Tree Sequence

AtCoder

IDarc206_c

Time2000ms

Memory256MB

Difficulty—

A sequence $B=(B_1,\ldots,B_N)$ of length $N$ consisting of integers between $1$ and $N$ (inclusive) is called a good sequence if and only if it satisfies the following condition: * For every interval $[l,r]\ (1\leq l \leq r \leq N)$, there exists an integer $x\ (l\leq x\leq r)$ such that the following condition holds: * Prepare a graph with $N$ vertices numbered from $1$ to $N$ and zero edges. For each $l \leq i \leq r$, if $i\neq x$, add an edge between vertices $i$ and $B_i$. Then, vertices $l, l+1,\ldots,r$ form a tree. That is, vertices $l, l+1,\ldots,r$ are connected. You are given a sequence $A=(A_1,\ldots,A_N)$ where each element is either an integer between $1$ and $N$, or $-1$. Find the number, modulo $998244353$, of ways to replace the elements that are $-1$ in $A$ with integers between $1$ and $N$ such that $A$ becomes a good sequence. ## Constraints * All input values are integers. * $1\leq N\leq 2\times 10^5$ * $1\leq A_i\leq N$ or $A_i=-1$ ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3
-1 -1 -1

Output #1

7

The replacements that satisfy the condition are the following seven:

*   $(1,1,2)$
*   $(2,1,2)$
*   $(2,2,2)$
*   $(2,3,1)$
*   $(2,3,2)$
*   $(2,3,3)$
*   $(3,1,2)$

For example, in the case of $A=(1,1,2)$, that the interval $[1,2]$ satisfies the condition can be confirmed by setting $x=1$, where vertices $2$ and $A_2=1$ are connected and vertices $1,2$ form a tree.
Similarly, intervals $[1,1],[2,2],[3,3],[2,3],[1,3]$ also satisfy the condition, so $(1,1,2)$ is a good sequence.

Input #2

3
-1 3 -1

Output #2

3

The replacements that satisfy the condition are the following three:

*   $(2,3,1)$
*   $(2,3,2)$
*   $(2,3,3)$

Input #3

6
2 3 -1 -1 -1 -1

Output #3

API Response (JSON)

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    "name": "Tree Sequence",
    "description": {
      "content": "A sequence $B=(B_1,\\ldots,B_N)$ of length $N$ consisting of integers between $1$ and $N$ (inclusive) is called a good sequence if and only if it satisfies the following condition: *   For every inter",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
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      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc206_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "A sequence $B=(B_1,\\ldots,B_N)$ of length $N$ consisting of integers between $1$ and $N$ (inclusive) is called a good sequence if and only if it satisfies the following condition:\n\n*   For every inter...",
      "is_translate": false,
      "language": "English"
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