A13997 | Minimum Path

You are given a weighted undirected connected graph consisting of $n$ vertices and $m$ edges. It is guaranteed that there are no self-loops or multiple edges in the given graph.

Let's define the weight of the path consisting of $k$ edges with indices $e_1, e_2, \dots, e_k$ as $\sum\limits_{i=1}^{k}{w_{e_i}} - \max\limits_{i=1}^{k}{w_{e_i}} + \min\limits_{i=1}^{k}{w_{e_i}}$ , where $w_i$ — weight of the $i$ -th edge in the graph.

Your task is to find the minimum weight of the path from the $1$ -st vertex to the $i$ -th vertex for each $i$ ( $2 \le i \le n$ ).

The first line contains two integers $n$ and $m$ ( $2 \le n \le 2 \cdot 10^5$ ; $1 \le m \le 2 \cdot 10^5$ ) — the number of vertices and the number of edges in the graph.

Following $m$ lines contains three integers $v_i, u_i, w_i$ ( $1 \le v_i, u_i \le n$ ; $1 \le w_i \le 10^9$ ; $v_i \neq u_i$ ) — endpoints of the $i$ -th edge and its weight respectively.

Print $n-1$ integers — the minimum weight of the path from $1$ -st vertex to the $i$ -th vertex for each $i$ ( $2 \le i \le n$ ).