势函数

初始状态的势函数减去终止状态的势函数即为期望。

1|0CF1349D Slime and Biscuits

设 $m=\sum a_i$ 。

\begin{aligned} f (x) & = \frac{x}{m} + \frac{x}{m} f (x - 1) + \frac{m - x}{m (n - 1)} f (x + 1) + \frac{(m - x) (n - 2)}{m (n - 1)} f (x) \end{aligned}

$\large\begin{aligned} f(x)&=\frac{x}{m}+\frac{x}{m}f(x-1)+\frac{m-x}{m(n-1)}f(x+1)+\frac{(m-x)(n-2)}{m(n-1)}f(x) \end{aligned}$

2|0CF850F Rainbow Balls

设 $m=\sum a_i$ 。

\begin{aligned} f (x) & = \frac{x}{m} + \frac{x (x - 1) + (m - x) (m - x - 1)}{m (m - 1)} f (x) + \frac{x (m - x)}{m (m - 1)} (f (x + 1) + f (x - 1)) \\ 0 & = \frac{x}{m} + \frac{x (m - x)}{m (m - 1)} (f (x + 1) + f (x - 1) - 2 f (x)) \end{aligned}

$\large\begin{aligned} f(x)&=\frac{x}{m}+\frac{x(x-1)+(m-x)(m-x-1)}{m(m-1)}f(x)+\frac{x(m-x)}{m(m-1)}(f(x+1)+f(x-1))\\ 0&=\frac{x}{m}+\frac{x(m-x)}{m(m-1)}(f(x+1)+f(x-1)-2f(x))\\ \end{aligned}$

3|0CF1025G Company Acquisitions

\begin{aligned} f (x) + f (y) & = 1 + \frac{1}{2} (f (x + 1) + y f (0)) + \frac{1}{2} (f (y + 1) + x f (0)) \\ f (x) + f (y) & = 1 + \frac{1}{2} f (x + 1) + \frac{1}{2} f (y + 1) \\ f (x) & = \frac{1}{2} + \frac{1}{2} f (x + 1) \\ f (x) & = 1 - 2^{x} \end{aligned}

$\large\begin{aligned} f(x)+f(y)&=1+\frac{1}{2}\left( f(x+1)+yf(0) \right)+\frac{1}{2}\left( f(y+1)+xf(0) \right)\\ f(x)+f(y)&=1+\frac{1}{2}f(x+1)+\frac{1}{2}f(y+1)\\ f(x)&=\frac{1}{2}+\frac{1}{2}f(x+1)\\ f(x)&=1-2^x\\ \end{aligned}$

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