Minimum Longest Trip

Bessie is going on a trip in Cowland, which has $N$ ($2\le N\le 2\cdot 10^5$) towns numbered from $1$ to $N$ and $M$ ($1\le M\le 4\cdot 10^5$) one-way roads. The $i$th road runs from town $a_i$ to town $b_i$ and has label $l_i$ ($1\le a_i,b_i\le N$, $1\le l_i\le 10^9$).

A trip of length $k$ starting at town $x_0$ is a sequence of towns $x_0,x_1,\dots ,x_k$, such that there is a road from town $x_i$ to town $x_{i+1}$ for all $0\le i<k$. It is guaranteed that there are no trips of infinite length in Cowland, and that no two roads connect the same pair of towns.

For each town, Bessie wants to know the longest possible trip starting at it. For some starting towns, there are multiple longest trips - out of these, she prefers the trip with the lexicographically minimum sequence of road labels. A sequence is lexicographically smaller than another sequence of the same length if, at the first position in which they differ, the first sequence has a smaller element than the second sequence.

Output the length and sum of road labels of Bessie's preferred trip starting at each town.

In the following explanation, we let $a_i\stackrel{l_i}{\to }{b}_i$ represent the road from $a_i$ to $b_i$ with label $l_i$.

There are several trips starting from vertex $4$, including $4\stackrel{4}{\to }3\stackrel{5}{\to }1$, $4\stackrel{1}{\to }1$, and $4\stackrel{2}{\to }2\stackrel{10}{\to }1$. Of these trips, $4\stackrel{4}{\to }3\stackrel{5}{\to }1$ and $4\stackrel{2}{\to }2\stackrel{10}{\to }1$ are the longest. These trips each have length 2, and their road label sequences are $[4,5]$ and $[2,10]$, respectively. $[2,10]$ is the lexicographically smaller sequence, and its sum is $12$.

Problem credits: Claire Zhang and Spencer Compton

Source: USA Computing Olympiad.

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