题目:http://www.lydsy.com/JudgeOnline/problem.php?id=1492
很经典的一道把决策看成点,然后动态维护凸壳的DP题目,可以用平衡树维护,据说还可以用神奇的CDQ分治水过去(看来真的应该去学学神级分治了啊)。
代码(SBT):
#include <cstdio>
#include <algorithm>
#include <cstring>
using namespace std ;
#define cal( p0 , p1 ) ( ( p0.y - p1.y ) / ( p0.x - p1.x ) )
#define MAXN 101000
#define X( x ) ( ( r[ x ] * f[ x ] ) / ( a[ x ] * r[ x ] + b[ x ] ) )
#define Y( x ) ( f[ x ] / ( a[ x ] * r[ x ] + b[ x ] ) )
#define fun( i , p ) ( a[ i ] * p.x + b[ i ] * p.y )
#define clear( x ) memset( x , 0 , sizeof( x ) )
#define update( t ) S( t ) = S( L( t ) ) + S( R( t ) ) + 1
#define L( t ) left[ t ]
#define R( t ) right[ t ]
#define K( t ) key[ t ]
#define S( t ) size[ t ]
const double inf = 100000000 ;
const double INF = inf * inf ;
struct itype {
double x , y ;
bool operator < ( const itype &a ) const {
return x < a.x ;
}
bool operator == ( const itype &a ) const {
return x == a.x ;
}
bool operator > ( const itype &a ) const {
return x > a.x ;
}
} key[ MAXN ] ;
int left[ MAXN ] , right[ MAXN ] , size[ MAXN ] , V = 0 , roof = 0 ;
double lk[ MAXN ] , rk[ MAXN ] ;
itype make( double x , double y ) {
itype u ;
u.x = x , u.y = y ;
return u ;
}
void Left( int &t ) {
int k = R( t ) ;
R( t ) = L( k ) ; update( t ) ;
L( k ) = t ; update( k ) ;
t = k ;
}
void Right( int &t ) {
int k = L( t ) ;
L( t ) = R( k ) ; update( t ) ;
R( k ) = t ; update( k ) ;
t = k ;
}
void maintain( int &t ) {
if ( S( L( L( t ) ) ) > S( R( t ) ) ) {
Right( t ) ;
maintain( R( t ) ) ; maintain( t ) ;
return ;
}
if ( S( R( L( t ) ) ) > S( R( t ) ) ) {
Left( L( t ) ) ; Right( t ) ;
maintain( L( t ) ) , maintain( R( t ) ) ; maintain( t ) ;
return ;
}
if ( S( R( R( t ) ) ) > S( L( t ) ) ) {
Left( t ) ;
maintain( L( t ) ) ; maintain( t ) ;
return ;
}
if ( S( L( R( t ) ) ) > S( L( t ) ) ) {
Right( R( t ) ) ; Left( t ) ;
maintain( L( t ) ) , maintain( R( t ) ) ; maintain( t ) ;
return ;
}
}
void Insert( itype k , int &t ) {
if ( ! t ) {
t = ++ V ;
S( t ) = 1 , K( t ) = k ;
return ;
}
Insert( k , k < K( t ) ? L( t ) : R( t ) ) ;
update( t ) ; maintain( t ) ;
}
void Delete( itype k , int &t ) {
if ( K( t ) == k ) {
if ( ! L( t ) ) {
t = R( t ) ; return ;
} else if ( ! R( t ) ) {
t = L( t ) ; return ;
} else {
Right( t ) ; Delete( k , R( t ) ) ;
}
} else Delete( k , k < K( t ) ? L( t ) : R( t ) ) ;
update( t ) ; maintain( t ) ;
}
itype Prefix( itype k , int t ) {
if ( ! t ) return make( - INF , - INF ) ;
if ( k > K( t ) ) return max( K( t ) , Prefix( k , R( t ) ) ) ;
return Prefix( k , L( t ) ) ;
}
itype Suffix( itype k , int t ) {
if ( ! t ) return make( INF , INF ) ;
if ( K( t ) > k ) return min( K( t ) , Suffix( k , L( t ) ) ) ;
return Suffix( k , R( t ) ) ;
}
int Find( itype k , int t ) {
if ( ! t ) return 0 ;
if ( k == K( t ) ) return t ;
return Find( k , k < K( t ) ? L( t ) : R( t ) ) ;
}
void Push( itype k ) {
int t = Find( k , roof ) ;
if ( t ) {
if ( K( t ).y >= k.y ) return ;
Delete( K( t ) , roof ) ;
}
itype pre = Prefix( k , roof ) , suff = Suffix( k , roof ) ;
if ( cal( pre , k ) > cal( pre , suff ) ) {
for ( ; pre.y != - INF ; ) {
itype ret = Prefix( pre , roof ) ;
if ( cal( ret , pre ) <= cal( pre , k ) ) {
Delete( pre , roof ) ;
pre = ret ;
} else break ;
}
for ( ; suff.y != - INF ; ) {
itype ret = Suffix( suff , roof ) ;
if ( cal( k , ret ) >= cal( k , suff ) ) {
Delete( suff , roof ) ;
suff = ret ;
} else break ;
}
Insert( k , roof ) ;
lk[ V ] = cal( pre , k ) , rk[ V ] = cal( k , suff ) ;
rk[ Find( pre , roof ) ] = cal( pre , k ) ;
lk[ Find( suff , roof ) ] = cal( k , suff ) ;
}
}
itype Select( double k , int t ) {
if ( K( t ).x == - inf ) return Select( k , R( t ) ) ;
if ( K( t ).x == inf ) return Select( k , L( t ) ) ;
if ( lk[ t ] >= k && k >= rk[ t ] ) return K( t ) ;
if ( rk[ t ] > k ) return Select( k , R( t ) ) ;
if ( lk[ t ] < k ) return Select( k , L( t ) ) ;
}
double a[ MAXN ] , b[ MAXN ] , r[ MAXN ] , f[ MAXN ] , money ;
int n ;
int main( ) {
clear( left ) , clear( right ) , clear( size ) ;
Insert( make( 0 , - INF ) , roof ) , Insert( make( inf , - INF ) , roof ) ;
scanf( "%d%lf" , &n , &money ) ;
for ( int i = 0 ; i ++ < n ; ) scanf( "%lf%lf%lf" , a + i , b + i , r + i ) ;
for ( int i = 0 ; i ++ < n ; ) {
f[ i ] = max( money , f[ i - 1 ] ) ;
if ( i > 1 ) {
itype p = Select( - a[ i ] / b[ i ] , roof ) ;
f[ i ] = max( f[ i ] , fun( i , p ) ) ;
}
Push( make( X( i ) , Y( i ) ) ) ;
}
printf( "%.3f\n" , f[ n ] ) ;
return 0 ;
}