Infinite Operations

AtCoder

IDarc126_e

Time5000ms

Memory256MB

Difficulty—

Given are a sequence of $N$ positive integers $A = (A_1, A_2, \ldots, A_N)$ and $Q$ queries. The $i$\-th query is as follows: * given integers $x_i$ and $y_i$ (where $1\leq x_i\leq N$), change $A_{x_i}$ to $y_i$. Each time the query changes the sequence, find the answer to the following problem modulo $998244353$ (see Notes). > Let $f(n)$ be the maximum total number of points when doing the following operation $n$ times on $A$. > > * Choose $i, j$ such that $A_i\leq A_j$ and choose **non-negative real number** $x$ such that $A_i + 2x \leq A_j$. > * Add $x$ to $A_i$ and subtract $x$ from $A_j$. > * Gain $x$ points. > > It can be proved that the limit $\displaystyle \lim_{n\to\infty} f(n)$ exists. Find this limit. ## Constraints * $2\leq N\leq 3\times 10^5$ * $1\leq Q\leq 3\times 10^5$ * $1\leq A_i \leq 10^9$ * $1\leq x_i\leq N$ * $1\leq y_i\leq 10^9$ ## Input Input is given from Standard Input in the following format: $N$ $Q$ $A_1$ $A_2$ $\ldots$ $A_N$ $x_1$ $y_1$ $\vdots$ $x_Q$ $y_Q$ [samples] ## Notes We can prove that the limit in question is always a rational number. Additionally, under the Constraints of this problem, when that number is represented as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R\times Q\equiv P\pmod{998244353}$ and $0\leq R < 998244353$. Find this $R$.

Samples

Input #1

Output #1

0
1
2
2

The first query changes the sequence to $(5, 5, 5)$. Here, we have $f(n) = 0$ for any $n$, so the answer is $0$.
The second query changes the sequence to $(5, 6, 5)$. Here, one possible way to do the operations is as follows.

*   Choose $(i,j,x) = (3,2,0.4)$, changing the sequence to $(5, 5.6, 5.4)$ and gaining $0.4$ points.
*   Choose $(i,j,x) = (1,2,0.3)$, changing the sequence to $(5.3, 5.3, 5.4)$ and gaining $0.3$ points.

The above two operations gain $0.7$ points, from which we can see that $f(2) \geq 0.7$.
We can prove that it is impossible to gain more than $1$ point in total, and that the total number of points that can be gained by the optimal way to do the operations tends to $1$ as the number of operations increases. Thus, we have $\displaystyle \lim_{n\to\infty} f(n) = 1$.

Input #2

Output #2

API Response (JSON)

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      "content": "Given are a sequence of $N$ positive integers $A = (A_1, A_2, \\ldots, A_N)$ and $Q$ queries. The $i$\\-th query is as follows: *   given integers $x_i$ and $y_i$ (where $1\\leq x_i\\leq N$), change $A_{",
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      "statement_type": "Markdown",
      "content": "Given are a sequence of $N$ positive integers $A = (A_1, A_2, \\ldots, A_N)$ and $Q$ queries. The $i$\\-th query is as follows:\n\n*   given integers $x_i$ and $y_i$ (where $1\\leq x_i\\leq N$), change $A_{...",
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