Array and Segments (Hard version)
Contest: Codeforces Round 535 (Div. 3)
Difficulty rating: 2100
Tags: data structures, implementation

The only difference between easy and hard versions is a number of elements in the array.

You are given an array $$$a$$$ consisting of $$$n$$$ integers. The value of the $$$i$$$-th element of the array is $$$a_i$$$.

You are also given a set of $$$m$$$ segments. The $$$j$$$-th segment is $$$[l_j; r_j]$$$, where $$$1 \le l_j \le r_j \le n$$$.

You can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $$$a = [0, 0, 0, 0, 0]$$$ and the given segments are $$$[1; 3]$$$ and $$$[2; 4]$$$ then you can choose both of them and the array will become $$$b = [-1, -2, -2, -1, 0]$$$.

You have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $$$a$$$ and obtain the array $$$b$$$ then the value $$$\max\limits_{i=1}^{n}b_i - \min\limits_{i=1}^{n}b_i$$$ will be maximum possible.

Note that you can choose the empty set.

If there are multiple answers, you can print any.

If you are Python programmer, consider using PyPy instead of Python when you submit your code.

Input format:
The first line of the input contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n \le 10^5, 0 \le m \le 300$$$) — the length of the array $$$a$$$ and the number of segments, respectively.

The second line of the input contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$-10^6 \le a_i \le 10^6$$$), where $$$a_i$$$ is the value of the $$$i$$$-th element of the array $$$a$$$.

The next $$$m$$$ lines are contain two integers each. The $$$j$$$-th of them contains two integers $$$l_j$$$ and $$$r_j$$$ ($$$1 \le l_j \le r_j \le n$$$), where $$$l_j$$$ and $$$r_j$$$ are the ends of the $$$j$$$-th segment.

Output format:
In the first line of the output print one integer $$$d$$$ — the maximum possible value $$$\max\limits_{i=1}^{n}b_i - \min\limits_{i=1}^{n}b_i$$$ if $$$b$$$ is the array obtained by applying some subset of the given segments to the array $$$a$$$.

In the second line of the output print one integer $$$q$$$ ($$$0 \le q \le m$$$) — the number of segments you apply.

In the third line print $$$q$$$ distinct integers $$$c_1, c_2, \dots, c_q$$$ in any order ($$$1 \le c_k \le m$$$) — indices of segments you apply to the array $$$a$$$ in such a way that the value $$$\max\limits_{i=1}^{n}b_i - \min\limits_{i=1}^{n}b_i$$$ of the obtained array $$$b$$$ is maximum possible.

If there are multiple answers, you can print any.

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

Note:
In the first example the obtained array $$$b$$$ will be $$$[0, -4, 1, 1, 2]$$$ so the answer is $$$6$$$.

In the second example the obtained array $$$b$$$ will be $$$[2, -3, 1, -1, 4]$$$ so the answer is $$$7$$$.

In the third example you cannot do anything so the answer is $$$0$$$.

Example input:
5 4
2 -2 3 1 2
1 3
4 5
2 5
1 3

Example output:
6
2
4 1

Example input:
5 4
2 -2 3 1 4
3 5
3 4
2 4
2 5

Example output:
7
2
3 2

Editorial:
This tutorial is based on the previous problem (easy version) tutorial.
At first, I want to say I know that this problem and this approach can be implemented in $$$O(m \log n)$$$ with segment tree.
So, we iterate over all supposed maximums in the array and trying to apply all segments not covering our current element. How do we can calculate the answer for each element if this element is the supposed maximum?
Let's divide all segments we apply into two parts: the first part consists of segments such that their right endpoints is less than the current position and the second part consists of segments such that their left endpoints is greater than the current position. Then let's independently calculate answers for the left and for the right parts and merge them to obtain the answer.
I will consider only first part of the solution (first part of segments) because the second part is absolutely symmetric with it.
Let's maintain the minimum value on prefix of the array (let it be $$$mn$$$ and initially it equals to $$$+\infty$$$), maintain the array $$$ansv$$$ of length $$$n$$$ (initially its values are $$$-\infty$$$ and $$$ansv_i$$$ means the answer if the $$$i$$$-th element of the array will be supposed maximum) and the array $$$add$$$ of length $$$n$$$, where $$$add_i$$$ will be the value for which we decrease the $$$i$$$-th element (in other words, the number of segments we apply to the $$$i$$$-th element).
What do we do for the current position $$$i$$$? Firstly, let's update the answer for it with the value $$$a_i + add_i - mn$$$ (in other words, set $$$ansv_i := max(ansv_i, a_i + add_i - mn$$$)). Then let's apply all segments with right endpoints equals to the current position straight-forward and update the value $$$mn$$$ with each new value of covered elements. Just iterate over all positions $$$j$$$ of each segment ends in the current position, make $$$add_j := add_j - 1$$$ and set $$$mn := min(mn, a_j + add_j)$$$. And don't forget to update the value $$$mn$$$ with the value $$$a_i + add_i$$$ after all changes (because we need to update this value with each element not covered by segments too).
So then let's do the same from right to left and then $$$ansv_i$$$ will mean the answer if the $$$i$$$-th element is the supposed maximum in the resulting array. Then we can find any position of the maximum in the array $$$ansv$$$ and apply all segments which don't cover this position.
What is time complexity of the solution above? We iterate over all elements in the array, this is $$$O(n)$$$ and apply each segment in $$$O(n)$$$, so the final time complexity is $$$O(n + nm) = O(nm)$$$.