Problem:
Score : 600 points

Problem Statement

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

Output

Print the sum of f(T) modulo 998244353.

Sample Input 1

3 4
2 2 4

Sample 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\})

Sample Input 2

5 8
9 9 9 9 9

Sample Output 2

0

Sample Input 3

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

Sample Output 3

3296

Solution in Python (3.8.2) (judge verdict: Time Limit Exceeded)


MOD = 998244353

def main():
    N, S = map(int, input().split())
    A = list(map(int, input().split()))
    print(solve(N, S, A))

def solve(N, S, A):
    dp = [[0]*(S+1) for _ in range(N+1)]
    dp[0][0] = 1
    # for line in dp:
    #     print(line)
    for n in range(1, N+1):
        an = A[n-1]
        # print(an)
        for s in range(S+1):
            part1 = dp[n-1][s]
            part2 = 0
            if s-an >= 0:
                part2 = dp[n-1][s-an]
            dp[n][s] = (2*part1 + part2) % MOD
        # for line in dp:
        #     print(line)
    return dp[N][S]

if __name__ == '__main__':
    main()