【BZOJ2874】训练士兵（主席树）

题意：有一个N*M的矩阵，给出一些形如(x1,y1,x2,y2,s)的操作，代表(x1,y1)到(x2,y2)都被加上了s这个数

现在有一些强制在线的询问，询问(x1,y1)到(x2,y2)的和

对于100%的数据 n,m<=10^8,k<=40000,q<=100000； 

思路：将操作(x1,y1,x2,y2,s)差分成

(x1,y1)+s

(x1,y2+1)-s

(x2+1,y1)-s

(x2+1,y2+1)+s 

四个点

询问(x,y)等价于

\[\sum_{i=1}^x  \sum_{j=1}^y  \ a[i,j]*(x+1)*(y+1)-a[i,j]*i*(y+1)-a[i,j]*j*(x+1)+a[i,j]*i*j \]

证明如下：

可以看出只需维护a[i,j],a[i,j]*i,a[i,j]*j,a[i,j]*i*j四个值的二维前缀和

从左到右，从上到下排序并离散化，对每一行建立一棵主席树表示左上角的前缀和，

询问时在主席树中查找一段从头开始的前缀和即可

1 var t:array[0..6000000,1..6]of int64;
2     root:array[0..600000]of int64;
3     c:array[1..300000,1..3]of int64;
4     cx,cy:array[1..300000]of int64;
5     n,m,i,j,u,k,cnt,u1,u2,k1,q1,tmp,x,y:longint;
6     ans,x1,y1,x2,y2:int64;
7
8 procedure swap(var x,y:int64);
9 var t:int64;
10 begin
11  t:=x; x:=y; y:=t;
12 end;
13
14 procedure qsortx(l,r:longint);
15 var i,j,mid:longint;
16 begin
17  i:=l; j:=r; mid:=cx[(l+r)>>1];
18  repeat
19   while mid>cx[i] do inc(i);
20   while mid<cx[j] do dec(j);
21   if i<=j then
22   begin
23    swap(cx[i],cx[j]);
24    inc(i); dec(j);
25   end;
26  until i>j;
27  if l<j then qsortx(l,j);
28  if i<r then qsortx(i,r);
29 end;
30
31 procedure qsorty(l,r:longint);
32 var i,j,mid:longint;
33 begin
34  i:=l; j:=r; mid:=cy[(l+r)>>1];
35  repeat
36   while mid>cy[i] do inc(i);
37   while mid<cy[j] do dec(j);
38   if i<=j then
39   begin
40    swap(cy[i],cy[j]);
41    inc(i); dec(j);
42   end;
43  until i>j;
44  if l<j then qsorty(l,j);
45  if i<r then qsorty(i,r);
46 end;
47
48 function hashx(x:longint):longint;
49 var l,r,mid,last:longint;
50 begin
51  l:=1; r:=u1; last:=0;
52  while l<=r do
53  begin
54   mid:=(l+r)>>1;
55   if x>=cx[mid] then begin last:=mid; l:=mid+1; end
56    else r:=mid-1;
57  end;
58  exit(last);
59 end;
60
61 function hashy(x:longint):longint;
62 var l,r,mid,last:longint;
63 begin
64  l:=1; r:=u2; last:=0;
65  while l<=r do
66  begin
67   mid:=(l+r)>>1;
68   if x>=cy[mid] then begin last:=mid; l:=mid+1; end
69    else r:=mid-1;
70  end;
71  exit(last);
72 end;
73
74 procedure qsort(l,r:longint);
75 var i,j,mid1,mid2:longint;
76 begin
77  i:=l; j:=r; mid1:=c[(l+r)>>1,1]; mid2:=c[(l+r)>>1,2];
78  repeat
79   while (mid1>c[i,1])or((mid1=c[i,1])and(mid2>c[i,2])) do inc(i);
80   while (mid1<c[j,1])or((mid1=c[j,1])and(mid2<c[j,2])) do dec(j);
81   if i<=j then
82   begin
83    swap(c[i,1],c[j,1]);
84    swap(c[i,2],c[j,2]);
85    swap(c[i,3],c[j,3]);
86    inc(i); dec(j);
87   end;
88  until i>j;
89  if l<j then qsort(l,j);
90  if i<r then qsort(i,r);
91 end;
92
93 function query(p,l,r,x,y:longint):int64;
94 var mid:longint;
95 begin
96  if x=0 then exit(0);
97  if p=0 then exit(0);
98  if x>=r then exit(t[p,y]);
99  mid:=(l+r)>>1;
100  if x<=mid then exit(query(t[p,5],l,mid,x,y))
101   else exit(t[t[p,5],y]+query(t[p,6],mid+1,r,x,y));
102 end;
103
104 function ask(x,y:longint):int64;
105 var x1,y1:longint;
106 begin
107  x1:=hashx(x); y1:=hashy(y);
108  ask:=0;
109  ask:=ask+query(root[x1],1,u2,y1,1)*(x+1)*(y+1);
110  ask:=ask-query(root[x1],1,u2,y1,2)*(y+1);
111  ask:=ask-query(root[x1],1,u2,y1,3)*(x+1);
112  ask:=ask+query(root[x1],1,u2,y1,4);
113 end;
114
115 procedure update(var p:int64;l,r,x:longint;v:int64;x1,y1:longint);
116 var mid:longint;
117 begin
118  inc(cnt); t[cnt]:=t
;
119  p:=cnt;
120  t[cnt,1]:=t[cnt,1]+v;
121  t[cnt,2]:=t[cnt,2]+v*x1;
122  t[cnt,3]:=t[cnt,3]+v*y1;
123  t[cnt,4]:=t[cnt,4]+v*x1*y1;
124  if l=r then exit;
125  mid:=(l+r)>>1;
126  if x<=mid then update(t[p,5],l,mid,x,v,x1,y1)
127   else update(t[p,6],mid+1,r,x,v,x1,y1);
128 end;
129
130 begin
131  assign(input,'bzoj2874.in'); reset(input);
132  assign(output,'bzoj2874.out'); rewrite(output);
133  readln(n,m,k1,q1);
134  k:=0;
135  for i:=1 to k1 do
136  begin
137   read(x1); read(x2); read(y1); read(y2); read(tmp);
138   inc(k); c[k,1]:=x1; c[k,2]:=y1; c[k,3]:=tmp;
139   inc(k); c[k,1]:=x1; c[k,2]:=y2+1; c[k,3]:=-tmp;
140   inc(k); c[k,1]:=x2+1; c[k,2]:=y1; c[k,3]:=-tmp;
141   inc(k); c[k,1]:=x2+1; c[k,2]:=y2+1; c[k,3]:=tmp;
142   inc(u); cx[u]:=x1; cy[u]:=y1;
143   inc(u); cx[u]:=x2+1; cy[u]:=y2+1;
144  end;
145  qsortx(1,u); qsorty(1,u);
146  u1:=1;
147  for i:=2 to u do
148   if cx[i]<>cx[u1] then begin inc(u1); cx[u1]:=cx[i]; end;
149  u2:=1;
150  for i:=2 to u do
151   if cy[i]<>cy[u2] then begin inc(u2); cy[u2]:=cy[i]; end;
152
153  qsort(1,k);
154  j:=1;
155  for i:=1 to u1 do
156  begin
157   root[i]:=root[i-1];
158   while (j<=k)and(hashx(c[j,1])=i) do
159   begin
160    update(root[i],1,u2,hashy(c[j,2]),c[j,3],c[j,1],c[j,2]);
161    inc(j);
162   end;
163  end;
164  for i:=1 to q1 do
165  begin
166   readln(x,y);
167   x1:=ans mod n+1; x2:=(ans+x) mod n+1; if x1>x2 then swap(x1,x2);
168   y1:=ans mod m+1; y2:=(ans+y) mod m+1; if y1>y2 then swap(y1,y2);
169   ans:=ask(x2,y2)-ask(x1-1,y2)-ask(x2,y1-1)+ask(x1-1,y1-1);
170   writeln(ans);
171  end;
172  close(input);
173  close(output);
174 end.