圆覆盖问题算法解析-优快云博客

【题意】

给出n个圆（n<=20），求有多少个不被圆覆盖的封闭区域

【输入】

第一行t，表示有t组数据

每组数据第一行为n表示有多少个圆

接下来n行分别描述n个圆，每行为x，y，r

【输出】

对于每组数据，输出有几个不被圆覆盖的封闭区域

Sample Input

Sample Output

1
0

离散化

07年论文有讲（高逸涵《与圆有关的离散化方法》）

对于任意一行y=k，都可以求出来不被圆覆盖的区间

观察可以很轻易的发现，区域连通性的变化，其实都发生在圆与圆交点所在的那一行或者是圆的上下处

我们将这些行称之为事件点

对于每个事件点，我们求他上下稍微偏移一点的区间

因为不在通过事件点，连通性不会变化

所以如果从下往上扫描的话，对于一个事件点的下偏移那行，跟上个事件点上偏移的那行，连通性是完全相同的

当然区间数也相同

接下来就是从一个事件点的下偏移转移到上偏移

如果他们之间区间有交集，那么这两个区间属于同一个联通块

一旦事件点下偏移有一个联通块没有上偏移的区间继承，那么说明有一个封闭的区间

程序写的十分乱。

program poj1688;
const
  esp:extended=1e-8;
type
  line=record
         l,r:extended;
       end;
  node=record
         x:extended;
         incr:boolean;
       end;
var
  n,i,j,k,w,t,tot,all,total,o,ans,xx,oo,count:longint;
  x,y,r:array [0..21] of longint;
  event:array [0..601] of extended;
  xl:array [0..61] of node;
  temp:node;
  ind:array [0..10001] of line;
  own:array [0..10001] of longint;
  sum:array [0..1] of longint;
  dl:array [0..1,0..10001] of longint;
  yes:array [0..10001]  of boolean;
  root,next,point:array [0..10001] of longint;

procedure swap_int (var a,b:longint);
var
  i:longint;
begin
  i:=a;
  a:=b;
  b:=i;
end;

procedure swap_dou (var a,b:extended);
var
  i:extended;
begin
  i:=a;
  a:=b;
  b:=i;
end;

procedure solve (i,j:longint);
var
  left,right,t,mid,leftr,rightr:extended;
begin
  t:=y[j]+(y[i]-y[j])*r[j]/(r[i]+r[j]);
  left:=y[j]-r[j];
  if y[i]-r[i]>left then left:=y[i]-r[i];
  right:=t;
  repeat
    mid:=(left+right)/2;
    leftr:=sqrt(sqr(r[j])-sqr(mid-y[j]));
    rightr:=sqrt(sqr(r[i])-sqr(mid-y[i]));
    if (x[j]+leftr<x[i]-rightr) or (x[j]-leftr>x[i]+rightr) then
      left:=mid
	                                                    else
      right:=mid;
  until abs(left-right)<1e-8;
  inc(tot);
  event[tot]:=left;

  left:=t;
  right:=y[j]+r[j];
  if y[i]+r[i]<right then right:=y[i]+r[i];
  repeat
    mid:=(left+right)/2;
    leftr:=sqrt(sqr(r[j])-sqr(mid-y[j]));
    rightr:=sqrt(sqr(r[i])-sqr(mid-y[i]));
    if (x[j]+leftr<x[i]-rightr) or (x[j]-leftr>x[i]+rightr) then
      right:=mid
                                                            else
      left:=mid;
  until abs(left-right)<1e-8;
  inc(tot);
  event[tot]:=left;
end;

procedure qsort (s,e:longint);
var
  i,j:longint;
  k:extended;
begin
  if s>=e then exit;
  i:=s;
  j:=e;
  k:=event[(s+e) div 2];
  while i<=j do
    begin
      while (i<=j)and(event[i]<k) do inc(i);
      while (i<=j)and(event[j]>k) do dec(j);
      if i>j then break;
      swap_dou(event[i],event[j]);
      inc(i);
      dec(j);
    end;
  qsort(s,j);
  qsort(i,e);
end;

procedure connect (s,e:longint);
begin
  inc(oo);
  point[oo]:=e;
  next[oo]:=root[s];;
  root[s]:=oo;
end;

procedure dfs (now:longint);
var
  i:longint;
begin
  yes[now]:=true;
  i:=root[now];
  while i<>0 do
    begin
      if not yes[point[i]] then
        begin
          own[point[i]]:=own[now];
          if point[i]>=dl[o,1] then inc(count);
          dfs(point[i]);
        end;
      i:=next[i];
    end;
end;

procedure merge (now:longint);
var
  i,j,k:longint;
  a,b,t:extended;
begin
  o:=1-o;
  t:=event[now]-esp;
  total:=0;
  for i:=1 to n do
    if sqr(y[i]-t)<sqr(r[i]) then
      begin
        inc(total);
        xl[total].x:=x[i]-sqrt(sqr(r[i])-sqr(y[i]-t));
        xl[total].incr:=true;
        inc(total);
        xl[total].x:=x[i]+sqrt(sqr(r[i])-sqr(y[i]-t));
        xl[total].incr:=false;
      end;
  for i:=1 to total-1 do
    for j:=i+1 to total do
      if xl[i].x>xl[j].x then
        begin
          temp:=xl[i];
          xl[i]:=xl[j];
          xl[j]:=temp;
        end;
  k:=0;
  w:=1;
  for i:=1 to total do
    if xl[i].incr then
      begin
        if k=0 then
          begin
            ind[dl[1-o,w]].r:=xl[i].x;
            inc(w);
          end;
        inc(k);
      end
                  else
      begin
        dec(k);
        if k=0 then ind[dl[1-o,w]].l:=xl[i].x;
      end;
  t:=event[now]+esp;
  total:=0;
  for i:=1 to n do
    if sqr(y[i]-t)<sqr(r[i]) then
      begin
        inc(total);
        xl[total].x:=x[i]-sqrt(sqr(r[i])-sqr(y[i]-t));
        xl[total].incr:=true;
        inc(total);
        xl[total].x:=x[i]+sqrt(sqr(r[i])-sqr(y[i]-t));
        xl[total].incr:=false;
      end;
  for i:=1 to total-1 do
    for j:=i+1 to total do
      if xl[i].x>xl[j].x then
        begin
          temp:=xl[i];
          xl[i]:=xl[j];
          xl[j]:=temp;
        end;
  sum[o]:=1;
  inc(all);
  dl[o,1]:=all;
  ind[all].l:=-1000000;
  k:=0;
  for i:=1 to total do
    if xl[i].incr then
      begin
        if k=0 then ind[all].r:=xl[i].x;
        inc(k);
      end
                  else
      begin
        dec(k);
        if k=0 then
          begin
            inc(all);
            inc(sum[o]);
            dl[o,sum[o]]:=all;
            ind[all].l:=xl[i].x;
          end;
      end;
  ind[all].r:=1000000;
  for i:=1 to sum[o] do
    yes[dl[o,i]]:=false;
  for i:=1 to sum[1-o] do
    yes[dl[1-o,i]]:=false;
  for i:=1 to sum[1-o] do
    root[dl[1-o,i]]:=0;
  oo:=0;
  for i:=1 to sum[1-o] do
    for j:=1 to sum[1-o] do
      if (i<>j)and(own[dl[1-o,i]]=own[dl[1-o,j]]) then
        begin
          connect(dl[1-o,i],dl[1-o,j]);
          connect(dl[1-o,j],dl[1-o,i]);
        end;
  for i:=1 to sum[1-o] do
    for j:=1 to sum[o] do
      if not ((ind[dl[1-o,i]].l>ind[dl[o,j]].r)or(ind[dl[1-o,i]].r<ind[dl[o,j]].l)) then
        begin
          connect(dl[1-o,i],dl[o,j]);
          connect(dl[o,j],dl[1-o,i]);
        end;
  for i:=1 to sum[1-o] do
    if not yes[dl[1-o,i]] then
      begin
        count:=0;
        dfs(dl[1-o,i]);
        if count=0 then inc(ans);
      end;
  for i:=1 to sum[o] do
    if not yes[dl[o,i]] then
      begin
        inc(xx);
        own[dl[o,i]]:=xx;
      end;
end;

begin
  read(t);
  while t<>0 do
    begin
      fillchar(root,sizeof(root),false);
      fillchar(next,sizeof(next),0);
      read(n);
      for i:=1 to n do
        read(x[i],y[i],r[i]);
      for i:=1 to n-1 do
        for j:=i+1 to n do
          if x[i]-r[i]>x[j]-r[j] then
            begin
              swap_int(x[i],x[j]);
              swap_int(y[i],y[j]);
              swap_int(r[i],r[j]);
            end;
      tot:=0;
      for i:=1 to n do
        begin
          inc(tot);
          event[tot]:=y[i]-r[i];
          inc(tot);
          event[tot]:=y[i]+r[i];
        end;
      for i:=1 to n-1 do
        for j:=i+1 to n do
          if sqr(x[i]-x[j])+sqr(y[i]-y[j])<sqr(r[i]+r[j]) then
            solve(i,j);
      qsort(1,tot);
      event[0]:=-10000000000;
      o:=0;
      sum[o]:=1;
      all:=1;
      ans:=0;
      xx:=1;
      dl[o,1]:=1;
      own[1]:=1;
      ind[1].l:=-1000000;
      ind[1].r:=1000000;
      for i:=1 to tot do
        if event[i]-event[i-1]>esp then
          merge(i);
      writeln(ans);
      dec(t);
    end;
end.