Maximum Number of Events That Can Be Attended II
<p>You are given an array of <code>events</code> where <code>events[i] = [startDay<sub>i</sub>, endDay<sub>i</sub>, value<sub>i</sub>]</code>. The <code>i<sup>th</sup></code> event starts at <code>startDay<sub>i</sub></code><sub> </sub>and ends at <code>endDay<sub>i</sub></code>, and if you attend this event, you will receive a value of <code>value<sub>i</sub></code>. You are also given an integer <code>k</code> which represents the maximum number of events you can attend.</p>

<p>You can only attend one event at a time. If you choose to attend an event, you must attend the <strong>entire</strong> event. Note that the end day is <strong>inclusive</strong>: that is, you cannot attend two events where one of them starts and the other ends on the same day.</p>

<p>Return <em>the <strong>maximum sum</strong> of values that you can receive by attending events.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1700-1799/1751.Maximum%20Number%20of%20Events%20That%20Can%20Be%20Attended%20II/images/screenshot-2021-01-11-at-60048-pm.png" style="width: 400px; height: 103px;" /></p>

<pre>
<strong>Input:</strong> events = [[1,2,4],[3,4,3],[2,3,1]], k = 2
<strong>Output:</strong> 7
<strong>Explanation: </strong>Choose the green events, 0 and 1 (0-indexed) for a total value of 4 + 3 = 7.</pre>

<p><strong class="example">Example 2:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1700-1799/1751.Maximum%20Number%20of%20Events%20That%20Can%20Be%20Attended%20II/images/screenshot-2021-01-11-at-60150-pm.png" style="width: 400px; height: 103px;" /></p>

<pre>
<strong>Input:</strong> events = [[1,2,4],[3,4,3],[2,3,10]], k = 2
<strong>Output:</strong> 10
<strong>Explanation:</strong> Choose event 2 for a total value of 10.
Notice that you cannot attend any other event as they overlap, and that you do <strong>not</strong> have to attend k events.</pre>

<p><strong class="example">Example 3:</strong></p>

<p><strong><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1700-1799/1751.Maximum%20Number%20of%20Events%20That%20Can%20Be%20Attended%20II/images/screenshot-2021-01-11-at-60703-pm.png" style="width: 400px; height: 126px;" /></strong></p>

<pre>
<strong>Input:</strong> events = [[1,1,1],[2,2,2],[3,3,3],[4,4,4]], k = 3
<strong>Output:</strong> 9
<strong>Explanation:</strong> Although the events do not overlap, you can only attend 3 events. Pick the highest valued three.</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= events.length</code></li>
	<li><code>1 &lt;= k * events.length &lt;= 10<sup>6</sup></code></li>
	<li><code>1 &lt;= startDay<sub>i</sub> &lt;= endDay<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= value<sub>i</sub> &lt;= 10<sup>6</sup></code></li>
</ul>
Array; Binary Search; Dynamic Programming; Sorting
class Solution:
    def maxValue(self, events: List[List[int]], k: int) -> int:
        @cache
        def dfs(i: int, k: int) -> int:
            if i >= len(events):
                return 0
            _, ed, val = events[i]
            ans = dfs(i + 1, k)
            if k:
                j = bisect_right(events, ed, lo=i + 1, key=lambda x: x[0])
                ans = max(ans, dfs(j, k - 1) + val)
            return ans

        events.sort()
        return dfs(0, k)
