POJ 2942(Tarjen的点双连通分量+交叉染色法)

内容目录

这题是点双连通分量,我一开始写成边的……

首先点双连通分量可能重叠……(1,2) (2,3) (3,1) (3,4) (4,5) (5,6) (3.6)

这时有(1,2,3)和(3,4,5,6)两组双连通分量

故一定要在Tarjen里判……另外Stack存边时注意特判(Stack不能为空)

当dfs[k]<=low[i] 时 就找到了点双连通分量 (这是点双连通)

请注意在推送(k,i)后遍历得到的一堆边(包括(k,i))组成了一组点双连通分量

两个重要定理:

(1)       如果一个双连通分量内的某些顶点在一个奇圈中(即双连通分量含有奇圈),那么这个双连通分量的其他顶点也在某个奇圈中;

(2)       如果一个双连通分量含有奇圈,则他必定不是一个二分图。反过来也成立,这是一个充要条件。

Program P2942;
const
   maxn=1050;
   maxm=1000000;
Var
   n,m,i,j,x,y:longint;
   b:array[1..maxn,1..maxn] of boolean;

   color,c,dfs,low:array[1..maxn] of longint;

   attend,flag:array[1..maxn] of boolean;

   size,time,totssc:longint;

   stack:array[0..maxm,1..2] of longint;

function min(a,b:longint):longint;
begin
   if a<b then exit(a) else exit(b);
end;
function is_Binary(k,col:longint):boolean;
var
   i,j:longint;
begin
   color[k]:=col;
   for i:=1 to n do
      if (flag[i]) and (b[k,i]) then
      begin
         if color[i]=0 then
         begin
            if not(is_binary(i,3-col)) then exit(false);
         end
         else
         if color[k]=color[i] then exit(false);
      end;

   exit(true);
end;

procedure tarjen(k,father:longint);
var
   i,j:longint;
begin
   inc(time);
   dfs[k]:=time;
   low[k]:=time;

   c[k]:=1;
   for i:=1 to n do
      if (b[k,i]) and (dfs[k]>dfs[i]) and (i<>father) then
      begin
         if dfs[i]=0 then
         begin

         inc(size);
         stack[size,1]:=k;
         stack[size,2]:=i;



         tarjen(i,k);
         low[k]:=min(low[k],low[i]);

         if dfs[k]<=low[i] then
         begin
            fillchar(flag,sizeof(flag),false);
            fillchar(color,sizeof(color),false);
            while (size>0) do
            begin
               flag[stack[size,1]]:=true;
               flag[stack[size,2]]:=true;
               dec(size);
               if (stack[size+1,1]=k) and (stack[size+1,2]=i) then break;

            end;

            if not(is_binary(k,1)) then
            begin
               for j:=1 to n do
                  if flag[j] then attend[j]:=true;


            end;


         end;


         end
         else
         if c[i]=1 then low[k]:=min(low[k],dfs[i]);
      end;





   c[k]:=2;

end;


function main:longint;
var
   i,j,tot:longint;
begin
   fillchar(dfs,sizeof(dfs),0);
   fillchar(low,sizeof(low),0);
   fillchar(c,sizeof(c),0);
   fillchar(attend,sizeof(attend),false);
   fillchar(flag,sizeof(flag),false);
   fillchar(color,sizeof(color),0);
   fillchar(stack,sizeof(stack),0);
   size:=0;time:=0;
   for i:=1 to n do
      if (dfs[i]=0) then
      begin
          tarjen(i,0);
      end;

   tot:=0;
   for i:=1 to n do if attend[i] then inc(tot);
   exit(n-tot);

end;

begin
   while not seekeof do
   begin
      read(n,m);
      if (n=0) and (m=0) then break;
      fillchar(b,sizeof(b),true);
      for i:=1 to n do b[i,i]:=false;
      for i:=1 to m do
      begin
         read(x,y);
         b[x,y]:=false;
         b[y,x]:=false;
      end;
      writeln(main);
   end;
end.