Knapsack for All Subsets

AtCoder

IDabc169_f

Time2000ms

Memory256MB

Difficulty—

Given are a sequence of $N$ positive integers $A_1$, $A_2$, $\ldots$, $A_N$ and another positive integer $S$. For a non-empty subset $T$ of the set ${1, 2, \ldots , N }$, let us define $f(T)$ as follows: * $f(T)$ is the number of different non-empty subsets ${x_1, x_2, \ldots , x_k }$ of $T$ such that $A_{x_1}+A_{x_2}+\cdots +A_{x_k} = S$. Find the sum of $f(T)$ over all $2^N-1$ subsets $T$ of ${1, 2, \ldots , N }$. Since the 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

6

For each $T$, the value of $f(T)$ is shown below. The sum of these values is $6$.

*   $f({1}) = 0$
*   $f({2}) = 0$
*   $f({3}) = 1$ (One subset ${3}$ satisfies the condition.)
*   $f({1, 2}) = 1$ (${1, 2}$)
*   $f({2, 3}) = 1$ (${3}$)
*   $f({1, 3}) = 1$ (${3}$)
*   $f({1, 2, 3}) = 2$ (${1, 2}, {3}$)

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 Subsets",
    "description": {
      "content": "Given are a sequence of $N$ positive integers $A_1$, $A_2$, $\\ldots$, $A_N$ and another positive integer $S$.   For a non-empty subset $T$ of the set ${1, 2, \\ldots , N }$, let us define $f(T)$ as fol",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc169_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are a sequence of $N$ positive integers $A_1$, $A_2$, $\\ldots$, $A_N$ and another positive integer $S$.  \nFor a non-empty subset $T$ of the set ${1, 2, \\ldots , N }$, let us define $f(T)$ as fol...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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