the baseline complexity and the optimized complexity. For most interview constraints, justify why the optimized approach meets the expected input size.

Edge Cases and Tests

Empty and singleton inputs, duplicates, ties, invalid inputs, boundary values, and tests that exercise the main invariant.