A14132 | Paired Payment

There are $n$ cities and $m$ bidirectional roads in the country. The roads in the country form an undirected weighted graph. The graph is not guaranteed to be connected. Each road has it's own parameter $w$ . You can travel through the roads, but the government made a new law: you can only go through two roads at a time (go from city $a$ to city $b$ and then from city $b$ to city $c$ ) and you will have to pay $(w_{ab} + w_{bc})^2$ money to go through those roads. Find out whether it is possible to travel from city $1$ to every other city $t$ and what's the minimum amount of money you need to get from $1$ to $t$ .

First line contains two integers $n$ , $m$ ( $2 \leq n \leq 10^5$ , $1 \leq m \leq min(\frac{n \cdot (n - 1)}{2}, 2 \cdot 10^5)$ ).

Next $m$ lines each contain three integers $v_i$ , $u_i$ , $w_i$ ( $1 \leq v_i, u_i \leq n$ , $1 \leq w_i \leq 50$ , $u_i \neq v_i$ ). It's guaranteed that there are no multiple edges, i.e. for any edge $(u_i, v_i)$ there are no other edges $(u_i, v_i)$ or $(v_i, u_i)$ .

For every city $t$ print one integer. If there is no correct path between $1$ and $t$ output $-1$ . Otherwise print out the minimum amount of money needed to travel from $1$ to $t$ .