Many Increasing Problems

AtCoder

IDarc163_f

Time5000ms

Memory256MB

Difficulty—

PCT-kun created the following problem. > **Increasing Problem**You are given a length-$N$ sequence of non-negative integers $A_1,A_2,\dots,A_N$. You can perform the following operation any number of times (possibly zero). > > * Choose an integer $i$ such that $1 \le i \le N$, and increase or decrease $A_i$ by $1$. > > Your goal is to make $A$ non-decreasing. Find the minimum number of operations required to achieve this goal. Thinking that this problem is too easy to be placed at the end of the contest, PCT-kun has revised it as follows. > **Many Increasing Problems**There are $M^N$ integer sequences $A$ of length $N$ where all elements are between $1$ and $M$, inclusive. Find the sum of the answers to **Increasing Problem** for all those sequences, modulo $998244353$. Solve **Many Increasing Problems**. ## Constraints * $1 \le N,M \le 10^5$ ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 2

Output #1

1

Let us solve **Increasing Problem** for all sequences of length $2$ where all elements are between $1$ and $2$, inclusive.

*   For $A=(1,1)$, the answer is $0$.
*   For $A=(1,2)$, the answer is $0$.
*   For $A=(2,1)$, the answer is $1$.
*   For $A=(2,2)$, the answer is $0$.

Therefore, the final answer is $0+0+1+0=1$.

Input #2

6 4

Output #2

Input #3

163 702

Output #3

20728656

Input #4

98765 99887

Output #4

103564942

API Response (JSON)

{
  "problem": {
    "name": "Many Increasing Problems",
    "description": {
      "content": "PCT-kun created the following problem. > **Increasing Problem**You are given a length-$N$ sequence of non-negative integers $A_1,A_2,\\dots,A_N$. You can perform the following operation any number of ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc163_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "PCT-kun created the following problem.\n\n> **Increasing Problem**You are given a length-$N$ sequence of non-negative integers $A_1,A_2,\\dots,A_N$. You can perform the following operation any number of ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments