Modulo MST

AtCoder

IDabc328_e

Time2000ms

Memory256MB

Difficulty—

You are given a weighted simple connected undirected graph with $N$ vertices and $M$ edges, where vertices are numbered $1$ to $N$, and edges are numbered $1$ to $M$. Additionally, a positive integer $K$ is given. Edge $i\ (1\leq i\leq M)$ connects vertices $u_i$ and $v_i$ and has a weight of $w_i$. For a spanning tree $T$ of this graph, the cost of $T$ is defined as the sum, modulo $K$, of the weights of the edges in $T$. Find the minimum cost of a spanning tree of this graph. ## Constraints * $2\leq N\leq8$ * $N-1\leq M\leq\dfrac{N(N-1)}2$ * $1\leq K\leq10^{15}$ * $1\leq u_i\lt v_i\leq N\ (1\leq i\leq M)$ * $0\leq w_i\lt K\ (1\leq i\leq M)$ * The given graph is simple and connected. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ $u_1$ $v_1$ $w_1$ $u_2$ $v_2$ $w_2$ $\vdots$ $u_M$ $v_M$ $w_M$ [samples]

Samples

Input #1

Output #1

33

The given graph is shown below:
![image](https://img.atcoder.jp/abc328/67d2cc2b93ec47687a733cd379c3c07c.png)
The cost of the spanning tree containing edges $1,3,5,6$ is $(99+86+81+95)\bmod{328}=361\bmod{328}=33$. The cost of every spanning tree of this graph is at least $33$, so print $33$.

Input #2

6 5 998244353
1 2 337361568
1 6 450343304
2 3 61477244
2 5 745383438
4 5 727360840

Output #2

325437688

Print the cost of the only spanning tree of this graph, which is $325437688$.

Input #3

8 28 936294041850197
1 2 473294720906780
1 3 743030800139244
1 4 709363019414774
1 5 383643612490312
1 6 557102781022861
1 7 623179288538138
1 8 739618599410809
2 3 857687812294404
2 4 893923168139714
2 5 581822471860662
2 6 740549363586558
2 7 307226438833222
2 8 447399029952998
3 4 636318083622768
3 5 44548707643622
3 6 307262781240755
3 7 12070267388230
3 8 700247263184082
4 5 560567890325333
4 6 704726113717147
4 7 588263818615687
4 8 549007536393172
5 6 779230871080408
5 7 825982583786498
5 8 713928998174272
6 7 751331074538826
6 8 449873635430228
7 8 11298381761479

Output #3

11360716373

Note that the input and the answer may not fit into a $32\operatorname{bit}$ integer.

API Response (JSON)

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