Broken Wheel

AtCoder

IDarc193_b

Time2000ms

Memory256MB

Difficulty—

You are given a positive integer $N$ and a length-$N$ string $s_0s_1\ldots s_{N-1}$ consisting only of `0` and `1`. Consider a simple undirected graph $G$ with $(N+1)$ vertices numbered $0, 1, 2, \ldots, N$, and the following edges: * For each $i = 0, 1, \ldots, N-1$, there is an undirected edge between vertices $i$ and $(i+1)\bmod N$. * For each $i = 0, 1, \ldots, N-1$, there is an undirected edge between vertices $i$ and $N$ if and only if $s_i = $ `1`. * There are no other edges. Furthermore, create a directed graph $G'$ by assigning a direction to each edge of $G$. That is, for each undirected edge $\lbrace u, v \rbrace$ in $G$, replace it with either a directed edge $(u, v)$ from $u$ to $v$ or a directed edge $(v, u)$ from $v$ to $u$. For each $i = 0, 1, \ldots, N$, let $d_i$ be the in-degree of vertex $i$ in $G'$. Print the number, modulo $998244353$, of distinct sequences $(d_0, d_1, \ldots, d_N)$ that can be obtained. ## Constraints * $3 \leq N \leq 10^6$ * $N$ is an integer. * Each $s_i$ is `0` or `1`. ## Input The input is given from Standard Input in the following format: $N$ $s_0s_1\ldots s_{N-1}$ [samples]

Samples

Input #1

3
010

Output #1

14

$G$ has four undirected edges: $\lbrace 0, 1 \rbrace, \lbrace 0, 2 \rbrace, \lbrace 1, 2 \rbrace, \lbrace 1, 3 \rbrace$. For example, if we assign directions to each edge as $0 \to 1$, $2 \to 0$, $2 \to 1$, $1 \to 3$, then $(d_0, d_1, d_2, d_3) = (1, 2, 0, 1)$ is obtained.
The possible sequences $(d_0, d_1, d_2, d_3)$ are $(0, 1, 2, 1)$, $(0, 2, 1, 1)$, $(0, 2, 2, 0)$, $(0, 3, 1, 0)$, $(1, 0, 2, 1)$, $(1, 1, 1, 1)$, $(1, 1, 2, 0)$, $(1, 2, 0, 1)$, $(1, 2, 1, 0)$, $(1, 3, 0, 0)$, $(2, 0, 1, 1)$, $(2, 1, 0, 1)$, $(2, 1, 1, 0)$, $(2, 2, 0, 0)$, for a total of $14$.

Input #2

20
00001100111010100101

Output #2

261339902

API Response (JSON)

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    "name": "Broken Wheel",
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      "content": "You are given a positive integer $N$ and a length-$N$ string $s_0s_1\\ldots s_{N-1}$ consisting only of `0` and `1`. Consider a simple undirected graph $G$ with $(N+1)$ vertices numbered $0, 1, 2, \\ldo",
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    "platform": "AtCoder",
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      "memory_limit": 262144
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      "content": "You are given a positive integer $N$ and a length-$N$ string $s_0s_1\\ldots s_{N-1}$ consisting only of `0` and `1`.\nConsider a simple undirected graph $G$ with $(N+1)$ vertices numbered $0, 1, 2, \\ldo...",
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Full JSON Raw Segments