【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

 var t:array[..,..]of int64;
     root:array[..]of int64;
     c:array[..,..]of int64;
     cx,cy:array[..]of int64;
     n,m,i,j,u,k,cnt,u1,u2,k1,q1,tmp,x,y:longint;
     ans,x1,y1,x2,y2:int64;

 procedure swap(var x,y:int64);
 var t:int64;
 begin
  t:=x; x:=y; y:=t;
 end;

 procedure qsortx(l,r:longint);
 var i,j,mid:longint;
 begin
  i:=l; j:=r; mid:=cx[(l+r)>>];
  repeat
   while mid>cx[i] do inc(i);
   while mid<cx[j] do dec(j);
   if i<=j then
   begin
    swap(cx[i],cx[j]);
    inc(i); dec(j);
   end;
  until i>j;
  if l<j then qsortx(l,j);
  if i<r then qsortx(i,r);
 end;

 procedure qsorty(l,r:longint);
 var i,j,mid:longint;
 begin
  i:=l; j:=r; mid:=cy[(l+r)>>];
  repeat
   while mid>cy[i] do inc(i);
   while mid<cy[j] do dec(j);
   if i<=j then
   begin
    swap(cy[i],cy[j]);
    inc(i); dec(j);
   end;
  until i>j;
  if l<j then qsorty(l,j);
  if i<r then qsorty(i,r);
 end;

 function hashx(x:longint):longint;
 var l,r,mid,last:longint;
 begin
  l:=; r:=u1; last:=;
  while l<=r do
  begin
   mid:=(l+r)>>;
   if x>=cx[mid] then begin last:=mid; l:=mid+; end
    else r:=mid-;
  end;
  exit(last);
 end;

 function hashy(x:longint):longint;
 var l,r,mid,last:longint;
 begin
  l:=; r:=u2; last:=;
  while l<=r do
  begin
   mid:=(l+r)>>;
   if x>=cy[mid] then begin last:=mid; l:=mid+; end
    else r:=mid-;
  end;
  exit(last);
 end;

 procedure qsort(l,r:longint);
 var i,j,mid1,mid2:longint;
 begin
  i:=l; j:=r; mid1:=c[(l+r)>>,]; mid2:=c[(l+r)>>,];
  repeat
   while (mid1>c[i,])or((mid1=c[i,])and(mid2>c[i,])) do inc(i);
   while (mid1<c[j,])or((mid1=c[j,])and(mid2<c[j,])) do dec(j);
   if i<=j then
   begin
    swap(c[i,],c[j,]);
    swap(c[i,],c[j,]);
    swap(c[i,],c[j,]);
    inc(i); dec(j);
   end;
  until i>j;
  if l<j then qsort(l,j);
  if i<r then qsort(i,r);
 end;

 function query(p,l,r,x,y:longint):int64;
 var mid:longint;
 begin
  if x= then exit();
  if p= then exit();
  if x>=r then exit(t[p,y]);
  mid:=(l+r)>>;
  if x<=mid then exit(query(t[p,],l,mid,x,y))
   else exit(t[t[p,],y]+query(t[p,],mid+,r,x,y));
 end;

 function ask(x,y:longint):int64;
 var x1,y1:longint;
 begin
  x1:=hashx(x); y1:=hashy(y);
  ask:=;
  ask:=ask+query(root[x1],,u2,y1,)*(x+)*(y+);
  ask:=ask-query(root[x1],,u2,y1,)*(y+);
  ask:=ask-query(root[x1],,u2,y1,)*(x+);
  ask:=ask+query(root[x1],,u2,y1,);
 end;

 procedure update(var p:int64;l,r,x:longint;v:int64;x1,y1:longint);
 var mid:longint;
 begin
  inc(cnt); t[cnt]:=t[p];
  p:=cnt;
  t[cnt,]:=t[cnt,]+v;
  t[cnt,]:=t[cnt,]+v*x1;
  t[cnt,]:=t[cnt,]+v*y1;
  t[cnt,]:=t[cnt,]+v*x1*y1;
  if l=r then exit;
  mid:=(l+r)>>;
  if x<=mid then update(t[p,],l,mid,x,v,x1,y1)
   else update(t[p,],mid+,r,x,v,x1,y1);
 end;

 begin
  assign(input,'bzoj2874.in'); reset(input);
  assign(output,'bzoj2874.out'); rewrite(output);
  readln(n,m,k1,q1);
  k:=;
  for i:= to k1 do
  begin
   read(x1); read(x2); read(y1); read(y2); read(tmp);
   inc(k); c[k,]:=x1; c[k,]:=y1; c[k,]:=tmp;
   inc(k); c[k,]:=x1; c[k,]:=y2+; c[k,]:=-tmp;
   inc(k); c[k,]:=x2+; c[k,]:=y1; c[k,]:=-tmp;
   inc(k); c[k,]:=x2+; c[k,]:=y2+; c[k,]:=tmp;
   inc(u); cx[u]:=x1; cy[u]:=y1;
   inc(u); cx[u]:=x2+; cy[u]:=y2+;
  end;
  qsortx(,u); qsorty(,u);
  u1:=;
  for i:= to u do
   if cx[i]<>cx[u1] then begin inc(u1); cx[u1]:=cx[i]; end;
  u2:=;
  for i:= to u do
   if cy[i]<>cy[u2] then begin inc(u2); cy[u2]:=cy[i]; end;

  qsort(,k);
  j:=;
  for i:= to u1 do
  begin
   root[i]:=root[i-];
   while (j<=k)and(hashx(c[j,])=i) do
   begin
    update(root[i],,u2,hashy(c[j,]),c[j,],c[j,],c[j,]);
    inc(j);
   end;
  end;
  for i:= to q1 do
  begin
   readln(x,y);
   x1:=ans mod n+; x2:=(ans+x) mod n+; if x1>x2 then swap(x1,x2);
   y1:=ans mod m+; y2:=(ans+y) mod m+; if y1>y2 then swap(y1,y2);
   ans:=ask(x2,y2)-ask(x1-,y2)-ask(x2,y1-)+ask(x1-,y1-);
   writeln(ans);
  end;
  close(input);
  close(output);
 end.