D - Shortest Path Queries 2

Score : $400$ points

Problem Statement

There are $N$ cities and $M$ roads in the Kingdom of Takahashi.

The cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ is a one-way road that leads from City $A_i$ to City $B_i$, and it takes $C_i$ minutes to go through it.

Let us define $f(s, t, k)$ as the answer to the following query.

• Compute the minimum time needed to get from City $s$ to City $t$. Here, other than the Cities $s$ and $t$, it is only allowed to pass through Cities $1$ through $k$. If City $t$ is unreachable or $s = t$, the answer should be $0$.

Compute $f(s,t,k)$ for all triples $s,t,k$ and print the sum of them. More formally, print $\displaystyle \sum_{s = 1}^N \sum_{t = 1}^N \sum_{k = 1}^N f(s, t, k)$.

Constraints

• $1 \leq N \leq 400$
• $0 \leq M \leq N(N-1)$
• $1 \leq A_i \leq N$ $(1 \leq i \leq M)$
• $1 \leq B_i \leq N$ $(1 \leq i \leq M)$
• $A_i \neq B_i$ $(1 \leq i \leq M)$
• $1 \leq C_i \leq 10^6$ $(1 \leq i \leq M)$
• $A_i \neq A_j$ or $B_i \neq B_j$, if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$ $C_1$

$\vdots$

$A_M$ $B_M$ $C_M$

Output

Print $\displaystyle \sum_{s = 1}^N \sum_{t = 1}^N \sum_{k = 1}^N f(s, t, k)$.

3 2
1 2 3
2 3 2

25

The triples $s,t,k$ such that $f(s,t,k) \neq 0$ are as follows.

• For $k = 1$: $f(1,2,1) = 3, f(2,3,1) = 2$.
• For $k = 2$: $f(1,2,2) = 3, f(2,3,2) = 2, f(1,3,2) = 5$.
• For $k = 3$: $f(1,2,3) = 3, f(2,3,3) = 2, f(1,3,3) = 5$.

3 0

0

We have $f(s,t,k) = 0$ for all $s,t,k$.

5 20
1 2 6
1 3 10
1 4 4
1 5 1
2 1 5
2 3 9
2 4 8
2 5 6
3 1 5
3 2 1
3 4 7
3 5 9
4 1 4
4 2 6
4 3 4
4 5 8
5 1 2
5 2 5
5 3 6
5 4 5

517