A12788. Into Blocks (hard version)

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.