Kolya and Movie Theatre
Contest: Codeforces Round 894 (Div. 3)
Difficulty rating: 1600
Tags: constructive algorithms, data structures, greedy

Recently, Kolya found out that a new movie theatre is going to be opened in his city soon, which will show a new movie every day for $$$n$$$ days. So, on the day with the number $$$1 \le i \le n$$$, the movie theatre will show the premiere of the $$$i$$$-th movie. Also, Kolya found out the schedule of the movies and assigned the entertainment value to each movie, denoted by $$$a_i$$$.

However, the longer Kolya stays without visiting a movie theatre, the larger the decrease in entertainment value of the next movie. That decrease is equivalent to $$$d \cdot cnt$$$, where $$$d$$$ is a predetermined value and $$$cnt$$$ is the number of days since the last visit to the movie theatre. It is also known that Kolya managed to visit another movie theatre a day before the new one opened — the day with the number $$$0$$$. So if we visit the movie theatre the first time on the day with the number $$$i$$$, then $$$cnt$$$ — the number of days since the last visit to the movie theatre will be equal to $$$i$$$.

For example, if $$$d = 2$$$ and $$$a = [3, 2, 5, 4, 6]$$$, then by visiting movies with indices $$$1$$$ and $$$3$$$, $$$cnt$$$ value for the day $$$1$$$ will be equal to $$$1 - 0 = 1$$$ and $$$cnt$$$ value for the day $$$3$$$ will be $$$3 - 1 = 2$$$, so the total entertainment value of the movies will be $$$a_1 - d \cdot 1 + a_3 - d \cdot 2 = 3 - 2 \cdot 1 + 5 - 2 \cdot 2 = 2$$$.

Unfortunately, Kolya only has time to visit at most $$$m$$$ movies. Help him create a plan to visit the cinema in such a way that the total entertainment value of all the movies he visits is maximized.

Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains three integers $$$n$$$, $$$m$$$, and $$$d$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le m \le n$$$, $$$1 \le d \le 10^9$$$).

The second line of each set of input data contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$-10^9 \le a_i \le 10^9$$$) — the entertainment values of the movies.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output format:
For each test case, output a single integer — the maximum total entertainment value that Kolya can get.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
The first test case is explained in the problem statement.

In the second test case, it is optimal not to visit any movies.

In the third test case, it is optimal to visit movies with numbers $$$2$$$, $$$3$$$, $$$5$$$, $$$6$$$, so the total entertainment value of the visited movies will be $$$45 - 6 \cdot 2 + 1 - 6 \cdot 1 + 39 - 6 \cdot 2 + 11 - 6 \cdot 1 = 60$$$.

Example input:
6
5 2 2
3 2 5 4 6
4 3 2
1 1 1 1
6 6 6
-82 45 1 -77 39 11
5 2 2
3 2 5 4 8
2 1 1
-1 2
6 3 2
-8 8 -2 -1 9 0

Example output:
2
0
60
3
0
7

Editorial:
Let's notice that if we visit the cinema on days with numbers $$$i_1, i_2, \ldots, i_k$$$, the total entertainment value of the visited movies will be equal to $$$(a_{i_1} - d \cdot i_1) + (a_{i_2} - d \cdot (i_2 - i_1)) + \ldots + (a_{i_k} - d \cdot (i_k - i_{k-1})) = (a_{i_1} + a_{i_2} + \ldots + a_{i_k}) - d \cdot i_k$$$.
Thus, it is sufficient to iterate over the day when Kolya will visit the cinema for the last time — $$$i_k$$$, and maintain the maximum $$$m - 1$$$ non-negative elements on the prefix $$$[1, i_k - 1]$$$. This can be done, for example, using std::multiset. The total complexity of the solution will be $$$O(n \log{n})$$$.