A11783 | Game

Allen and Bessie are playing a simple number game. They both know a function $f: \{0, 1\}^n \to \mathbb{R}$ , i. e. the function takes $n$ binary arguments and returns a real value. At the start of the game, the variables $x_1, x_2, \dots, x_n$ are all set to $-1$ . Each round, with equal probability, one of Allen or Bessie gets to make a move. A move consists of picking an $i$ such that $x_i = -1$ and either setting $x_i \to 0$ or $x_i \to 1$ .

After $n$ rounds all variables are set, and the game value resolves to $f(x_1, x_2, \dots, x_n)$ . Allen wants to maximize the game value, and Bessie wants to minimize it.

Your goal is to help Allen and Bessie find the expected game value! They will play $r+1$ times though, so between each game, exactly one value of $f$ changes. In other words, between rounds $i$ and $i+1$ for $1 \le i \le r$ , $f(z_1, \dots, z_n) \to g_i$ for some $(z_1, \dots, z_n) \in \{0, 1\}^n$ . You are to find the expected game value in the beginning and after each change.

The first line contains two integers $n$ and $r$ ( $1 \le n \le 18$ , $0 \le r \le 2^{18}$ ).

The next line contains $2^n$ integers $c_0, c_1, \dots, c_{2^n-1}$ ( $0 \le c_i \le 10^9$ ), denoting the initial values of $f$ . More specifically, $f(x_0, x_1, \dots, x_{n-1}) = c_x$ , if $x = \overline{x_{n-1} \ldots x_0}$ in binary.

Each of the next $r$ lines contains two integers $z$ and $g$ ( $0 \le z \le 2^n - 1$ , $0 \le g \le 10^9$ ). If $z = \overline{z_{n-1} \dots z_0}$ in binary, then this means to set $f(z_0, \dots, z_{n-1}) \to g$ .

Print $r+1$ lines, the $i$ -th of which denotes the value of the game $f$ during the $i$ -th round. Your answer must have absolute or relative error within $10^{-6}$ .

Formally, let your answer be $a$ , and the jury's answer be $b$ . Your answer is considered correct if $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$ .