[NOIP2013] 花匠，noip2013花匠
初看這道題想到O(n2) 的暴力dp
用f[i][0]表示取第i個點為最低點時的答案, f[i][1]為最高點，且f[i][0] = max( f[j][1] ) +1
這樣每次都要查詢前面區間滿足 h[i]>h[j] 的最大值， 可以考慮 線段樹區間查詢 或者 BIT 或者BST ， 時間降至O(nlogn)
但是BIT時要注意查詢h[i]<h[j] 條件時涉及到 j ~ maxheight 的最值查詢， 可以把maxheight -h[i] +2 (BIT下標不為0) 存入樹狀數組
RE 代碼：BIT 誤取最大下標為n！！ 實際上應該在讀入時求出maxheight！！
![]()
#include<iostream>
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<vector>
#include<queue>
#include<stack>
#include<map>
#include<set>
#include<string>
#include<iomanip>
#include<ctime>
#include<climits>
#include<cctype>
#include<algorithm>
#ifdef WIN32
#define AUTO "%I64d"
#else
#define AUTO "%lld"
#endif
using namespace std;
#define lowbit(x) x&-x
#define smax(x,tmp) x=max(x,tmp)
const int maxn=100005;
const int INF=0x3f3f3f3f;
int high[maxn],low[maxn];//!! LAST RE!!! BIT must be bigger!! maxheight!!!
int f[maxn][2];//0: low point ; 1: high point
int a[maxn];
int n;
int maxheight=-INF;
inline void add_low(int x,int val)
{
for(int i=x;i<=maxheight+1;i+=lowbit(i)) smax(low[i],val);//also easy to consider as n!!
}
inline void add_high(int x,int val)
{
for(int i=x;i<=maxheight+1;i+=lowbit(i)) smax(high[i],val);
}
inline int query_low(int x)
{
int ret=0;
for(int i=x;i;i-=lowbit(i)) smax(ret,low[i]);
return ret;
}
inline int query_high(int x)
{
int ret=0;
for(int i=x;i;i-=lowbit(i)) smax(ret,high[i]);
return ret;
}
int main()
{
freopen("flower.in","r",stdin);
freopen("flower.out","w",stdout);
scanf("%d",&n);
for(int i=1;i<=n;i++) scanf("%d",a+i),smax(maxheight,a[i]);
int ans=1;
f[1][0]=f[1][1]=1;
add_low(a[1]+1,f[1][0]);add_high(maxheight-a[1]+2,f[1][1]);//add reversely, to query the max !!
for(int i=2;i<=n;i++)
{
f[i][0]=query_high(maxheight-a[i]+1)+1;
f[i][1]=query_low(a[i])+1;
add_low(a[i]+1,f[i][0]);//can't either!!
add_high(maxheight-a[i]+2,f[i][1]);// mustn't use the 0 point !!
smax(ans,max(f[i][0],f[i][1]));
}
printf("%d",ans);
return 0;
}
View Code
AC代碼：（BIT）
![]()
#include<iostream>
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<vector>
#include<queue>
#include<stack>
#include<map>
#include<set>
#include<string>
#include<iomanip>
#include<ctime>
#include<climits>
#include<cctype>
#include<algorithm>
#ifdef WIN32
#define AUTO "%I64d"
#else
#define AUTO "%lld"
#endif
using namespace std;
#define lowbit(x) x&-x
#define smax(x,tmp) x=max(x,tmp)
const int maxn=100005;
const int maxh=1000005;
const int INF=0x3f3f3f3f;
int high[maxh],low[maxh];//for BIT
int f[maxn][2];//0: low point ; 1: high point
int a[maxn];
int n;
int maxheight=-INF;
inline void add_low(int x,int val)
{
for(int i=x;i<=maxheight+1;i+=lowbit(i)) smax(low[i],val);
}
inline void add_high(int x,int val)
{
for(int i=x;i<=maxheight+1;i+=lowbit(i)) smax(high[i],val);
}
inline int query_low(int x)
{
int ret=0;
for(int i=x;i;i-=lowbit(i)) smax(ret,low[i]);
return ret;
}
inline int query_high(int x)
{
int ret=0;
for(int i=x;i;i-=lowbit(i)) smax(ret,high[i]);
return ret;
}
int main()
{
freopen("flower.in","r",stdin);
freopen("flower.out","w",stdout);
scanf("%d",&n);
for(int i=1;i<=n;i++) scanf("%d",a+i),smax(maxheight,a[i]);
int ans=1;
f[1][0]=f[1][1]=1;
add_low(a[1]+1,f[1][0]);add_high(maxheight-a[1]+2,f[1][1]);//add reversely, to query the max !!
for(int i=2;i<=n;i++)
{
f[i][0]=query_high(maxheight-a[i]+1)+1;
f[i][1]=query_low(a[i])+1;
add_low(a[i]+1,f[i][0]);//can't either!!
add_high(maxheight-a[i]+2,f[i][1]);// mustn't use the 0 point !!
smax(ans,max(f[i][0],f[i][1]));
}
printf("%d",ans);
return 0;
}
//O(n): f[i][0,1] indicates that don't need to stop at i, but had the previous cases
//O(n): find the corners with the tendency
View Code
現慮另一種 O(n) 的dp
用f[i][0,1] 表示 i 及其以前所有高度的最大值，但是0 表示之前出於下降階段而並非之前的上一個節點為轉折點， 僅僅表示一個趨勢
另一種 o(n) 算法
由於一段相同變化趨勢的區段內只能留下一個端點
故只需要統計出所有的”拐點“即可！
WA代碼： 只考慮到相鄰的幾個數，但是缺乏長遠的考慮！！！！應用趨勢來判斷！！
![]()
#include<iostream>
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<vector>
#include<queue>
#include<stack>
#include<map>
#include<set>
#include<string>
#include<iomanip>
#include<ctime>
#include<climits>
#include<cctype>
#include<algorithm>
#ifdef WIN32
#define AUTO "%I64d"
#else
#define AUTO "%lld"
#endif
using namespace std;
#define lowbit(x) x&-x
#define smax(x,tmp) x=max(x,tmp)
const int maxn=100005;
const int INF=0x3f3f3f3f;
int n;
int a[maxn];
int main()
{
freopen("flower.in","r",stdin);
freopen("flower.out","w",stdout);
scanf("%d",&n);
for(int i=1;i<=n;i++) scanf("%d",a+i);
int ans=2;
for(int i=2;i^n;i++)
{
if(a[i-1]>a[i] && a[i]<a[i+1]) ans++;//Last WA!! not only the near one, but long trems
if(a[i-1]<a[i] && a[i]>a[i+1]) ans++;
}
printf("%d",ans);
return 0;
}
View Code
AC代碼：
![]()
#include<iostream>
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<vector>
#include<queue>
#include<stack>
#include<map>
#include<set>
#include<string>
#include<iomanip>
#include<ctime>
#include<climits>
#include<cctype>
#include<algorithm>
#ifdef WIN32
#define AUTO "%I64d"
#else
#define AUTO "%lld"
#endif
using namespace std;
#define lowbit(x) x&-x
#define smax(x,tmp) x=max(x,tmp)
const int maxn=100005;
const int INF=0x3f3f3f3f;
int n;
int a[maxn];
int main()
{
freopen("flower.in","r",stdin);
freopen("flower.out","w",stdout);
scanf("%d",&n);
for(int i=1;i<=n;i++) scanf("%d",a+i);
int ans=1;//indicates the first one to start!!
int flag=0;//indicates start!!
for(int i=1;i^n;i++)
{
if(a[i]<a[i+1] && (flag==0 || flag==-1)) flag=1,ans++;
if(a[i]>a[i+1] && (flag==0 || flag==1)) flag=-1,ans++;
}
printf("%d",ans);
return 0;
}
View Code
