BZOJ-1063: [Noi2008]道路设计(树DP)

题目:http://www.lydsy.com/JudgeOnline/problem.php?id=1063

首先想了好久才发现这是一棵树,每个点最多连一条边,最右的点不连,所以n-1条边,然后连通,所以是树。

接下来就很好做啦,dp(v,0),dp(v,1),dp(v,2)分别表示在v不在链上,在链一端,在链中间,v子树的最小不便利度,然后就可以直接dp掉了,然后考虑第二问,我们发现最小不便利度一定不是超过log n(轻重树链剖分都log n了嘛),所以f(i,j,0),f(i,j,1),f(i,j,2)分别表示三种情况下不便利值为j的情况,然后处理个前缀和dp一下,总复杂度O(n + n log n)。

代码(细节超多QAQ):

#include <cstdio>

#include <algorithm>

#include <cstring>

  

using namespace std ;

  

#define DOWN( i , r , l ) for ( int i = r ; i >= l ; -- i )

#define Rep( i , x ) for ( int i = 0 ; i < x ; ++ i )

#define rep( i , x ) for ( int i = 0 ; i ++ < x ; )

#define REP( i , l , r ) for ( int i = l ; i <= r ; ++ i )

  

#define travel( x ) for ( edge *p = head[ x ] ; p ; p = p -> next )

  

const int maxn = 101000 ;

const int maxm = 201000 ;

  

struct edge {

        edge *next ;

        int t ;

} E[ maxm ] ;

  

edge *pt = E , *head[ maxn ] ;

  

inline void add( int s , int t ) {

        pt -> t = t , pt -> next = head[ s ] ; head[ s ] = pt ++ ;

}

  

inline void addedge( int s , int t ) {

        add( s , t ) , add( t , s ) ;

}

  

bool used[ maxn ] ;

int n , m , q ;

  

void dfs( int now ) {

        used[ now ] = true ;

        travel( now ) if ( ! used[ p -> t ] ) dfs( p -> t ) ;

}

  

int dp[ maxn ][ 3 ] , ch[ maxn ] , f[ maxn ][ 3 ] , ans ;

  

const int inf = 0x7fffffff ;

  

void Dp0( int now , int fa ) {

        travel( now ) if ( p -> t != fa ) Dp0( p -> t , now ) ;

        int cnt = 0 ;

        dp[ now ][ 0 ] = 0 ;

        travel( now ) if ( p -> t != fa ) ch[ ++ cnt ] = p -> t ;

        f[ 0 ][ 0 ] = 0 , f[ 0 ][ 1 ] = f[ 0 ][ 2 ] = inf ;

        rep( i , cnt ) {

                int v = ch[ i ] ;

                f[ i ][ 0 ] = max( f[ i - 1 ][ 0 ] , min( min( dp[ v ][ 0 ] , dp[ v ][ 1 ] ) , dp[ v ][ 2 ] ) + 1 ) ;

                f[ i ][ 1 ] = min( max( f[ i - 1 ][ 1 ] , min( min( dp[ v ][ 0 ] , dp[ v ][ 1 ] ) , dp[ v ][ 2 ] ) + 1 ) , max( f[ i - 1 ][ 0 ] , min( dp[ v ][ 0 ] , dp[ v ][ 1 ] ) ) ) ;

                f[ i ][ 2 ] = min( max( f[ i - 1 ][ 2 ] , min( min( dp[ v ][ 0 ] , dp[ v ][ 1 ] ) , dp[ v ][ 2 ] ) + 1 ) , max( f[ i - 1 ][ 1 ] , min( dp[ v ][ 0 ] , dp[ v ][ 1 ] ) ) ) ;

        }

        dp[ now ][ 0 ] = f[ cnt ][ 0 ] , dp[ now ][ 1 ] = f[ cnt ][ 1 ] , dp[ now ][ 2 ] = f[ cnt ][ 2 ] ;

}

  

const int maxd = 20 ;

  

typedef long long ll ;

  

int Dp[ maxn ][ maxd ][ 3 ] , sum[ maxn ][ maxd ][ 3 ] , g[ maxn ][ 3 ][ 2 ] ;

  

void Dp1( int now , int fa ) {

        int cnt = 0 , v , a , b , c , d ;

        travel( now ) if ( p -> t != fa ) {

                ++ cnt ; Dp1( p -> t , now ) ;

        }

        if ( ! cnt ) {

                REP( i , 0 , ans ) {

                        Dp[ now ][ i ][ 0 ] = Dp[ now ][ i ][ 1 ] = Dp[ now ][ i ][ 2 ] = 0 ;

                }

                Dp[ now ][ 0 ][ 0 ] = 1 ; return ;

        }

        cnt = 0 ;

        travel( now ) if ( p -> t != fa ) ch[ ++ cnt ] = p -> t ;

        Dp[ now ][ 0 ][ 0 ] = 0 ;

        if ( cnt == 1 ) {

                Dp[ now ][ 0 ][ 2 ] = 0 ;

                Dp[ now ][ 0 ][ 1 ] = ( Dp[ ch[ 1 ] ][ 0 ][ 0 ] + Dp[ ch[ 1 ] ][ 0 ][ 1 ] ) % q ;

        } else if ( cnt == 2 ) {

                Dp[ now ][ 0 ][ 1 ] = 0 ;

                Dp[ now ][ 0 ][ 2 ] = ( ll ) ( Dp[ ch[ 1 ] ][ 0 ][ 0 ] + Dp[ ch[ 1 ] ][ 0 ][ 1 ] ) * ( Dp[ ch[ 2 ] ][ 0 ][ 0 ] + Dp[ ch[ 2 ] ][ 0 ][ 1 ] ) % q ;

        } else {

                Dp[ now ][ 0 ][ 1 ] = Dp[ now ][ 0 ][ 2 ] = 0 ;

        }

        rep( i , cnt ) {

                int v = ch[ i ] ;

                sum[ i ][ 0 ][ 0 ] = Dp[ v ][ 0 ][ 0 ] , sum[ i ][ 0 ][ 1 ] = Dp[ v ][ 0 ][ 1 ] , sum[ i ][ 0 ][ 2 ] = Dp[ v ][ 0 ][ 2 ] ;

                REP( j , 1 , ans ) {

                        sum[ i ][ j ][ 0 ] = ( sum[ i ][ j - 1 ][ 0 ] + Dp[ v ][ j ][ 0 ] ) % q ;

                        sum[ i ][ j ][ 1 ] = ( sum[ i ][ j - 1 ][ 1 ] + Dp[ v ][ j ][ 1 ] ) % q ;

                        sum[ i ][ j ][ 2 ] = ( sum[ i ][ j - 1 ][ 2 ] + Dp[ v ][ j ][ 2 ] ) % q ;

                }

        }

        REP( j , 1 , ans ) {

                g[ 0 ][ 0 ][ 0 ] = g[ 0 ][ 0 ][ 1 ] = g[ 0 ][ 1 ][ 0 ] = g[ 0 ][ 1 ][ 1 ] = g[ 0 ][ 2 ][ 0 ] = g[ 0 ][ 2 ][ 1 ] = 0 ;

                g[ 0 ][ 0 ][ 0 ] = 1 ;

                rep( i , cnt ) {

                        int v = ch[ i ] ;

                        g[ i ][ 0 ][ 0 ] = ( ll ) g[ i - 1 ][ 0 ][ 0 ] * ( j - 2 >= 0 ? ( sum[ i ][ j - 2 ][ 0 ] + sum[ i ][ j - 2 ][ 1 ] + sum[ i ][ j - 2 ][ 2 ] ) : 0 ) % q ;

                        a = ( ll ) g[ i - 1 ][ 0 ][ 0 ] * ( Dp[ v ][ j - 1 ][ 0 ] + Dp[ v ][ j - 1 ][ 1 ] + Dp[ v ][ j - 1 ][ 2 ] ) % q ;

                        b = ( ll ) g[ i - 1 ][ 0 ][ 1 ] * ( sum[ i ][ j - 1 ][ 0 ] + sum[ i ][ j - 1 ][ 1 ] + sum[ i ][ j - 1 ][ 2 ] ) % q ;

                        g[ i ][ 0 ][ 1 ] = ( a + b ) % q ;

                        a = ( ll ) g[ i - 1 ][ 0 ][ 0 ] * ( sum[ i ][ j - 1 ][ 0 ] + sum[ i ][ j - 1 ][ 1 ] ) % q ;

                        b = ( ll ) g[ i - 1 ][ 1 ][ 0 ] * ( j - 2 >= 0 ? ( sum[ i ][ j - 2 ][ 0 ] + sum[ i ][ j - 2 ][ 1 ] + sum[ i ][ j - 2 ][ 2 ] ) : 0 ) % q ;

                        g[ i ][ 1 ][ 0 ] = ( a + b ) % q ;

                        a = ( ll ) g[ i - 1 ][ 0 ][ 0 ] * ( Dp[ v ][ j ][ 0 ] + Dp[ v ][ j ][ 1 ] ) % q ;

                        b = ( ll ) g[ i - 1 ][ 0 ][ 1 ] * ( sum[ i ][ j ][ 0 ] + sum[ i ][ j ][ 1 ] ) % q ;

                        c = ( ll ) g[ i - 1 ][ 1 ][ 0 ] * ( Dp[ v ][ j - 1 ][ 0 ] + Dp[ v ][ j - 1 ][ 1 ] + Dp[ v ][ j - 1 ][ 2 ] ) % q ;

                        d = ( ll ) g[ i - 1 ][ 1 ][ 1 ] * ( sum[ i ][ j - 1 ][ 0 ] + sum[ i ][ j - 1 ][ 1 ] + sum[ i ][ j - 1 ][ 2 ] ) % q ;

                        g[ i ][ 1 ][ 1 ] = ( a + b + c + d ) % q ;

                        a = ( ll ) g[ i - 1 ][ 1 ][ 0 ] * ( sum[ i ][ j - 1 ][ 0 ] + sum[ i ][ j - 1 ][ 1 ] ) % q ;

                        b = ( ll ) g[ i - 1 ][ 2 ][ 0 ] * ( j - 2 >= 0 ? ( sum[ i ][ j - 2 ][ 0 ] + sum[ i ][ j - 2 ][ 1 ] + sum[ i ][ j - 2 ][ 2 ] ) : 0 ) % q ;

                        g[ i ][ 2 ][ 0 ] = ( a + b ) % q ;

                        a = ( ll ) g[ i - 1 ][ 1 ][ 0 ] * ( Dp[ v ][ j ][ 0 ] + Dp[ v ][ j ][ 1 ] ) % q ;

                        b = ( ll ) g[ i - 1 ][ 1 ][ 1 ] * ( sum[ i ][ j ][ 0 ] + sum[ i ][ j ][ 1 ] ) % q ;

                        c = ( ll ) g[ i - 1 ][ 2 ][ 0 ] * ( Dp[ v ][ j - 1 ][ 0 ] + Dp[ v ][ j - 1 ][ 1 ] + Dp[ v ][ j - 1 ][ 2 ] ) % q ;

                        d = ( ll ) g[ i - 1 ][ 2 ][ 1 ] * ( sum[ i ][ j - 1 ][ 0 ] + sum[ i ][ j - 1 ][ 1 ] + sum[ i ][ j - 1 ][ 2 ] ) % q ;

                        g[ i ][ 2 ][ 1 ] = ( a + b + c + d ) % q ;

                }

                Dp[ now ][ j ][ 0 ] = g[ cnt ][ 0 ][ 1 ] ;

                Dp[ now ][ j ][ 1 ] = g[ cnt ][ 1 ][ 1 ] ;

                Dp[ now ][ j ][ 2 ] = g[ cnt ][ 2 ][ 1 ] ;

        }

}

  

int main(  ) {

        memset( head , 0 , sizeof( head ) ) ;

        scanf( "%d%d%d" , &n , &m , &q ) ;

        while ( m -- ) {

                int s , t ; scanf( "%d%d" , &s , &t ) ; addedge( s , t ) ;

        }

        memset( used , false , sizeof( used ) ) ;

        dfs( 1 ) ;

        rep( i , n ) if ( ! used[ i ] ) {

                printf( "-1\n-1\n" ) ; return 0 ;

        }

        Dp0( 1 , 0 ) ;

        printf( "%d\n" , ans = min( dp[ 1 ][ 0 ] , min( dp[ 1 ][ 1 ] , dp[ 1 ][ 2 ] ) ) ) ;

        Dp1( 1 , 0 ) ;

        printf( "%d\n" , ( Dp[ 1 ][ ans ][ 0 ] + Dp[ 1 ][ ans ][ 1 ] + Dp[ 1 ][ ans ][ 2 ] ) % q ) ;

        return 0 ;

}
    原文作者:AmadeusChan
    原文地址: https://www.jianshu.com/p/09fa1a84fe19
    本文转自网络文章,转载此文章仅为分享知识,如有侵权,请联系博主进行删除。
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