zoj 2112 （主席树，树状数组套线段树）

Dynamic Rankings

---

Time Limit: 10 Seconds       Memory Limit: 32768 KB

---

The Company Dynamic Rankings has developed a new kind of computer that is no longer satisfied with the query like to simply find the k-th smallest number of the given N numbers. They have developed a more powerful system such that for N numbers a[1], a[2], ..., a[N], you can ask it like: what is the k-th smallest number of a[i], a[i+1], ..., a[j]? (For some i<=j, 0<k<=j+1-i that you have given to it). More powerful, you can even change the value of some a[i], and continue to query, all the same.

Your task is to write a program for this computer, which

- Reads N numbers from the input (1 <= N <= 50,000)

- Processes M instructions of the input (1 <= M <= 10,000). These instructions include querying the k-th smallest number of a[i], a[i+1], ..., a[j] and change some a[i] to t.

Input

The first line of the input is a single number X (0 < X <= 4), the number of the test cases of the input. Then X blocks each represent a single test case.http://www.sn180.com/buyer/buyview/1115832741.html

The first line of each block contains two integers N and M, representing N numbers and M instruction. It is followed by N lines. The (i+1)-th line represents the number a[i]. Then M lines that is in the following format

Q i j k or
C i t

It represents to query the k-th number of a[i], a[i+1], ..., a[j] and change some a[i] to t, respectively. It is guaranteed that at any time of the operation. Any number a[i] is a non-negative integer that is less than 1,000,000,000.

There're NO breakline between two continuous test cases.

Output

For each querying operation, output one integer to represent the result. (i.e. the k-th smallest number of a[i], a[i+1],..., a[j])

There're NO breakline between two continuous test cases.http://www.sn180.com/buyer/buyview/1115821323.html

Sample Input

2
5 3
3 2 1 4 7
Q 1 4 3
C 2 6
Q 2 5 3
5 3
3 2 1 4 7
Q 1 4 3
C 2 6
Q 2 5 3

Sample Output

3
6
3
6

题意：查找区间第K大值，值可修改

分析：

每一棵线段树是维护每一个序列前缀的值在任意区间的个数，
如果还是按照静态的来做的话，那么每一次修改都要遍历O(n)棵树，
时间就是O(2*M*nlogn)->TLE
考虑到前缀和，我们通过树状数组来优化，即树状数组套主席树，
每个节点都对应一棵主席树，那么修改操作就只要修改logn棵树，
o(nlognlogn+Mlognlogn)时间是可以的，
但是直接建树要nlogn*logn（10^7）会MLE
我们发现对于静态的建树我们只要nlogn个节点就可以了，
而且对于修改操作，只是修改M次，每次改变俩个值（减去原先的，加上现在的）
也就是说如果把所有初值都插入到树状数组里是不值得的，
所以我们分两部分来做，所有初值按照静态来建，内存O(nlogn)，
而修改部分保存在树状数组中，每次修改logn棵树，每次插入增加logn个节点

代码：

[cpp]  view plain  copy

1. //#pragma comment(linker,"/STACK:102400000,102400000")  
2. #include <iostream>  
3. #include <cstdio>  
4. #include <cstring>  
5. #include <cmath>  
6. #include <algorithm>  
7. #include <stack>  
8. #include <map>  
9. #include <set>  
10. #include <vector>  
11. #include <queue>  
12. #define mem(p,k) memset(p,k,sizeof(p));  
13. #define pb push_back  
14. //#define lson l,m,rt<<1  
15. //#define rson m+1,r,rt<<1|1  
16. #define inf 0x3f3f3f3f  
17. #define ll long long  
18. using namespace std;  
19. const int N=50010 ;  
20. struct SD{  
21.     int flag,l,r,k;  
22. }que[10010];  
23. int T,n,q,len,tot;  
24. int num[N],head1[N],head2[N];  
25. int tree[N*50],lson[N*50],rson[N*50],use[N];  
26. char s[2];  
27. vector<int> vec;  
28.   
29. int Hash(int k){  
30.     //cout<<lower_bound(vec.begin(),vec.end(),k)-vec.begin()+1;  
31.     return lower_bound(vec.begin(),vec.end(),k)-vec.begin()+1;  
32. }  
33. void update(int pre,int &now,int k,int val,int l,int r){  
34.     now=tot++;  
35.     tree[now]=tree[pre]+val;  
36.     lson[now]=lson[pre];rson[now]=rson[pre];  
37.     if(l==r)return;  
38.     int m=(l+r)>>1;  
39.     if(k<=m)update(lson[pre],lson[now],k,val,l,m);  
40.     else update(rson[pre],rson[now],k,val,m+1,r);  
41. }  
42. int lowbit(int i){ return -i&i; }  
43.   
44. void add(int k,int f,int val){  
45.     for(int i=k;i<=n;i+=lowbit(i)){  
46.         update(head2[i],head2[i],f,val,1,len);  
47.     }  
48. }  
49.   
50. int sum(int k){  
51.     int s=0;  
52.     for(int i=k;i>0;i-=lowbit(i)){  
53.         s+=tree[lson[use[i]]];  
54.     }  
55.     return s;  
56. }  
57.   
58. int Query(int L,int R,int k){  
59.     int s1=head1[L-1],s2=head1[R],l=1,r=len;  
60.     for(int i=L-1;i>0;i-=lowbit(i))use[i]=head2[i];  
61.     for(int i=R;i>0;i-=lowbit(i))use[i]=head2[i];  
62.     //cout<<L<<R<<"==="<<endl;  
63.     while(l<r){  
64.         int tmp=sum(R)-sum(L-1)+tree[lson[s2]]-tree[lson[s1]];  
65.         int m=(l+r)>>1;//cout<<tmp<<"  "<<k<<endl;  
66.   
67.         if(k<=tmp){  
68.             r=m;  
69.             s1=lson[s1];s2=lson[s2];  
70.             for(int i=L-1;i>0;i-=lowbit(i))use[i]=lson[use[i]];  
71.             for(int i=R;i>0;i-=lowbit(i))use[i]=lson[use[i]];  
72.         }  
73.         else{  
74.             k-=tmp;  
75.             l=m+1;  
76.             s1=rson[s1],s2=rson[s2];  
77.             for(int i=L-1;i>0;i-=lowbit(i))use[i]=rson[use[i]];  
78.             for(int i=R;i>0;i-=lowbit(i))use[i]=rson[use[i]];  
79.         }//cout<<l<<r<<endl;  
80.     }  
81.     //cout<<"  ~~~~~~ "<<endl;  
82.     return vec[l-1];  
83. }  
84.   
85. int main()  
86. {  
87.     cin>>T;  
88.     while(T--){  
89.         cin>>n>>q;  
90.         tot=1;  
91.         vec.clear();  
92.         mem(tree,0);  
93.         mem(lson,0);  
94.         mem(rson,0);  
95.         for(int i=1;i<=n;i++)scanf("%d",num+i),vec.pb(num[i]);  
96.         for(int i=1;i<=q;i++){  
97.             scanf("%s",s);  
98.             if(s[0]=='Q'){  
99.                 que[i].flag=0;  
100.                 scanf("%d%d%d",&que[i].l,&que[i].r,&que[i].k);  
101.             }  
102.             else{  
103.                 que[i].flag=1;  
104.                 scanf("%d%d",&que[i].l,&que[i].k);  
105.                 vec.pb(que[i].k);  
106.             }  
107.         }  
108.         sort(vec.begin(),vec.end());  
109.         vec.erase(unique(vec.begin(),vec.end()),vec.end());  
110.         len=vec.size();  
111.         //for(int i=0;i<vec.size();i++)cout<<vec[i]<<endl;  
112.         head1[0]=0;  
113.         for(int i=1;i<=n;i++){  
114.             update(head1[i-1],head1[i],Hash(num[i]),1,1,len);  
115.         }  
116.   
117.         for(int i=1;i<=n;i++){  
118.             head2[i]=0;  
119.         }  
120.         for(int i=1;i<=q;i++){  
121.             if(que[i].flag){  
122.                 add(que[i].l,Hash(num[que[i].l]),-1);  
123.                 add(que[i].l,Hash(que[i].k),1);  
124.                 num[que[i].l]=que[i].k;  
125.             }  
126.             else{  
127.                 printf("%d\n",Query(que[i].l,que[i].r,que[i].k));  
128.             }  
129.         }  
130.   
131.     }  
132.   
133.     return 0;  
134. }