A14258 | Subset Trick

Vanya invented an interesting trick with a set of integers.

Let an illusionist have a set of positive integers $S$ . He names a positive integer $x$ . Then an audience volunteer must choose some subset (possibly, empty) of $S$ without disclosing it to the illusionist. The volunteer tells the illusionist the size of the chosen subset. And here comes the trick: the illusionist guesses whether the sum of the subset elements does not exceed $x$ . The sum of elements of an empty subset is considered to be $0$ .

Vanya wants to prepare the trick for a public performance. He prepared some set of distinct positive integers $S$ . Vasya wants the trick to be successful. He calls a positive number $x$ unsuitable, if he can't be sure that the trick would be successful for every subset a viewer can choose.

Vanya wants to count the number of unsuitable integers for the chosen set $S$ .

Vanya plans to try different sets $S$ . He wants you to write a program that finds the number of unsuitable integers for the initial set $S$ , and after each change to the set $S$ . Vanya will make $q$ changes to the set, and each change is one of the following two types:

- add a new integer $a$ to the set $S$ , or
- remove some integer $a$ from the set $S$ .

The first line contains two integers $n$ , $q$ ( $1 \leq n, q \leq 200\,000$ ) — the size of the initial set $S$ and the number of changes.

The next line contains $n$ distinct integers $s_1, s_2, \ldots, s_n$ ( $1 \leq s_i \leq 10^{13}$ ) — the initial elements of $S$ .

Each of the following $q$ lines contain two integers $t_i$ , $a_i$ ( $1 \leq t_i \leq 2$ , $1 \leq a_i \leq 10^{13}$ ), describing a change:

- If $t_i = 1$ , then an integer $a_i$ is added to the set $S$ . It is guaranteed that this integer is not present in $S$ before this operation.
- If $t_i = 2$ , then an integer $a_i$ is removed from the set $S$ . In is guaranteed that this integer is present in $S$ before this operation.

Print $q + 1$ lines.

In the first line print the number of unsuitable integers for the initial set $S$ . In the next $q$ lines print the number of unsuitable integers for $S$ after each change.