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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]pub mod pcl{pub mod compat{pub mod std{pub mod ops{pub use std::ops::{Bound,RangeBounds};}}pub mod num{pub use num::{One,Zero};}}pub mod macros{#[macro_export]macro_rules!rtl{(@ lhs($($ lhs:tt)*)@ rest+=$($ rest:tt)*)=>{rtl!(@ op+=@ lhs($($ lhs)*)@ rhs()@ rest $($ rest)*)};(@ lhs($($ lhs:tt)*)@ rest-=$($ rest:tt)*)=>{rtl!(@ op-=@ lhs($($ lhs)*)@ rhs()@ rest $($ rest)*)};(@ lhs($($ lhs:tt)*)@ rest*=$($ rest:tt)*)=>{rtl!(@ op*=@ lhs($($ lhs)*)@ rhs()@ rest $($ rest)*)};(@ lhs($($ lhs:tt)*)@ rest/=$($ rest:tt)*)=>{rtl!(@ op/=@ lhs($($ lhs)*)@ rhs()@ rest $($ rest)*)};(@ lhs($($ lhs:tt)*)@ rest%=$($ rest:tt)*)=>{rtl!(@ op%=@ lhs($($ lhs)*)@ rhs()@ rest $($ rest)*)};(@ lhs($($ lhs:tt)*)@ rest $ head:tt $($ rest:tt)*)=>{rtl!(@ lhs($($ lhs)*$ head)@ rest $($ rest)*);};(@ op $ op:tt @ lhs($($ lhs:tt)*)@ rhs($($ rhs:tt)*)@ rest)=>{let __rhs=$($ rhs)*;$($ lhs)*$ op __rhs;};(@ op $ op:tt @ lhs($($ lhs:tt)*)@ rhs($($ rhs:tt)*)@ rest $ head:tt $($ rest:tt)*)=>{rtl!(@ op $ op @ lhs($($ lhs)*)@ rhs($($ rhs)*$ head)@ rest $($ rest)*);};($($ rest:tt)*)=>{rtl!(@ lhs()@ rest $($ rest)*)};}}pub mod math{pub mod modint{#[macro_use]pub mod consts{use super::ModintInnerType;pub trait ModintConst{const MOD:ModintInnerType;}#[macro_export]macro_rules!define_modint_const{($(#[doc=$ doc:expr])*pub const $ name:ident=$ value:literal;)=>{$(#[doc=$ doc])*pub enum $ name{}impl $ crate::pcl::math::modint::consts::ModintConst for $ name{const MOD:$ crate::pcl::math::modint::ModintInnerType=$ value;}};}}use num::Num;use self::consts::ModintConst;use super::super::compat::num::{One,Zero};use std::cmp::Ordering;use std::fmt;use std::hash::{Hash,Hasher};use std::iter::{Product,Sum};use std::marker::PhantomData;use std::mem;use std::ops::{Add,AddAssign,Div,DivAssign,Mul,MulAssign,Neg,Rem,RemAssign,Sub,SubAssign,};pub type ModintInnerType=i64;define_modint_const!{#[doc="1e9+7 を表す定数。"]pub const MOD17=1_000_000_007;}pub type Modint17=Modint<MOD17>;pub struct Modint<C>{value:ModintInnerType,marker:PhantomData<C>,}impl<C>Modint<C>{pub unsafe fn new_unchecked(value:ModintInnerType)->Modint<C>{#[allow(unknown_lints,renamed_and_removed_lints,redundant_field_names)]Modint{value:value,marker:PhantomData,}}pub fn inner(self)->ModintInnerType{self.value}}impl<C:ModintConst>Modint<C>{pub fn new(mut value:ModintInnerType)->Modint<C>{let modulus=C::MOD;assert_ne!(modulus,0,"MOD is 0");if value<0{let m=(-value)/modulus;value+=(m+1)*modulus;}unsafe{Modint::new_unchecked(value%modulus)}}pub fn inv(self)->Modint<C>{let mut modulus=C::MOD;let mut a=self.value;let mut u=1;let mut v=0;while modulus>0{let t=a/modulus;a-=t*modulus;u-=t*v;mem::swap(&mut a,&mut modulus);mem::swap(&mut u,&mut v);}Modint::new(u)}}impl<C:ModintConst>PartialEq for Modint<C>{fn eq(&self,other:&Self)->bool{self.inner()==other.inner()}}impl<C:ModintConst>PartialOrd for Modint<C>{fn partial_cmp(&self,other:&Self)->Option<Ordering>{self.inner().partial_cmp(&other.inner())}}impl<C:ModintConst>Eq for Modint<C>{}impl<C:ModintConst>Ord for Modint<C>{fn cmp(&self,other:&Self)->Ordering{self.inner().cmp(&other.inner())}}impl<C:ModintConst>Hash for Modint<C>{fn hash<H:Hasher>(&self,state:&mut H){self.inner().hash(state);}}impl<C>fmt::Debug for Modint<C>{fn fmt<'a>(&self,f:&mut fmt::Formatter<'a>)->fmt::Result{write!(f,"{}",self.inner())}}impl<C>Clone for Modint<C>{fn clone(&self)->Self{*self}}impl<C>Copy for Modint<C>{}impl<C:ModintConst>AddAssign for Modint<C>{fn add_assign(&mut self,rhs:Modint<C>){let modulus=C::MOD;self.value+=rhs.value;if self.value>=modulus{self.value-=modulus;}}}impl<C:ModintConst>SubAssign for Modint<C>{fn sub_assign(&mut self,rhs:Modint<C>){let modulus=C::MOD;self.value-=rhs.value;if self.value<0{self.value+=modulus;}}}impl<C:ModintConst>MulAssign for Modint<C>{fn mul_assign(&mut self,rhs:Modint<C>){let modulus=C::MOD;self.value*=rhs.value;self.value%=modulus;}}impl<C:ModintConst>DivAssign for Modint<C>{fn div_assign(&mut self,rhs:Modint<C>){if rhs.value==0{panic!("attempted to divide by zero");}*self*=rhs.inv();}}impl<C:ModintConst>RemAssign for Modint<C>{fn rem_assign(&mut self,rhs:Modint<C>){if rhs.value==0{panic!("attempted to divide by zero.")}self.value%=rhs.value;}}impl<C:ModintConst>Neg for Modint<C>{type Output=Self;fn neg(self)->Self{Self::zero()-self}}macro_rules!impl_arith_by_assign{(impl $ traitname:ident::$ fnname:ident{use $ op:tt;})=>{impl<C:ModintConst>$ traitname for Modint<C>{type Output=Modint<C>;fn $ fnname(mut self,rhs:Modint<C>)->Modint<C>{self $ op rhs;self}}};}impl_arith_by_assign!(impl Add::add{use+=;});impl_arith_by_assign!(impl Sub::sub{use-=;});impl_arith_by_assign!(impl Mul::mul{use*=;});impl_arith_by_assign!(impl Div::div{use/=;});impl_arith_by_assign!(impl Rem::rem{use%=;});impl<C:ModintConst>One for Modint<C>{fn one()->Modint<C>{let modulus=C::MOD;assert_ne!(modulus,1,"one()is called for Modint with MOD=1");unsafe{Modint::new_unchecked(1)}}}impl<C:ModintConst>Sum for Modint<C>{fn sum<I:Iterator<Item=Modint<C>>>(iter:I)->Modint<C>{iter.fold(Modint::zero(),Add::add)}}impl<C:ModintConst>Product for Modint<C>{fn product<I:Iterator<Item=Modint<C>>>(iter:I)->Self{iter.fold(Modint::one(),Mul::mul)}}impl<C:ModintConst>Zero for Modint<C>{fn zero()->Modint<C>{unsafe{Modint::new_unchecked(0)}}fn is_zero(&self)->bool{self.inner()==0}}impl<C>fmt::Display for Modint<C>{fn fmt(&self,b:&mut fmt::Formatter)->fmt::Result{write!(b,"{}",self.inner())}}impl<C:ModintConst>Num for Modint<C>{type FromStrRadixErr=<ModintInnerType as Num>::FromStrRadixErr;fn from_str_radix(src:&str,radix:u32)->Result<Self,Self::FromStrRadixErr>{ModintInnerType::from_str_radix(src,radix).map(Modint::new)}}}pub mod sum{use super::super::compat::std::ops::RangeBounds;use super::super::traits::Group;use super::super::utils::range;pub struct CumSum<T>{psum:Vec<T>,}#[allow(unknown_lints,renamed_and_removed_lints,len_without_is_empty)]impl<T:Group+Copy>CumSum<T>{pub fn from_array<A:AsRef<[T]>>(array:A)->CumSum<T>{let array=array.as_ref();let mut psum=vec![T::id();array.len()+1];for i in 0..array.len(){let i=i+1;psum[i]=T::op(psum[i-1],array[i-1]);}#[allow(unknown_lints,renamed_and_removed_lints,redundant_field_names)]CumSum{psum:psum}}pub fn sum<R:RangeBounds<usize>>(&self,range:R)->T{let orig_len=self.psum.len()-1;let start=range::range_start(&range,0);let end=range::range_end(&range,orig_len);if end<=start{return T::id();}T::op(self.psum[end],T::inv(self.psum[start]))}pub fn len(&self)->usize{self.psum.len()-1}}pub struct CumSum2D<T>{psum:Vec<Vec<T>>,}impl<T:Group+Copy>CumSum2D<T>{pub fn from_matrix<M,A>(matrix:M)->CumSum2D<T>where M:AsRef<[A]>,A:AsRef<[T]>,{let array=matrix.as_ref();let height=array.len();if height==0{return CumSum2D{psum:vec![vec![T::id()]],};}let width=array[0].as_ref().len();let mut psum=vec![vec![T::id();width+1];height+1];for i in 0..height{let i=i+1;for j in 0..width{let j=j+1;assert_eq!(array[i-1].as_ref().len(),width,"the array's length is differ line by line");psum[i][j]=T::op(T::op(T::op(psum[i-1][j],psum[i][j-1]),T::inv(psum[i-1][j-1]),),array[i-1].as_ref()[j-1],)}}#[allow(unknown_lints,renamed_and_removed_lints,redundant_field_names)]CumSum2D{psum:psum}}pub fn sum<RY,RX>(&self,yrange:RY,xrange:RX)->T where RY:RangeBounds<usize>,RX:RangeBounds<usize>,{let orig_height=self.psum.len()-1;let orig_width=unsafe{self.psum.get_unchecked(0).len()}-1;let ystart=range::range_start(&yrange,0);let yend=range::range_end(&yrange,orig_height);let xstart=range::range_start(&xrange,0);let xend=range::range_end(&xrange,orig_width);if yend<=ystart||xend<=xstart{return T::id();}T::op(T::op(T::op(self.psum[yend][xend],self.psum[ystart][xstart]),T::inv(self.psum[ystart][xend]),),T::inv(self.psum[yend][xstart]),)}pub fn size(&self)->(usize,usize){(self.psum.len()-1,self.psum[0].len()-1)}}}pub use self::modint::{Modint,Modint17};pub use self::sum::{CumSum,CumSum2D};}pub mod prelude{pub use std::cmp::*;pub use std::collections::*;pub use std::f64::consts::*;pub use std::{f32,f64,i16,i32,i64,i8,isize,u16,u32,u64,u8,usize};mod prelude_ext{pub use itertools::Itertools;pub use maplit::{btreemap,btreeset,convert_args,hashmap,hashset};pub use proconio::fastout;pub use proconio::input;pub use proconio::marker::*;}pub use self::prelude_ext::*;}pub mod structure{pub mod disjoint_sets{use std::mem::swap;pub struct DisjointSets{par:Vec<i64>,size:usize,}impl DisjointSets{pub fn new(n:usize)->DisjointSets{DisjointSets{par:vec![-1;n],size:n,}}pub fn merge(&mut self,mut x:usize,mut y:usize)->bool{let len=self.par.len();assert!(x<len,"index out of range:x is{}but len is{}",x,len);assert!(y<len,"index out of range:y is{}but len is{}",y,len);x=self.root(x);y=self.root(y);if x==y{return false;}if self.par[x]<self.par[y]{swap(&mut x,&mut y);}debug_assert!(self.par[x]<0&&self.par[y]<0,"critical error:parent has invalid value for rank");self.par[x]+=self.par[y];self.par[y]=x as i64;self.size-=1;true}pub fn in_same(&mut self,x:usize,y:usize)->bool{self.root(x)==self.root(y)}pub fn root(&mut self,x:usize)->usize{let parx=self.par[x];if parx<0{x}else{let root=self.root(parx as usize);self.par[x]=root as i64;root}}pub fn size_of(&mut self,mut x:usize)->usize{x=self.root(x);debug_assert!(self.par[x]<0,"critical error:parent has invalid value for rank");-self.par[x]as usize}pub fn size(&self)->usize{self.size}}}pub mod segment_tree{use super::super::compat::std::ops::RangeBounds;use super::super::traits::math::Monoid;use super::super::utils::range;use std::fmt;pub struct SegmentTree<T>{data:Vec<T>,lenexp2:usize,len:usize,}impl<T:fmt::Debug>fmt::Debug for SegmentTree<T>{fn fmt<'a>(&self,f:&mut fmt::Formatter<'a>)->fmt::Result{f.debug_struct("SegmentTree").field("data",&self.data).finish()}}impl<T>SegmentTree<T>where T:Monoid+Copy,{pub fn from_array<A:AsRef<[T]>>(arr:A)->SegmentTree<T>{let arr=arr.as_ref();let len=arr.len();let lenexp2=calc_lenexp2(len);let data={let mut v=vec![T::id();lenexp2*2];v[lenexp2..(lenexp2+len)].copy_from_slice(arr);v};#[allow(unknown_lints,renamed_and_removed_lints,redundant_field_names)]SegmentTree{data:data,lenexp2:lenexp2,len:len,}}pub fn update(&mut self,mut idx:usize,value:T){assert!(idx<=self.len);idx+=self.lenexp2;self.data[idx]=value;#[allow(unknown_lints,renamed_and_removed_lints,while_immutable_condition)]while{idx>>=1;idx!=0}{self.data[idx]=T::op(self.data[idx*2],self.data[idx*2+1]);}}pub fn query<R:RangeBounds<usize>>(&self,range:R)->T{let mut start=range::range_start(&range,0);let mut end=range::range_end(&range,self.len);start+=self.lenexp2;end+=self.lenexp2;let mut res1=T::id();let mut res2=T::id();while start<end{if start&1!=0{res1=T::op(res1,self.data[start]);start+=1;}if end&1!=0{end-=1;res2=T::op(self.data[end],res2);}start>>=1;end>>=1;}T::op(res1,res2)}}fn calc_lenexp2(mut len:usize)->usize{len-=1;len|=len>>1;len|=len>>2;len|=len>>4;len|=len>>8;len|=len>>16;len|=len>>32;len+1}}pub use self::disjoint_sets::DisjointSets;pub use self::segment_tree::SegmentTree;}pub mod traits{pub mod math{pub mod group{use super::monoid::Monoid;pub trait Group:Monoid{fn inv(x:Self)->Self;}use super::super::super::compat::num::Zero;use std::ops::{Add,Neg};#[derive(Debug)]pub struct Additive<T>(pub T);impl<T:Clone>Clone for Additive<T>{fn clone(&self)->Self{Additive(self.0.clone())}}impl<T:Copy>Copy for Additive<T>{}impl<T>Monoid for Additive<T>where T:Zero+Add<Output=T>,{fn op(x:Self,y:Self)->Self{Additive(x.0+y.0)}fn id()->Self{Additive(T::zero())}}impl<T>Group for Additive<T>where T:Zero+Add<Output=T>+Neg<Output=T>,{fn inv(x:Self)->Self{Additive(-x.0)}}}pub mod monoid{pub trait Monoid{fn op(x:Self,y:Self)->Self;fn id()->Self;}use super::super::utils::num::{MaxValue,MinValue};use std::cmp::Ord;use std::cmp::{max,min};use std::fmt;pub struct Min<T>(pub T);impl<T:fmt::Debug>fmt::Debug for Min<T>{fn fmt<'a>(&self,f:&mut fmt::Formatter<'a>)->fmt::Result{f.debug_tuple("Min").field(&self.0).finish()}}impl<T:Clone>Clone for Min<T>{fn clone(&self)->Self{Min(self.0.clone())}}impl<T:Copy>Copy for Min<T>{}impl<T:Ord+MaxValue>Monoid for Min<T>{fn op(x:Self,y:Self)->Self{Min(min(x.0,y.0))}fn id()->Self{Min(T::max_value())}}pub struct Max<T>(pub T);impl<T:fmt::Debug>fmt::Debug for Max<T>{fn fmt<'a>(&self,f:&mut fmt::Formatter<'a>)->fmt::Result{f.debug_tuple("Max").field(&self.0).finish()}}impl<T:Clone>Clone for Max<T>{fn clone(&self)->Self{Max(self.0.clone())}}impl<T:Copy>Copy for Max<T>{}impl<T:Ord+MinValue>Monoid for Max<T>{fn op(x:Self,y:Self)->Self{Max(max(x.0,y.0))}fn id()->Self{Max(T::min_value())}}}pub use self::group::Group;pub use self::monoid::Monoid;}pub mod utils{pub mod num{pub trait MaxValue{fn max_value()->Self;}pub trait MinValue{fn min_value()->Self;}macro_rules!impl_minmax_value_for_primitives{($($ ty:tt)*)=>{$(impl MaxValue for $ ty{fn max_value()->$ ty{::std::$ ty::MAX}}impl MinValue for $ ty{fn min_value()->$ ty{::std::$ ty::MIN}})*};}impl_minmax_value_for_primitives!{u8 u16 u32 u64 usize i8 i16 i32 i64 isize}}}pub use self::math::*;}pub mod utils{pub mod range{use std::cmp;use super::super::compat::std::ops::{Bound,RangeBounds};pub fn range_start<R:RangeBounds<usize>>(range:&R,min:usize)->usize{let start=match range.start_bound(){Bound::Included(&x)=>x,Bound::Excluded(&x)=>x+1,Bound::Unbounded=>0,};cmp::max(start,min)}pub fn range_end<R:RangeBounds<usize>>(range:&R,max:usize)->usize{let end=match range.end_bound(){Bound::Included(&x)=>x+1,Bound::Excluded(&x)=>x,Bound::Unbounded=>max,};cmp::min(end,max)}}}}
use self::pcl::compat::num::{One, Zero};
use self::pcl::math::modint::Modint;
use self::pcl::prelude::*;
define_modint_const! { pub const MOD = 998_244_353 ; }
type M = Modint<MOD>;
fn main() {
    input! { n : usize , s : usize , a : [ usize ; n ] , }
    let mut dp = vec![vec![M::zero(); s + 1]; n + 1];
    dp[0][0] = M::one();
    for i in 0..n {
        for j in 0..=s {
            rtl!(dp[i + 1][j] += dp[i][j]);
            rtl!(dp[i + 1][j] += dp[i][j]);
            if j >= a[i] {
                rtl!(dp[i + 1][j] += dp[i][j - a[i]]);
            }
        }
    }
    println!("{}", dp[n][s]);
}