// codesnip-guard: SqrtDecomposition
#[cfg_attr(any(),rustfmt::skip)]pub use self::sqrt_decomposition::{RangeUpdateRangeFoldSqrtDecomposition,SqrtDecomposition,SqrtDecompositionBuckets};#[cfg_attr(any(),rustfmt::skip)]mod sqrt_decomposition{use super::{Magma,Monoid,RangeBoundsExt,Unital};use std::{marker::PhantomData,ops::RangeBounds};pub trait SqrtDecomposition:Sized{type M:Monoid;type B;fn bucket(bsize:usize)->Self::B;fn update_bucket(bucket:&mut Self::B,x:&<Self::M as Magma>::T);fn update_cell(bucket:&mut Self::B,cell:&mut<Self::M as Magma>::T,x:&<Self::M as Magma>::T);fn fold_bucket(bucket:&Self::B)-><Self::M as Magma>::T;fn fold_cell(bucket:&Self::B,cell:&<Self::M as Magma>::T)-><Self::M as Magma>::T;fn sqrt_decomposition(n:usize,bucket_size:Option<usize>)->SqrtDecompositionBuckets<Self>{let bucket_size=bucket_size.unwrap_or((n as f64).sqrt().ceil()as usize).max(1);let mut buckets=vec![];for l in(0..n).step_by(bucket_size){let bsize=(l+bucket_size).min(n)-l;let bucket=Self::bucket(bsize);buckets.push(Bucket{cells:vec![Self::M::unit();bsize],bucket});}SqrtDecompositionBuckets{n,bucket_size,buckets,_marker:PhantomData}}}struct Bucket<T,B>{cells:Vec<T>,bucket:B}pub struct SqrtDecompositionBuckets<S>where S:SqrtDecomposition{n:usize,bucket_size:usize,buckets:Vec<Bucket<<S::M as Magma>::T,S::B>>,_marker:PhantomData<fn()->S>}impl<S>SqrtDecompositionBuckets<S>where S:SqrtDecomposition{pub fn update_cell(&mut self,i:usize,x:<S::M as Magma>::T){let Bucket{cells,bucket}=&mut self.buckets[i/self.bucket_size];let j=i%self.bucket_size;S::update_cell(bucket,&mut cells[j],&x);}pub fn update<R>(&mut self,range:R,x:<S::M as Magma>::T)where R:RangeBounds<usize>{let range=range.to_range_bounded(0,self.n).expect("invalid range");for(i,Bucket{cells,bucket})in self.buckets.iter_mut().enumerate(){let s=i*self.bucket_size;let t=s+cells.len();if t<=range.start||range.end<=s{}else if range.start<=s&&t<=range.end{S::update_bucket(bucket,&x);}else{for cell in&mut cells[range.start.max(s)-s..range.end.min(t)-s]{S::update_cell(bucket,cell,&x);}}}}pub fn get(&self,i:usize)-><S::M as Magma>::T{let Bucket{cells,bucket}=&self.buckets[i/self.bucket_size];let j=i%self.bucket_size;S::fold_cell(bucket,&cells[j])}pub fn fold<R>(&self,range:R)-><S::M as Magma>::T where R:RangeBounds<usize>{let range=range.to_range_bounded(0,self.n).expect("invalid range");let mut res=S::M::unit();for(i,Bucket{cells,bucket})in self.buckets.iter().enumerate(){let s=i*self.bucket_size;let t=s+cells.len();if t<=range.start||range.end<=s{}else if range.start<=s&&t<=range.end{<S::M as Magma>::operate_assign(&mut res,&S::fold_bucket(bucket));}else{for cell in&cells[range.start.max(s)-s..range.end.min(t)-s]{<S::M as Magma>::operate_assign(&mut res,&S::fold_cell(bucket,cell));}}}res}}pub struct RangeUpdateRangeFoldSqrtDecomposition<M>where M:Monoid{_marker:PhantomData<fn()->M>}impl<M>SqrtDecomposition for RangeUpdateRangeFoldSqrtDecomposition<M>where M:Monoid{type M=M;type B=(M::T,M::T,usize);fn bucket(bsize:usize)->Self::B{(M::unit(),M::unit(),bsize)}fn update_bucket(bucket:&mut Self::B,x:&<Self::M as Magma>::T){M::operate_assign(&mut bucket.1,x);}fn update_cell(bucket:&mut Self::B,cell:&mut<Self::M as Magma>::T,x:&<Self::M as Magma>::T){M::operate_assign(&mut bucket.0,x);M::operate_assign(cell,x);}fn fold_bucket(bucket:&Self::B)-><Self::M as Magma>::T{M::operate(&bucket.0,&M::pow(bucket.1 .clone(),bucket.2))}fn fold_cell(bucket:&Self::B,cell:&<Self::M as Magma>::T)-><Self::M as Magma>::T{M::operate(cell,&bucket.1)}}}
// codesnip-guard: algebra
#[cfg_attr(any(),rustfmt::skip)]pub use self::magma::*;#[cfg_attr(any(),rustfmt::skip)]mod magma{#![doc=" algebraic traits"]#[doc=" binary operaion: $T \\circ T \\to T$"]pub trait Magma{#[doc=" type of operands: $T$"]type T:Clone;#[doc=" binary operaion: $\\circ$"]fn operate(x:&Self::T,y:&Self::T)->Self::T;fn reverse_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(y,x)}fn operate_assign(x:&mut Self::T,y:&Self::T){*x=Self::operate(x,y);}}#[doc=" $\\forall a,\\forall b,\\forall c \\in T, (a \\circ b) \\circ c = a \\circ (b \\circ c)$"]pub trait Associative:Magma{}#[doc=" associative binary operation"]pub trait SemiGroup:Magma+Associative{}impl<S>SemiGroup for S where S:Magma+Associative{}#[doc=" $\\exists e \\in T, \\forall a \\in T, e \\circ a = a \\circ e = a$"]pub trait Unital:Magma{#[doc=" identity element: $e$"]fn unit()->Self::T;fn is_unit(x:&Self::T)->bool where Self::T:PartialEq{x==&Self::unit()}fn set_unit(x:&mut Self::T){*x=Self::unit();}}pub trait ExpBits{type Iter:Iterator<Item=bool>;fn bits(self)->Self::Iter;}pub trait SignedExpBits{type T:ExpBits;fn neg_and_bits(self)->(bool,Self::T);}pub struct Bits<T>{n:T}macro_rules!impl_exp_bits_for_uint{($($t:ty)*)=>{$(impl Iterator for Bits<$t>{type Item=bool;fn next(&mut self)->Option<bool>{if self.n==0{None}else{let bit=(self.n&1)==1;self.n>>=1;Some(bit)}}}impl ExpBits for$t{type Iter=Bits<$t>;fn bits(self)->Self::Iter{Bits{n:self}}}impl SignedExpBits for$t{type T=$t;fn neg_and_bits(self)->(bool,Self::T){(false,self)}})*};}impl_exp_bits_for_uint!(u8 u16 u32 u64 u128 usize);macro_rules!impl_signed_exp_bits_for_sint{($($s:ty,$u:ty;)*)=>{$(impl SignedExpBits for$s{type T=$u;fn neg_and_bits(self)->(bool,Self::T){(self<0,self.unsigned_abs())}})*};}impl_signed_exp_bits_for_sint!(i8,u8;i16,u16;i32,u32;i64,u64;i128,u128;isize,usize;);#[doc=" associative binary operation and an identity element"]pub trait Monoid:SemiGroup+Unital{#[doc=" binary exponentiation: $x^n = x\\circ\\ddots\\circ x$"]fn pow<E>(mut x:Self::T,exp:E)->Self::T where E:ExpBits{let mut bits=exp.bits();let Some(first)=bits.next()else{return Self::unit();};let mut res=if first{x.clone()}else{Self::unit()};for bit in bits{x=Self::operate(&x,&x);if bit{res=Self::operate(&res,&x);}}res}fn fold<I>(iter:I)->Self::T where I:IntoIterator<Item=Self::T>{let mut iter=iter.into_iter();if let Some(item)=iter.next(){iter.fold(item,|acc,x|Self::operate(&acc,&x))}else{Self::unit()}}}impl<M>Monoid for M where M:SemiGroup+Unital{}#[doc=" $\\exists e \\in T, \\forall a \\in T, \\exists b,c \\in T, b \\circ a = a \\circ c = e$"]pub trait Invertible:Magma+Unital{#[doc=" $a$ where $a \\circ x = e$"]fn inverse(x:&Self::T)->Self::T;fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(x,&Self::inverse(y))}fn rinv_operate_assign(x:&mut Self::T,y:&Self::T){*x=Self::rinv_operate(x,y);}}#[doc=" associative binary operation and an identity element and inverse elements"]pub trait Group:Monoid+Invertible{fn signed_pow<E>(x:Self::T,exp:E)->Self::T where E:SignedExpBits{let(neg,exp)=E::neg_and_bits(exp);let res=Self::pow(x,exp);if neg{Self::inverse(&res)}else{res}}}impl<G>Group for G where G:Monoid+Invertible{}#[doc=" $\\forall a,\\forall b \\in T, a \\circ b = b \\circ a$"]pub trait Commutative:Magma{}#[doc=" commutative monoid"]pub trait AbelianMonoid:Monoid+Commutative{}impl<M>AbelianMonoid for M where M:Monoid+Commutative{}#[doc=" commutative group"]pub trait AbelianGroup:Group+Commutative{}impl<G>AbelianGroup for G where G:Group+Commutative{}#[doc=" $\\forall a \\in T, a \\circ a = a$"]pub trait Idempotent:Magma{}#[doc=" idempotent monoid"]pub trait IdempotentMonoid:Monoid+Idempotent{}impl<M>IdempotentMonoid for M where M:Monoid+Idempotent{}#[macro_export]macro_rules!monoid_fold{($m:ty)=>{<$m as Unital>::unit()};($m:ty,)=>{<$m as Unital>::unit()};($m:ty,$f:expr)=>{$f};($m:ty,$f:expr,$($ff:expr),*)=>{<$m as Magma>::operate(&($f),&monoid_fold!($m,$($ff),*))};}#[macro_export]macro_rules!define_monoid{($Name:ident,$t:ty,|$x:ident,$y:ident|$op:expr,$unit:expr)=>{struct$Name;impl Magma for$Name{type T=$t;fn operate($x:&Self::T,$y:&Self::T)->Self::T{$op}}impl Unital for$Name{fn unit()->Self::T{$unit}}impl Associative for$Name{}};}}
// codesnip-guard: bounded
#[cfg_attr(any(),rustfmt::skip)]pub use self::bounded::Bounded;#[cfg_attr(any(),rustfmt::skip)]mod bounded{#[doc=" Trait for max/min bounds"]pub trait Bounded:Sized+PartialOrd{fn maximum()->Self;fn minimum()->Self;fn is_maximum(&self)->bool{self==&Self::maximum()}fn is_minimum(&self)->bool{self==&Self::minimum()}fn set_maximum(&mut self){*self=Self::maximum()}fn set_minimum(&mut self){*self=Self::minimum()}}macro_rules!impl_bounded_num{($($t:ident)*)=>{$(impl Bounded for$t{fn maximum()->Self{$t::MAX}fn minimum()->Self{$t::MIN}})*};}impl_bounded_num!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize f32 f64);macro_rules!impl_bounded_tuple{(@impl$($T:ident)*)=>{impl<$($T:Bounded),*>Bounded for($($T,)*){fn maximum()->Self{($(<$T as Bounded>::maximum(),)*)}fn minimum()->Self{($(<$T as Bounded>::minimum(),)*)}}};(@inner$($T:ident)*,)=>{impl_bounded_tuple!(@impl$($T)*);};(@inner$($T:ident)*,$U:ident$($Rest:ident)*)=>{impl_bounded_tuple!(@impl$($T)*);impl_bounded_tuple!(@inner$($T)*$U,$($Rest)*);};($T:ident$($Rest:ident)*)=>{impl_bounded_tuple!(@inner$T,$($Rest)*);};}impl_bounded_tuple!(A B C D E F G H I J);impl Bounded for(){fn maximum()->Self{}fn minimum()->Self{}}impl Bounded for bool{fn maximum()->Self{true}fn minimum()->Self{false}}impl<T>Bounded for Option<T>where T:Bounded{fn maximum()->Self{Some(<T as Bounded>::maximum())}fn minimum()->Self{None}}impl<T>Bounded for std::cmp::Reverse<T>where T:Bounded{fn maximum()->Self{std::cmp::Reverse(<T as Bounded>::minimum())}fn minimum()->Self{std::cmp::Reverse(<T as Bounded>::maximum())}}}
// codesnip-guard: discrete_steps
#[cfg_attr(any(),rustfmt::skip)]pub use self::discrete_steps::{DiscreteSteps,RangeBoundsExt};#[cfg_attr(any(),rustfmt::skip)]mod discrete_steps{use super::Bounded;use std::{convert::TryFrom,ops::{Bound,Range,RangeBounds,RangeInclusive}};pub trait DiscreteSteps<Delta>:Clone{fn delta()->Delta;fn steps_between(start:&Self,end:&Self)->Option<Delta>;fn forward_checked(start:Self,delta:Delta)->Option<Self>;fn backward_checked(start:Self,delta:Delta)->Option<Self>;fn forward(start:Self,delta:Delta)->Self{Self::forward_checked(start,delta).expect("overflow in `DiscreteSteps::forward`")}fn backward(start:Self,delta:Delta)->Self{Self::backward_checked(start,delta).expect("overflow in `DiscreteSteps::backward`")}fn forward_delta_checked(start:Self)->Option<Self>{Self::forward_checked(start,Self::delta())}fn backward_delta_checked(start:Self)->Option<Self>{Self::backward_checked(start,Self::delta())}fn forward_delta(start:Self)->Self{Self::forward(start,Self::delta())}fn backward_delta(start:Self)->Self{Self::backward(start,Self::delta())}}macro_rules!impl_discrete_steps_integer{(@common$u_source:ident)=>{fn delta()->$u_source{1}fn forward(start:Self,delta:$u_source)->Self{assert!(Self::forward_checked(start,delta).is_some(),"attempt to add with overflow");start.wrapping_add(delta as Self)}fn backward(start:Self,delta:$u_source)->Self{assert!(Self::backward_checked(start,delta).is_some(),"attempt to subtract with overflow");start.wrapping_sub(delta as Self)}};($u_source:ident$i_source:ident;$($u_narrower:ident$i_narrower:ident),*;$($u_wider:ident$i_wider:ident),*)=>{$(impl DiscreteSteps<$u_source>for$u_narrower{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(*end-*start).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{Self::try_from(delta).ok().and_then(|delta|start.checked_add(delta))}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{Self::try_from(delta).ok().and_then(|delta|start.checked_sub(delta))}}impl DiscreteSteps<$u_source>for$i_narrower{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(end.abs_diff(*start)).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{$u_narrower::try_from(delta).ok().and_then(|delta|{let wrapped=start.wrapping_add(delta as Self);if wrapped>=start{Some(wrapped)}else{None}})}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{$u_narrower::try_from(delta).ok().and_then(|delta|{let wrapped=start.wrapping_sub(delta as Self);if wrapped<=start{Some(wrapped)}else{None}})}})*$(impl DiscreteSteps<$u_source>for$u_wider{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(*end-*start).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_add(delta as Self)}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_sub(delta as Self)}}impl DiscreteSteps<$u_source>for$i_wider{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{end.checked_sub(*start).and_then(|result|$u_source::try_from(result).ok())}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_add(delta as Self)}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_sub(delta as Self)}})*};}impl_discrete_steps_integer!(u16 i16;u8 i8,u16 i16,usize isize;u32 i32,u64 i64,u128 i128);impl_discrete_steps_integer!(u32 i32;u8 i8,u16 i16,u32 i32,usize isize;u64 i64,u128 i128);impl_discrete_steps_integer!(u64 i64;u8 i8,u16 i16,u32 i32,u64 i64,usize isize;u128 i128);impl_discrete_steps_integer!(u128 i128;u8 i8,u16 i16,u32 i32,u64 i64,u128 i128,usize isize;);impl_discrete_steps_integer!(usize isize;u8 i8,u16 i16,u32 i32,u64 i64,usize isize;u128 i128);pub trait RangeBoundsExt<T>{fn start_bound_included_checked(&self)->Option<T>;fn start_bound_excluded_checked(&self)->Option<T>;fn end_bound_included_checked(&self)->Option<T>;fn end_bound_excluded_checked(&self)->Option<T>;fn start_bound_included(&self)->T;fn start_bound_excluded(&self)->T;fn end_bound_included(&self)->T;fn end_bound_excluded(&self)->T;fn start_bound_included_bounded(&self,lb:T)->Option<T>where T:Ord;fn start_bound_excluded_bounded(&self,lb:T)->Option<T>where T:Ord;fn end_bound_included_bounded(&self,ub:T)->Option<T>where T:Ord;fn end_bound_excluded_bounded(&self,ub:T)->Option<T>where T:Ord;fn to_range_checked(&self)->Option<Range<T>>{match(self.start_bound_included_checked(),self.end_bound_excluded_checked()){(Some(start),Some(end))=>Some(start..end),_=>None,}}fn to_range(&self)->Range<T>{self.start_bound_included()..self.end_bound_excluded()}fn to_range_bounded(&self,min:T,max:T)->Option<Range<T>>where T:Ord{Some(self.start_bound_included_bounded(min)?..self.end_bound_excluded_bounded(max)?)}fn to_range_inclusive_checked(&self)->Option<RangeInclusive<T>>{match(self.start_bound_included_checked(),self.end_bound_included_checked()){(Some(start),Some(end))=>Some(start..=end),_=>None,}}fn to_range_inclusive(&self)->RangeInclusive<T>{self.start_bound_included()..=self.end_bound_included()}fn to_range_inclusive_bounded(&self,min:T,max:T)->Option<RangeInclusive<T>>where T:Ord{Some(self.start_bound_included_bounded(min)?..=self.end_bound_included_bounded(max)?)}}macro_rules!impl_range_bounds_ext{($($source:ident=>$($target:ident)+);*$(;)?)=>{$($(impl<R>RangeBoundsExt<$target>for R where R:RangeBounds<$target>,{fn start_bound_included_checked(&self)->Option<$target>{match self.start_bound(){Bound::Included(x)=>Some(*x),Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x),Bound::Unbounded=>Some(Bounded::minimum()),}}fn start_bound_excluded_checked(&self)->Option<$target>{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x),Bound::Excluded(x)=>Some(*x),Bound::Unbounded=>None,}}fn end_bound_included_checked(&self)->Option<$target>{match self.end_bound(){Bound::Included(x)=>Some(*x),Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x),Bound::Unbounded=>Some(Bounded::maximum()),}}fn end_bound_excluded_checked(&self)->Option<$target>{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x),Bound::Excluded(x)=>Some(*x),Bound::Unbounded=>None,}}fn start_bound_included(&self)->$target{match self.start_bound(){Bound::Included(x)=>*x,Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta(*x),Bound::Unbounded=>Bounded::minimum(),}}fn start_bound_excluded(&self)->$target{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta(*x),Bound::Excluded(x)=>*x,Bound::Unbounded=>DiscreteSteps::<$source>::backward_delta(Bounded::minimum()),}}fn end_bound_included(&self)->$target{match self.end_bound(){Bound::Included(x)=>*x,Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta(*x),Bound::Unbounded=>Bounded::maximum(),}}fn end_bound_excluded(&self)->$target{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta(*x),Bound::Excluded(x)=>*x,Bound::Unbounded=>DiscreteSteps::<$source>::forward_delta(Bounded::maximum()),}}fn start_bound_included_bounded(&self,lb:$target)->Option<$target>where$target:Ord{match self.start_bound(){Bound::Included(x)=>Some(*x).filter(|&x|lb<=x),Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x).filter(|&x|lb<=x),Bound::Unbounded=>Some(lb),}}fn start_bound_excluded_bounded(&self,lb:$target)->Option<$target>where$target:Ord{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x).filter(|&x|lb<=x),Bound::Excluded(x)=>Some(*x).filter(|&x|lb<=x),Bound::Unbounded=>Some(lb),}}fn end_bound_included_bounded(&self,ub:$target)->Option<$target>where$target:Ord{match self.end_bound(){Bound::Included(x)=>Some(*x).filter(|&x|x<=ub),Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x).filter(|&x|x<=ub),Bound::Unbounded=>Some(ub),}}fn end_bound_excluded_bounded(&self,ub:$target)->Option<$target>where$target:Ord{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x).filter(|&x|x<=ub),Bound::Excluded(x)=>Some(*x).filter(|&x|x<=ub),Bound::Unbounded=>Some(ub),}}})+)*};}impl_range_bounds_ext!(u16=>u8 i8 u16 i16;u32=>u32 i32;u64=>u64 i64;u128=>u128 i128;usize=>isize usize;);}