From b47faa3762bbdf9745b1aeef894857e8fbd01620 Mon Sep 17 00:00:00 2001
From: Himanshu Kumar <97390553+agarwalhimanshugaya@users.noreply.github.com>
Date: Wed, 26 Jun 2024 03:19:29 +0000
Subject: [PATCH] add ques no 630

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+---
+id: course-schedule-III
+title: Course Schedule III
+sidebar_label: 0630 - Course Schedule III
+tags:
+  - Heap
+  - Array
+  - Greedy
+  - Sorting
+description: "This is a solution to the Course Schedule III problem on LeetCode."
+---
+
+## Problem Description
+
+There are n different online courses numbered from `1 to n`. You are given an array courses where `courses[i] = [durationi, lastDayi]` indicate that the `ith` course should be taken continuously for durationi days and must be finished before or on lastDayi.
+
+You will start on the 1st day and you cannot take two or more courses simultaneously.
+
+Return the maximum number of courses that you can take.
+
+### Examples
+
+**Example 1:**
+
+```
+Input: courses = [[100,200],[200,1300],[1000,1250],[2000,3200]]
+Output: 3
+Explanation: 
+There are totally 4 courses, but you can take 3 courses at most:
+First, take the 1st course, it costs 100 days so you will finish it on the 100th day, and ready to take the next course on the 101st day.
+Second, take the 3rd course, it costs 1000 days so you will finish it on the 1100th day, and ready to take the next course on the 1101st day. 
+Third, take the 2nd course, it costs 200 days so you will finish it on the 1300th day. 
+The 4th course cannot be taken now, since you will finish it on the 3300th day, which exceeds the closed date
+
+```
+**Example 2:**
+```
+Input: courses = [[1,2]]
+Output: 1
+
+```
+### Constraints
+
+- `1 <= courses.length <= 10^4`
+- `1 <= durationi, lastDayi <= 10^4`
+
+## Solution for Course Schedule III
+
+### Approach 
+
+Sort all the courses by their ending time. When considering the first `K` courses, they all end before end. A necessary and sufficient condition for our schedule to be valid, is that `(for all K)`, the courses we choose to take within the first K of them, have total duration less than end.
+
+For each `K`, we will greedily remove the largest-length course until the total duration `start is <= end`. To select these largest-length courses, we will use a max heap. start will maintain the loop invariant that it is the sum of the lengths of the courses we have currently taken.
+
+Clearly, this greedy choice makes the number of courses used maximal for each K. When considering potential future K, there's never a case where we preferred having a longer course to a shorter one, so indeed our greedy choice dominates all other candidates.
+
+
+## Code in Different Languages
+
+<Tabs>
+<TabItem value="cpp" label="C++">
+  <SolutionAuthor name="@agarwalhimanshugaya"/>
+
+```cpp
+class Solution {
+public:
+    int scheduleCourse(vector<vector<int>>& courses) {
+        if(courses.size() <= 0) return 0;
+        sort(courses.begin(), courses.end(), [](const vector<int>& a, vector<int>& b) {
+            return a[1] < b[1];
+        });
+        priority_queue<int> q;
+        int sum = 0;
+        for(auto i : courses) {
+            sum += i[0];
+            q.push(i[0]);
+            if(sum > i[1]) {
+                sum -= q.top();
+                q.pop();
+            }
+        }
+        return q.size();
+    }
+};
+```
+</TabItem>
+<TabItem value="java" label="Java">
+  <SolutionAuthor name="@agarwalhimanshugaya"/>
+
+```java
+public class Solution {
+    public int scheduleCourse(int[][] courses) {
+        Arrays.sort(courses,(a,b)->a[1]-b[1]); //Sort the courses by their deadlines (Greedy! We have to deal with courses with early deadlines first)
+        PriorityQueue<Integer> pq=new PriorityQueue<>((a,b)->b-a);
+        int time=0;
+        for (int[] c:courses) 
+        {
+            time+=c[0]; // add current course to a priority queue
+            pq.add(c[0]);
+            if (time>c[1]) time-=pq.poll(); //If time exceeds, drop the previous course which costs the most time. (That must be the best choice!)
+        }        
+        return pq.size();
+    }
+}
+```
+
+</TabItem>
+<TabItem value="python" label="Python">
+  <SolutionAuthor name="@agarwalhimanshugaya"/>
+
+```python
+def scheduleCourse(self, A):
+    pq = []
+    start = 0
+    for t, end in sorted(A, key = lambda (t, end): end):
+        start += t
+        heapq.heappush(pq, -t)
+        while start > end:
+            start += heapq.heappop(pq)
+    return len(pq)
+```
+</TabItem>
+</Tabs>
+
+## Complexity Analysis
+
+### Time Complexity: $O(N⋅log(N))$
+
+### Space Complexity: $O(N)$
+
+## References
+
+- **LeetCode Problem**: [Kth Largest Element in a Stream](https://leetcode.com/problems/course-schedule-iii/description/)
+
+- **Solution Link**: [Kth Largest Element in a Stream](https://leetcode.com/problems/course-schedule-iii/solutions/)