P9716 [EC Final 2022] Coloring

You are given $n$ elements numbered from $1$ to $n$. Element $i$ has value $w_i$ and color $c_i$. Each element also has a pointer $a_i$ to some other element. Initially, the color of element $s$ is $1$, while the color of all the other elements is $0$. More formally, $c_s=1$ and $c_i=0$ for all $i\neq s$ $(1 \le i \le n)$. You can perform the following operation for any number of times: - Assign $c_i\leftarrow c_{a_i}$ at a cost of $p_i$. Your score is equal to the sum of values of all the elements with color $1$ after the operations minus the sum of costs of the operations. Find the maximum possible score you can obtain.

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(There won’t be extra line breakers in the actual test cases.) In the first sample, you can successively perform the following operations: - Assign $c_2\leftarrow c_{a_2}$ at a cost of $p_2$, then $c=[1,1,0]$; - Assign $c_1\leftarrow c_{a_1}$ at a cost of $p_1$, then $c=[0,1,0]$; - Assign $c_3\leftarrow c_{a_3}$ at a cost of $p_3$, then $c=[0,1,1]$; - Assign $c_2\leftarrow c_{a_2}$ at a cost of $p_2$, then $c=[0,0,1]$. After the operations, only the color of element $3$ is $1$, so your score is equal to $w_3-(p_2+p_1+p_3+p_2)=1$. It can be shown that it is impossible to obtain a score greater than $1$.