E. Building an Aquarium
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

You love fish, that's why you have decided to build an aquarium. You have a piece of coral made of $$$n$$$ columns, the $$$i$$$-th of which is $$$a_i$$$ units tall. Afterwards, you will build a tank around the coral as follows:

  • Pick an integer $$$h \geq 1$$$ — the height of the tank. Build walls of height $$$h$$$ on either side of the tank.
  • Then, fill the tank up with water so that the height of each column is $$$h$$$, unless the coral is taller than $$$h$$$; then no water should be added to this column.
For example, with $$$a=[3,1,2,4,6,2,5]$$$ and a height of $$$h=4$$$, you will end up using a total of $$$w=8$$$ units of water, as shown.
You can use at most $$$x$$$ units of water to fill up the tank, but you want to build the biggest tank possible. What is the largest value of $$$h$$$ you can select?

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The first line of each test case contains two positive integers $$$n$$$ and $$$x$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$; $$$1 \leq x \leq 10^9$$$) — the number of columns of the coral and the maximum amount of water you can use.

The second line of each test case contains $$$n$$$ space-separated integers $$$a_i$$$ ($$$1 \leq a_i \leq 10^9$$$) — the heights of the coral.

The sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.


For each test case, output a single positive integer $$$h$$$ ($$$h \geq 1$$$) — the maximum height the tank can have, so you need at most $$$x$$$ units of water to fill up the tank.

We have a proof that under these constraints, such a value of $$$h$$$ always exists.

7 9
3 1 2 4 6 2 5
3 10
1 1 1
4 1
1 4 3 4
6 1984
2 6 5 9 1 8
1 1000000000

The first test case is pictured in the statement. With $$$h=4$$$ we need $$$8$$$ units of water, but if $$$h$$$ is increased to $$$5$$$ we need $$$13$$$ units of water, which is more than $$$x=9$$$. So $$$h=4$$$ is optimal.

In the second test case, we can pick $$$h=4$$$ and add $$$3$$$ units to each column, using a total of $$$9$$$ units of water. It can be shown that this is optimal.

In the third test case, we can pick $$$h=2$$$ and use all of our water, so it is optimal.