HDU 3726 Graph and Queries 平衡树+前向星+并查集+离线操作+逆向思维 数据结构大综合题

Graph and Queries

Time Limit: 10000/5000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)

【Problem Description】
You are given an undirected graph with N vertexes and M edges. Every vertex in this graph has an integer value assigned to it at the beginning. You're also given a sequence of operations and you need to process them as requested. Here's a list of the possible operations that you might encounter:
1)  Deletes an edge from the graph. The format is [D X], where X is an integer from 1 to M, indicating the ID of the edge that you should delete. It is guaranteed that no edge will be deleted more than once.
2)  Queries the weight of the vertex with K-th maximum value among all vertexes currently connected with vertex X (including X itself). The format is [Q X K], where X is an integer from 1 to N, indicating the id of the vertex, and you may assume that K will always fit into a 32-bit signed integer. In case K is illegal, the value for that query will be considered as undefined, and you should return 0 as the answer to that query.
3)  Changes the weight of a vertex. The format is [C X V], where X is an integer from 1 to N, and V is an integer within the range [-106, 106].

The operations end with one single character, E, which indicates that the current case has ended. For simplicity, you only need to output one real number - the average answer of all queries.
 
【Input】
There are multiple test cases in the input file. Each case starts with two integers N and M (1 <= N <= 2 * 104, 0 <= M <= 6 * 104), the number of vertexes in the graph. The next N lines describes the initial weight of each vertex (-106 <= weight[i] <= 106). The next part of each test case describes the edges in the graph at the beginning. Vertexes are numbered from 1 to N. The last part of each test case describes the operations to be performed on the graph. It is guaranteed that the number of query operations [Q X K] in each case will be in the range [1, 2 * 105], and there will be no more than 2 * 105 operations that change the values of the vertexes [C X V].
There will be a blank line between two successive cases. A case with N = 0, M = 0 indicates the end of the input file and this case should not be processed by your program.
 
【Output】
For each test case, output one real number – the average answer of all queries, in the format as indicated in the sample output. Please note that the result is rounded to six decimal places.
 
【Sample Input】
3 3
10
20
30
1 2
2 3
1 3
D 3
Q 1 2
Q 2 1
D 2
Q 3 2
C 1 50
Q 1 1
E

3 3
10
20
20
1 2
2 3
1 3
Q 1 1
Q 1 2
Q 1 3
E

0 0

【Sample Output】

Case 1: 25.000000 
Case 2: 16.666667

【Hint】

For the first sample:
D 3 -- deletes the 3rd edge in the graph (the remaining edges are (1, 2) and (2, 3))
Q 1 2 -- finds the vertex with the second largest value among all vertexes connected with 1. The answer is 20.
Q 2 1 -- finds the vertex with the largest value among all vertexes connected with 2. The answer is 30.
D 2 -- deletes the 2nd edge in the graph (the only edge left after this operation is (1, 2))
Q 3 2 -- finds the vertex with the second largest value among all vertexes connected with 3.  The answer is 0 (Undefined).
C 1 50 -- changes the value of vertex 1 to 50.
Q 1 1 -- finds the vertex with the largest value among all vertex connected with 1. The answer is 50.
E -- This is the end of the current test case. Four queries have been evaluated, and the answer to this case is (20 + 30 + 0 + 50) / 4 = 25.000.

For the second sample, caution about the vertex with same weight:
Q 1 1 – the answer is 20
Q 1 2 – the answer is 20
Q 1 3 – the answer is 10

【题意】

给出一张无向图，并在图中进行多种操作：

1.删除一条边；2.改变一个点的权值；3.询问x能够到达的所有点中，第k大的是多少

【分析】

花费了好多时间在这道题上，算是这段时间中做到的最综合的数据结构题了。

首先本题的无向图一开始就是个陷阱，如果单纯地从图的角度来考虑，每次询问都需要遍历全图来找第k大的值，这显然是不可取的，而中间又存在删边操作，图的连通性不是稳定的，结点的权值会变，而且可能多次改变，所以整个图是完全不稳定的，没办法用图论的方法来解决。

考虑倒过来操作，如果我们从做完所有操作之后最后的结果图出发，逆序回去，则原本的删边可以看成是将两个连通块连接到一起，询问第k值是在x点当前所属的连通块中进行，对点权的修改也是，而对于每一个独立的连通块，最后这两步可以用平衡树来实现。

所以算法的雏形就有了，询问连通块k值、修改连通块中点的点权操作——平衡树，维护点的连通性——并查集，保存点权的修改信息——前向星

完整过程：

1.首先从完整图出发，读入所有操作，记录删掉的边，按顺序记录点权的变化情况，记录其他信息；

2.用删边信息建立终图的连通性，并查集维护，对于每一个独立的连通块，建立一个独立的平衡树（这里我用的是SBT，网上题解搜出来好多人用的Splay，我其实有点不太理解，感觉这里没有需要用到Splay特殊结构的地方，单纯的维护平衡树的话Splay的稳定性和效率应该是不如SBT的。有大神路过看到这个的话，希望能交流下~~~）；

3.从最后一条操作开始逆序回来：

     i.  询问，则在x所属的平衡树中找第k值；

     ii. 修改，则在x所属的平衡树中删掉原始的值，插入新值，这里对点权的顺序维护我用了前向星，要保证点权的操作也是要有序的；

     iii.删边，在这里就是如果两个点属于两个不同的连通块，则将两个连通块连起来，并查集合并，同时平衡树合并。平衡树合并的时候只能把小的那棵树一个一个加到大的树中去，貌似Splay有个启发式合并，用了finger search神马的东西，可以把合并在O（nlogn）内完成，不会写，ORZ。

【后记】

写这道题的时候，SBT模板改了两次，-_-///，然后中间有SBT结合并查集结合前向星的，代码里面就是数组套数组套数组套数组......好多地方写着写着就写乱了，教训是如果能简单，一定不要往复杂了写。

然后并查集的教训：father[]绝对不能直接引用，必须调用getfather（）

  1 /* ***********************************************
  2 MYID    : Chen Fan
  3 LANG    : G++
  4 PROG    : HDU3726
  5 ************************************************ */
  6 
  7 #include <iostream>
  8 #include <cstdio>
  9 #include <cstring>
 10 #include <algorithm>
 11 
 12 using namespace std;
 13 
 14 #define MAXN 20010
 15 
 16 typedef struct enod
 17 {
 18     int p,q;
 19     bool enable;
 20 } enode;
 21 enode e[60010];
 22 
 23 typedef struct qnod
 24 {
 25     int x,k;
 26 } qnode;
 27 qnode lisq[200010];
 28 
 29 typedef struct nod
 30 {
 31     int no,value,time;
 32 } node;
 33 node lis[300010];
 34 
 35 int sons[MAXN][2],size[MAXN],data[MAXN],sbt[MAXN],sbttail;
 36 int lisd[60010],taild,tailq,tail,tailtot,lisc[200010],tailc=0;
 37 int start[MAXN],num[MAXN],father[MAXN];
 38 char listot[500010];
 39 
 40 void rotate(int &t,int w) //rotate(&node,0/1)
 41 {
 42     int k=sons[t][1-w];
 43     if (!k) return ;
 44     sons[t][1-w]=sons[k][w];
 45     sons[k][w]=t;
 46     size[k]=size[t];
 47     size[t]=size[sons[t][0]]+size[sons[t][1]]+1;
 48     t=k;
 49 }
 50 
 51 void maintain(int& t,bool flag) //maintain(&node,flag)
 52 {
 53     if (!t) return ;
 54     if (!flag)
 55         if (size[sons[sons[t][0]][0]]>size[sons[t][1]]) rotate(t,1);
 56         else if (size[sons[sons[t][0]][1]]>size[sons[t][1]]) 
 57         {
 58             rotate(sons[t][0],0);
 59             rotate(t,1);
 60         } else return ;
 61     else
 62         if (size[sons[sons[t][1]][1]]>size[sons[t][0]]) rotate(t,0);
 63         else if (size[sons[sons[t][1]][0]]>size[sons[t][0]])
 64         {
 65             rotate(sons[t][1],1);
 66             rotate(t,0);
 67         } else return ;
 68     
 69     maintain(sons[t][0],false);
 70     maintain(sons[t][1],true);
 71     maintain(t,false);
 72     maintain(t,true);
 73 }
 74 
 75 void insert(int& t,int v,int pos) //insert(&root,value)
 76 {
 77     if (!size[t])
 78     {
 79         if (!pos)
 80         {
 81             sbttail++;
 82             pos=sbttail;
 83         }
 84         data[pos]=v;
 85         size[pos]=1;
 86         sons[pos][0]=0;
 87         sons[pos][1]=0;
 88         t=pos;
 89     } else 
 90     {
 91         size[t]++;
 92         if (v<data[t]) insert(sons[t][0],v,pos);
 93         else insert(sons[t][1],v,pos);
 94         maintain(t,v>=data[t]);
 95     }
 96 }
 97 
 98 int last;
 99 int del(int& t,int v) //node=del(&root,key)
100 {
101     size[t]--;
102     if (v==data[t]||(v<data[t]&&sons[t][0]==0)||(v>data[t]&&sons[t][1]==0))
103     {
104         int ret=data[t];
105         if (sons[t][0]==0||sons[t][1]==0) 
106         {
107             last=t;
108             t=sons[t][1]+sons[t][0];
109         }
110         else data[t]=del(sons[t][0],data[t]+1);
111         return ret;
112     } else
113     if (v<data[t]) return del(sons[t][0],v);
114     else return del(sons[t][1],v);
115 }
116 
117 int select(int t,int k)
118 {
119     if (k==size[sons[t][0]]+1) return t;
120     if (k<=size[sons[t][0]]) return select(sons[t][0],k);
121     else return select(sons[t][1],k-1-size[sons[t][0]]);
122 }
123 
124 void clean_father(int n)
125 {
126     for (int i=1;i<=n;i++)
127     {
128         father[i]=i;
129         sbt[i]=i;
130     }
131 } 
132 
133 int getfather(int x)
134 {
135     if (father[x]!=x) father[x]=getfather(father[x]);
136     return father[x];
137 }
138 
139 void link(int x,int y)
140 {
141     int xx=getfather(x),yy=getfather(y);
142     if (size[sbt[xx]]>size[sbt[yy]]) 
143     {
144         father[yy]=xx;
145         while(size[sbt[yy]]>0) 
146         {
147             int temp=del(sbt[yy],data[sbt[yy]]);
148             insert(sbt[xx],temp,last);    
149         }
150     }
151     else 
152     {
153         father[xx]=yy;
154         while(size[sbt[xx]]>0) 
155         {
156             int temp=del(sbt[xx],data[sbt[xx]]);
157             insert(sbt[yy],temp,last);
158         }
159     }
160 }
161 
162 bool op(node a,node b)
163 {
164     if (a.no==b.no) return a.time<b.time;
165     else return a.no<b.no;
166 }
167 
168 int main()
169 {
170     freopen("3726.txt","r",stdin);
171     
172     int n,m,tt=0;
173     while(scanf("%d%d",&n,&m)==2&&(n+m>0))
174     {
175         for (int i=1;i<=n;i++)
176         {
177             lis[i].no=i;
178             lis[i].time=i;
179             scanf("%d",&lis[i].value);
180         }
181         for (int i=1;i<=m;i++)
182         {
183             scanf("%d%d",&e[i].p,&e[i].q);
184             e[i].enable=true;
185         }
186         taild=0;tailq=0;tailc=0;tail=n;tailtot=0;
187         bool doit=true;
188         while(doit)
189         {
190             char c=getchar();
191             while(c!='D'&&c!='Q'&&c!='C'&&c!='E') c=getchar();
192             tailtot++;
193             listot[tailtot]=c;
194             switch(c)
195             {
196                 case 'D':
197                     taild++;
198                     scanf("%d",&lisd[taild]);
199                     e[lisd[taild]].enable=false;
200                     break;
201                 case 'Q':
202                     tailq++;
203                     scanf("%d%d",&lisq[tailq].x,&lisq[tailq].k);
204                     break;
205                 case 'C':
206                     tail++;
207                     lis[tail].time=tail;
208                     scanf("%d%d",&lis[tail].no,&lis[tail].value);
209                     tailc++;
210                     lisc[tailc]=lis[tail].no;
211                     break;
212                 default:
213                     doit=false;
214             }
215         }
216 
217         sort(&lis[1],&lis[tail+1],op);
218         int o=0;
219         memset(num,0,sizeof(num));
220         for (int i=1;i<=tail;i++)
221         {
222             if (o!=lis[i].no)
223             {
224                 o=lis[i].no;
225                 start[o]=i;
226             }
227             num[o]++;
228         }
229 
230         clean_father(n);
231         sbttail=0;
232         memset(size,0,sizeof(size));
233         for (int i=1;i<=n;i++) insert(sbt[i],lis[start[i]+num[i]-1].value,0);
234         for (int i=1;i<=m;i++)
235         if (e[i].enable)
236         if (getfather(e[i].p)!=getfather(e[i].q)) 
237         link(e[i].p,e[i].q);
238 
239         int ansq=tailq;
240         double ans=0;
241         for (int i=tailtot-1;i>=1;i--)
242         switch(listot[i])
243         {
244             case 'Q':
245                 if (lisq[tailq].k>0&&size[sbt[getfather(lisq[tailq].x)]]>=lisq[tailq].k) 
246                     ans+=data[select(sbt[getfather(lisq[tailq].x)],size[sbt[getfather(lisq[tailq].x)]]-lisq[tailq].k+1)];
247                 tailq--;
248                 break;
249             case 'D':
250                 if (getfather(e[lisd[taild]].p)!=getfather(e[lisd[taild]].q))
251                     link(e[lisd[taild]].p,e[lisd[taild]].q);
252                 taild--;
253                 break;
254             case 'C':
255                 num[lisc[tailc]]--;
256                 del(sbt[getfather(lisc[tailc])],lis[start[lisc[tailc]]+num[lisc[tailc]]].value);
257                 insert(sbt[getfather(lisc[tailc])],lis[start[lisc[tailc]]+num[lisc[tailc]]-1].value,last);
258                 tailc--;
259         }
260         tt++;
261         printf("Case %d: %.6f
",tt,ans/ansq);
262     }
263 
264     return 0;
265 }
View Code
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原文地址：https://www.cnblogs.com/jcf94/p/4360785.html