Knapsack for All Segments

AtCoder

IDabc159_f

Time2000ms

Memory256MB

Difficulty—

Given are a sequence of $N$ integers $A_1$, $A_2$, $\ldots$, $A_N$ and a positive integer $S$. For a pair of integers $(L, R)$ such that $1\leq L \leq R \leq N$, let us define $f(L, R)$ as follows: * $f(L, R)$ is the number of sequences of integers $(x_1, x_2, \ldots , x_k)$ such that $L \leq x_1 < x_2 < \cdots < x_k \leq R$ and $A_{x_1}+A_{x_2}+\cdots +A_{x_k} = S$. Find the sum of $f(L, R)$ over all pairs of integers $(L, R)$ such that $1\leq L \leq R\leq N$. Since this sum can be enormous, print it modulo $998244353$. ## Constraints * All values in input are integers. * $1 \leq N \leq 3000$ * $1 \leq S \leq 3000$ * $1 \leq A_i \leq 3000$ ## Input Input is given from Standard Input in the following format: $N$ $S$ $A_1$ $A_2$ $...$ $A_N$ [samples]

Samples

Input #1

3 4
2 2 4

Output #1

5

The value of $f(L, R)$ for each pair is as follows, for a total of $5$.

*   $f(1,1) = 0$
*   $f(1,2) = 1$ (for the sequence $(1, 2)$)
*   $f(1,3) = 2$ (for $(1, 2)$ and $(3)$)
*   $f(2,2) = 0$
*   $f(2,3) = 1$ (for $(3)$)
*   $f(3,3) = 1$ (for $(3)$)

Input #2

5 8
9 9 9 9 9

Output #2

Input #3

10 10
3 1 4 1 5 9 2 6 5 3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Knapsack for All Segments",
    "description": {
      "content": "Given are a sequence of $N$ integers $A_1$, $A_2$, $\\ldots$, $A_N$ and a positive integer $S$.   For a pair of integers $(L, R)$ such that $1\\leq L \\leq R \\leq N$, let us define $f(L, R)$ as follows: ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc159_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are a sequence of $N$ integers $A_1$, $A_2$, $\\ldots$, $A_N$ and a positive integer $S$.  \nFor a pair of integers $(L, R)$ such that $1\\leq L \\leq R \\leq N$, let us define $f(L, R)$ as follows: ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments