MST on Line++

AtCoder

IDarc167_c

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $K$, and a sequence of $N$ positive integers: $A=(A_{1},A_{2},\dots,A_{N})$. For a permutation $P=(P_{1},P_{2},\dots,P_{N})$ of $(1,2,\dots,N)$, consider the following problem "MST on Line," and let $f(P)$ the answer. > **Problem: MST on Line** > We have a weighted undirected graph $G$ with $N$ vertices numbered $1$ to $N$. For every integer pair $(i,j)$ such that $1\leq i\lt j\leq N$, the following holds for $G$. > > * If $j-i\leq K$, there is an edge between vertex $i$ and vertex $j$, whose weight is **$\max(A_{P_{i}},A_{P_{j}})$**. > * If $j-i\gt K$, there is no edge between vertex $i$ and vertex $j$. > > Find the total weight of the edges of a minimum spanning tree of $G$. Find the sum, modulo $998244353$, of $f(P)$ over all permutations $P=(P_{1},P_{2},\dots ,P_{N})$ of $(1,2,\dots,N)$. ## Constraints * $1\leq K\lt N\leq 5000$ * $1\leq A_{i}\leq 10^{9}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $A_{1}$ $A_{2}$ $\cdots$ $A_{N}$ [samples]

Samples

Input #1

5 2
3 4 5 2 1

Output #1

1740

For $P=(1,2,3,4,5)$, the following edges form a spanning tree of $G$:
the edge between vertices $1$ and $2$ of weight $4$,
the edge between vertices $2$ and $3$ of weight $5$,
the edge between vertices $2$ and $4$ of weight $4$, and
the edge between vertices $4$ and $5$ of weight $2$,
for a total weight of $15$.
It is impossible to take a spanning tree with a smaller total edge weight, so $f(P)=15$.
In this way, it can be seen that the sum of $f(P)$ over all permutations $P$ of $(1,2,3,4,5)$ is $1740$, which should be printed.

Input #2

2 1
167 924

Output #2

Input #3

12 9
22847 98332 854 68844 81080 46058 40949 62493 76561 52907 88628 99740

Output #3

660459584

API Response (JSON)

{
  "problem": {
    "name": "MST on Line++",
    "description": {
      "content": "You are given positive integers $N$ and $K$, and a sequence of $N$ positive integers: $A=(A_{1},A_{2},\\dots,A_{N})$. For a permutation $P=(P_{1},P_{2},\\dots,P_{N})$ of $(1,2,\\dots,N)$, consider the fo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc167_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $K$, and a sequence of $N$ positive integers: $A=(A_{1},A_{2},\\dots,A_{N})$.\nFor a permutation $P=(P_{1},P_{2},\\dots,P_{N})$ of $(1,2,\\dots,N)$, consider the fo...",
      "is_translate": false,
      "language": "English"
    }
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}

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