Vaccination
Contest: Nebius Welcome Round (Div. 1 + Div. 2)
Difficulty rating: 1000
Tags: greedy, implementation

Ethan runs a vaccination station to help people combat the seasonal flu. He analyses the historical data in order to develop an optimal strategy for vaccine usage.

Consider there are $$$n$$$ patients coming to the station on a particular day. The $$$i$$$-th patient comes at the moment $$$t_i$$$. We know that each of these patients can be asked to wait for no more than $$$w$$$ time moments. That means the $$$i$$$-th patient can get vaccine at moments $$$t_i, t_i + 1, \ldots, t_i + w$$$.

Vaccines come in packs, each pack consists of $$$k$$$ doses. Each patient needs exactly one dose. Packs are stored in a special fridge. After a pack is taken out of the fridge and opened, it can no longer be put back. The lifetime of the vaccine outside the fridge is $$$d$$$ moments of time. Thus, if the pack was taken out of the fridge and opened at moment $$$x$$$, its doses can be used to vaccinate patients at moments $$$x, x + 1, \ldots, x + d$$$. At moment $$$x + d + 1$$$ all the remaining unused doses of this pack are thrown away.

Assume that the vaccination station has enough staff to conduct an arbitrary number of operations at every moment of time. What is the minimum number of vaccine packs required to vaccinate all $$$n$$$ patients?

Input format:
The first line of the input contains the number of test cases $$$t$$$ ($$$1 \leq t \leq 10^4$$$). Then follow $$$t$$$ descriptions of the test cases.

The first line of each test case contains four integers $$$n$$$, $$$k$$$, $$$d$$$ and $$$w$$$ ($$$1 \leq n, k \leq 2 \cdot 10^5$$$, $$$0 \leq d, w \leq 10^6$$$). They are the number of patients, the number of doses per vaccine pack, the number of moments of time the vaccine can live outside the fridge, and the number of moments of time each of the patients can wait, respectively.

The second line of each test case contains a non-decreasing sequence $$$t_1, t_2, \ldots, t_n$$$ ($$$0 \leq t_1 \leq t_2 \leq \ldots \leq t_n \leq 10^6$$$). The $$$i$$$-th element of this sequence is the moment when the $$$i$$$-th patient comes to the vaccination station.

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

Output format:
Output one integer, the minimum number of vaccine packs required to vaccinate all $$$n$$$ patients.

Constraints:
Time limit: 2.0 s
Memory limit: 512.0 MB

Note:
In the first example, the first pack can be opened at moment $$$1$$$ to vaccinate patient $$$1$$$. The vaccine is durable enough to be used at moments $$$2$$$ and $$$3$$$ for patients $$$2$$$ and $$$3$$$, respectively. Then the staff needs to ask patients $$$4$$$ and $$$5$$$ to wait for moment $$$13$$$. At moment $$$13$$$ the staff opens the second vaccine pack and serves patients $$$4$$$ and $$$5$$$. Finally, the last patient comes at moment $$$18$$$ and immediately gets the last dose of the second pack while it is still fine.

In the second example, the vaccine should be used exactly at the moment it is taken out of the fridge. Moreover, all the patients want to be served at exactly the same moment they come. That means the staff needs to open two packs at moment $$$3$$$ and use five doses on patients $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, and $$$5$$$. There will be three doses left ouf of these two packs but they can't be used for patient $$$6$$$. When patient $$$6$$$ comes at moment $$$4$$$ the staff needs to open a new pack just to use only one dose out of it.

Example input:
5
6 3 5 3
1 2 3 10 11 18
6 4 0 0
3 3 3 3 3 4
9 10 2 2
0 1 2 3 4 5 6 7 8
3 10 3 6
10 20 30
5 5 4 4
0 2 4 6 8

Example output:
2
3
2
3
1

Editorial:
Observation 1. There exists an optimal answer where each pack of vaccine is used for a consecutive segment of patients. Indeed, if there are three patients $$$a < b < c$$$ such that $$$a$$$ and $$$c$$$ are vaccinated using the dose from one pack and $$$b$$$ is vaccinated using the dose from the other pack we can swap the packs used for $$$b$$$ and $$$c$$$ and the answer will still be valid.
Observation 2. It always makes sense to ask new patients to wait as long as possible before opening a new pack.
From these two observations we derive a very easy strategy. Consider patients one by one in order of non-decreasing $$$t_i$$$. If we consider some patient $$$i$$$ and there is an open pack that still valid and still has some doses remaining, use it immediately. If there is no valid open pack of vaccines and there is no one waiting, ask patient $$$i$$$ to wait till $$$t_i + w$$$ moment of time. If there is no valid pack of vaccines, but there is someone already waiting for moment $$$x$$$, ask patient $$$i$$$ to wait for moment $$$x$$$ as well.
As a courtesy to our participants the values of $$$d$$$ and $$$w$$$ are limited by $$$10^6$$$ to avoid a potential overflow of a signed 32-bit integer type.