01 Abs Sum

AtCoder

IDttpc2024_2_k

Time8000ms

Memory256MB

Difficulty—

You are given a sequence $A = (A_1, A_2, \cdots, A_N)$ of length $N$, consisting of $0$, $1$, and $-1$. There are $2^q$ ways to replace all $-1$s in $A$ with either $0$ or $1$, where $q$ is the number of $-1$s in $A$. Among these, for all sequences that contain both $0$ and $1$, calculate the sum of the answers to the following problem modulo $998244353$: > Let $\alpha$ and $\beta$ be the number of $0$s and $1$s in $A$, respectively. > Also, let $X = (X_0, X_1, \cdots, X_{\alpha-1})$ be the sequence of indices of $A_i = 0$, arranged in increasing order, and let $Y = (Y_0, Y_1, \cdots, Y_{\beta-1})$ be the sequence of indices of $A_i = 1$, arranged in increasing order (note that the indices of $X$ and $Y$ start from $0$). > > $\displaystyle\sum_{i=0}^{\mathrm{LCM}(\alpha, \beta)-1} |X_{(i \bmod \alpha)} - Y_{(i \bmod \beta)}|$ > > Calculate the value of the above expression. Here, $\mathrm{LCM}(x, y)$ represents the least common multiple of $x$ and $y$, and $x \bmod y$ denotes the remainder when $x$ is divided by $y$. ## Constraints * All input values are integers. * $2 \leq N \leq 2250$ * Each $A_i$ is either $0$, $1$, or $-1$ $(i = 1, 2, \cdots, N)$. ## Input The input is given in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ ## Partial Score * $30$ points will be awarded for passing the test set satisfying the additional constraint $N \leq 100$. [samples]

Samples

Input #1

4
1 1 -1 -1

Output #1

14

For $A = (1, 1, 0, 0)$, $|3 - 1| + |4 - 2| = 4$.  
For $A = (1, 1, 0, 1)$, $|3 - 1| + |3 - 2| + |3 - 4| = 4$.  
For $A = (1, 1, 1, 0)$, $|4 - 1| + |4 - 2| + |4 - 3| = 6$.  
Thus, the answer is $4 + 4 + 6 = 14$.

Input #2

3
0 0 0

Output #2

0

Since it is not possible to form a sequence containing both $0$ and $1$, the answer is $0$.

Input #3

10
-1 -1 0 -1 -1 1 -1 -1 -1 -1

Output #3

API Response (JSON)

{
  "problem": {
    "name": "01 Abs Sum",
    "description": {
      "content": "You are given a sequence $A = (A_1, A_2, \\cdots, A_N)$ of length $N$, consisting of $0$, $1$, and $-1$. There are $2^q$ ways to replace all $-1$s in $A$ with either $0$ or $1$, where $q$ is the number",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "ttpc2024_2_k"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence $A = (A_1, A_2, \\cdots, A_N)$ of length $N$, consisting of $0$, $1$, and $-1$.\nThere are $2^q$ ways to replace all $-1$s in $A$ with either $0$ or $1$, where $q$ is the number...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments