#P1209G2. Into Blocks (hard version)
Into Blocks (hard version)
Description
This is a harder version of the problem. In this version $q \le 200\,000$.
A sequence of integers is called nice if its elements are arranged in blocks like in $[3, 3, 3, 4, 1, 1]$. Formally, if two elements are equal, everything in between must also be equal.
Let's define difficulty of a sequence as a minimum possible number of elements to change to get a nice sequence. However, if you change at least one element of value $x$ to value $y$, you must also change all other elements of value $x$ into $y$ as well. For example, for $[3, 3, 1, 3, 2, 1, 2]$ it isn't allowed to change first $1$ to $3$ and second $1$ to $2$. You need to leave $1$'s untouched or change them to the same value.
You are given a sequence of integers $a_1, a_2, \ldots, a_n$ and $q$ updates.
Each update is of form "$i$ $x$" — change $a_i$ to $x$. Updates are not independent (the change stays for the future).
Print the difficulty of the initial sequence and of the sequence after every update.
The first line contains integers $n$ and $q$ ($1 \le n \le 200\,000$, $0 \le q \le 200\,000$), the length of the sequence and the number of the updates.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 200\,000$), the initial sequence.
Each of the following $q$ lines contains integers $i_t$ and $x_t$ ($1 \le i_t \le n$, $1 \le x_t \le 200\,000$), the position and the new value for this position.
Print $q+1$ integers, the answer for the initial sequence and the answer after every update.
Input
The first line contains integers $n$ and $q$ ($1 \le n \le 200\,000$, $0 \le q \le 200\,000$), the length of the sequence and the number of the updates.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 200\,000$), the initial sequence.
Each of the following $q$ lines contains integers $i_t$ and $x_t$ ($1 \le i_t \le n$, $1 \le x_t \le 200\,000$), the position and the new value for this position.
Output
Print $q+1$ integers, the answer for the initial sequence and the answer after every update.
Samples
5 6
1 2 1 2 1
2 1
4 1
5 3
2 3
4 2
2 1
2
1
0
0
2
3
0