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This approach uses a greedy algorithm in conjunction with priority queues to select the projects maximizing profits. The core idea is to iteratively choose the most profitable project that can be initiated with the current available capital.
Initially, projects are stored in a list along with their capital and profit requirements. We prioritize by try fetching the most profitable projects. We maintain available projects in a max-heap (or priority queue) to achieve this efficiently.
We attempt to start at most k projects, updating the available capital every time we include a new project. For every iteration, once we identify the most feasible projects that can be executed given the current capital, we pick and execute the one with the highest profit and update our available capital. This process repeats until no more projects can be started or we've reached k projects.
Time Complexity: O(N log N + K log N), where N is the number of projects. This involves an initial sort and effectively K max operations on a priority-like structure.
Space Complexity: O(N) due to storage used for auxiliary lists and heaps.
1import heapq
2
3class Solution:
4 def findMaximizedCapital(self, k, w, Profits, Capital):
5 n = len(Profits)
Python's solution takes advantage of its heapq module to function as a max-heap by negating values. It begins by sorting merged project lists on capital, allowing for easy discovery of executable projects by the current capital. Feasible projects are added to a max heap where the most profitable one is repeatedly accessed, increasing capital available for the next iteration.
This dynamic programming approach delineates with the constraints arising from the number of projects executable. We maintain an accruing structure documenting the maximum possible capital achievable for the number of projects done. The dynamic programming strategy opts for inherent flexibility where feasible, owing to recursive checks across all projects capable of initiating.
Memoization ensures repeated subproblem results are accessed without re-computation, improving overall efficiency particularly in scenarios involving expedient overlap of capital ranges per project.
Despite the computationally exhaustive exploration of all project combinations, dynamic memorized caching dramatically curtails avoidant efforts during feasibility analysis—gaining crucial strategic leverage with convenience in considering bounded project completion restrictions.
Time Complexity: O(N * W * K) — represents total options reviewed in DP tableau per states, characterized by projects count, capital capability, projects aimed for completion.
Space Complexity: O(N * W) — retains memory for tailored subproblems caching overseeing potential states across projects under capital explored known projects.
1#include <iostream>
2#include <vector>
3#include <algorithm>
4using namespace std;
5
6class Solution {
7private:
8 vector<vector<int>> dp;
9
10 int findMaximum(int k, int w, vector<pair<int, int>>& projects, int idx) {
11 if (k == 0 || idx >= projects.size())
12 return w;
13
14 if (dp[idx][k] != -1)
15 return dp[idx][k];
16
17 int notTake = findMaximum(k, w, projects, idx + 1);
18 int take = 0;
19
20 if (w >= projects[idx].first)
21 take = findMaximum(k - 1, w + projects[idx].second, projects, idx + 1);
22
23 return dp[idx][k] = max(take, notTake);
24 }
25
26public:
27 int findMaximizedCapital(int k, int w, vector<int>& profits, vector<int>& capital) {
28 vector<pair<int, int>> projects;
29 int n = profits.size();
30 for (int i = 0; i < n; ++i) {
31 projects.push_back({capital[i], profits[i]});
32 }
33
34 dp.assign(n, vector<int>(k + 1, -1));
35
36 return findMaximum(k, w, projects, 0);
37 }
38};
39
40int main() {
41 Solution sol;
42 int k = 2;
43 int w = 0;
44 vector<int> profits = {1, 2, 3};
45 vector<int> capital = {0, 1, 1};
46 int result = sol.findMaximizedCapital(k, w, profits, capital);
47 cout << "The final maximized capital is: " << result << endl;
48 return 0;
49}C++ solution here uses dynamic programming by caching possible results, maintaining these within a vector in which outcomes are kept respective to the projects acquired and capital deployed therewith.
Each scenario considers the possibilities to take or not take a project, updates potential optimal results through memoization—results in non-taken or taken pathways tendered cumulatively towards maximizing capital deployment.