关于fft的一点总结

好吧，其实我并没有深入运用fft，只会优化卷积

听说fft经常和生成函数结合在一起………………oi真是迅猛发展，我真是与时代脱节了……

关于fft的学习推荐直接去看算法导论，写得非常清楚

主要弄懂n次单位根的相关性质定理（消去定理，折半定理）即可，当然也可以直接背代码……

bzoj2179

模板题，fft可以优化高精度乘法

顺便说一句，pascal可以定义operator，但跑得慢

这题我跑了10s……

 uses math;
 type point=record
        x,y:double;
      end;
      arr=array[..] of point;

 var a,b,c:arr;
     d,r:array[..] of longint;
     i,n,m,l:longint;
     s:ansistring;

 operator +(a,b:point)c:point;
   begin
     c.x:=a.x+b.x;
     c.y:=a.y+b.y;
   end;

 operator -(a,b:point)c:point;
   begin
     c.x:=a.x-b.x;
     c.y:=a.y-b.y;
   end;

 operator *(a,b:point)c:point;
   begin
     c.x:=a.x*b.x-a.y*b.y;
     c.y:=a.x*b.y+a.y*b.x;
   end;

 procedure fft(var a:arr; ty:longint);
   var i,j,s,k:longint;
       w,p,u,v:point;
   begin
     for i:= to n- do
       if i<r[i] then
       begin
         w:=a[r[i]];
         a[r[i]]:=a[i];
         a[i]:=w;
       end;
     i:=;
     while i<n do
     begin
       p.x:=cos(pi/i);
       p.y:=ty*sin(pi/i);
       s:=i shl ;
       j:=;
       while j<n do
       begin
         w.x:=; w.y:=;
         k:=;
         while k<i do
         begin
           u:=a[j+k];
           v:=w*a[j+k+i];
           a[j+k]:=u+v;
           a[j+k+i]:=u-v;
           w:=w*p;
           inc(k);
         end;
         inc(j,s);
       end;
       i:=i shl ;
     end;
   end;

 begin
   readln(n);
   readln(s);
   for i:= to n- do
     a[n--i].x:=ord(s[i+])-;
   readln(s);
   for i:= to n- do
     b[n--i].x:=ord(s[i+])-;
   dec(n);
   m:=*n;
   n:=;
   l:=;
   while n<=m do
   begin
     n:=n shl ;
     inc(l);
   end;
   for i:= to n- do
     r[i]:=(r[i shr ] shr ) or ((i and ) shl (l-));
   fft(a,); fft(b,);
   for i:= to n- do
     c[i]:=a[i]*b[i];
   fft(c,-);
   i:=;
   while i<=m do
   begin
     d[i]:=d[i]+trunc(round(c[i].x/n));
     if d[i]>= then
     begin
       d[i+]:=d[i+]+d[i] div ;
       d[i]:=d[i] mod ;
     end;
     if d[m+]> then inc(m);
     inc(i);
   end;
   for i:=m downto  do
     write(d[i]);
   writeln;
 end.

bzoj2194

模板题，倒过来就变成卷积了

不用operator就跑的快多了

 type point=record
        x,y:double;
      end;
      arr=array[..] of point;

 var a,b,c:arr;
     r:array[..] of longint;
     h,i,n,l,m:longint;

 procedure fft(var a:arr; ty:longint);
   var i,j,k,s:longint;
       u,v,p,w:point;
   begin
     for i:= to n- do
       if i<r[i] then
       begin
         w:=a[r[i]];
         a[r[i]]:=a[i];
         a[i]:=w;
       end;

     i:=;
     while i<n do
     begin
       p.x:=cos(pi/i);
       p.y:=ty*sin(pi/i);
       s:=i shl ;
       j:=;
       while j<n do
       begin
         w.x:=;
         w.y:=;
         for k:= to i- do
         begin
           u:=a[j+k];
           v.x:=w.x*a[j+k+i].x-w.y*a[j+k+i].y;
           v.y:=w.x*a[j+k+i].y+w.y*a[j+k+i].x;
           a[j+k].x:=u.x+v.x;
           a[j+k].y:=u.y+v.y;
           a[j+k+i].x:=u.x-v.x;
           a[j+k+i].y:=u.y-v.y;
           u:=w;
           w.x:=u.x*p.x-u.y*p.y;
           w.y:=u.x*p.y+u.y*p.x;
         end;
         inc(j,s);
       end;
       i:=i shl ;
     end;
   end;

 begin
   readln(n);
   h:=n;
   for i:= to n- do
     readln(a[i].x,b[n--i].x);
   m:=*n-;
   n:=;
   while n<=m do
   begin
     n:=n*;
     inc(l);
   end;
   for i:= to n- do
     r[i]:=(r[i shr ] shr ) or ((i and ) shl (l-));
   fft(a,);
   fft(b,);
   for i:= to n- do
   begin
     c[i].x:=a[i].x*b[i].x-a[i].y*b[i].y;
     c[i].y:=a[i].x*b[i].y+a[i].y*b[i].x;
   end;
   fft(c,-);
   for i:=h- to m do
     writeln(round(c[i].x/n));

 end.

bzoj3527

给一下题面吧，， 令 Ei=Fi/qi，求Ei

带进去稍微弄弄就是卷积啊……

 type point=record
        x,y:extended;
      end;
      arr=array[..] of point;

 var a,b,c:arr;
     ans,d:array[..] of extended;
     r:array[..] of longint;
     i,n,m,l:longint;

 procedure fft(var a:arr; ty:longint);
   var i,j,k,s:longint;
       w,p,u,v:point;
   begin
     for i:= to n- do
       if i<r[i] then
       begin
         w:=a[r[i]];
         a[r[i]]:=a[i];
         a[i]:=w;
       end;

     i:=;
     while i<n do
     begin
       p.x:=cos(pi/i);
       p.y:=ty*sin(pi/i);
       s:=i shl ;
       j:=;
       while j<n do
       begin
         w.x:=;
         w.y:=;
         for k:= to i- do
         begin
           u:=a[j+k];
           v.x:=w.x*a[j+k+i].x-w.y*a[j+k+i].y;
           v.y:=w.x*a[j+k+i].y+w.y*a[j+k+i].x;
           a[j+k].x:=u.x+v.x;
           a[j+k].y:=u.y+v.y;
           a[j+k+i].x:=u.x-v.x;
           a[j+k+i].y:=u.y-v.y;
           u:=w;
           w.x:=u.x*p.x-u.y*p.y;
           w.y:=u.x*p.y+u.y*p.x;
         end;
         inc(j,s);
       end;
       i:=i shl ;
     end;
     if ty=- then
     begin
       for i:= to n- do
         a[i].x:=a[i].x/extended(n);
      end;
   end;

 begin
   readln(n);
   m:=n;
   n:=;
   while n<=*(m-) do
   begin
     n:=n shl ;
     inc(l);
   end;
   for i:= to n- do
     r[i]:=(r[i shr ] shr ) or ((i and ) shl (l-));
   for i:= to m- do
     readln(d[i]);
   for i:= to m- do
   begin
     a[i].x:=d[i];
     if i<> then b[i].x:=/i/i;
   end;
   fft(a,); fft(b,);
   for i:= to n- do
   begin
     c[i].x:=a[i].x*b[i].x-a[i].y*b[i].y;
     c[i].y:=a[i].x*b[i].y+a[i].y*b[i].x;
   end;
   fft(c,-);
   for i:= to m- do
     ans[i]:=c[i].x;

   for i:= to n- do
   begin
     a[i].x:=;
     a[i].y:=;
     if i<m then a[i].x:=d[m--i];
   end;
   fft(a,);
   for i:= to n- do
   begin
     c[i].x:=a[i].x*b[i].x-a[i].y*b[i].y;
     c[i].y:=a[i].x*b[i].y+a[i].y*b[i].x;
   end;
   fft(c,-);
   for i:= to m- do
   begin
     ans[i]:=ans[i]-c[m-i-].x;
     writeln(ans[i]::);
   end;
 end.

bzoj3160

好题，首先合法方案=不连续间隔相等的回文-连续的回文

连续回文的方案显然可以manacher搞出来

连续的呢？考虑穷举中心点，我们只要快算计算中间点两边的相等间隔字符相等的对数s

则此中心点贡献的方案数即为2^s-1

设中心点为k，则s=sigama([a(i)=a(k-i)]) 这里k代表的是manacher翻倍后的下标，i为原串的下标

考虑字符只有a，b两种，我们只要分别统计即可

当统计a时，把a标为1，b标为0，则[a(i)=a(k-i)]=a(i)*a(k-i) 卷积就出现了，fft搞搞即可

注意inline可以加快operator的速度，但要慎用，有时候会出错

 uses math;
 const mo=;
 type point=record
        x,y:double;
      end;
      arr=array[..] of point;

 var a,b:arr;
     p,d,r:array[..] of longint;
     c:array[..] of char;
     s:ansistring;
     x,n,l,len,i,ans:longint;

 operator +(a,b:point)c:point;inline;
   begin
     c.x:=a.x+b.x;
     c.y:=a.y+b.y;
   end;

 operator -(a,b:point)c:point;inline;
   begin
     c.x:=a.x-b.x;
     c.y:=a.y-b.y;
   end;

 operator *(a,b:point)c:point;inline;
   begin
     c.x:=a.x*b.x-a.y*b.y;
     c.y:=a.x*b.y+a.y*b.x;
   end;

 function min(a,b:longint):longint;
   begin
     if a>b then exit(b) else exit(a);
   end;

 function manacher:longint;
   var w,t,i,right,k:longint;
   begin
     c[]:='$';
     t:=;
     for i:= to len do
     begin
       inc(t); c[t]:='#';
       inc(t); c[t]:=s[i];
     end;
     c[t+]:='#';
     right:=; k:=;
     w:=;
     for i:= to t do
     begin
       if right>i then p[i]:=min(p[*k-i],right-i)
       else p[i]:=;
       while c[i+p[i]]=c[i-p[i]] do inc(p[i]);
       w:=(w+p[i] div ) mod mo;
       if p[i]+i>right then
       begin
         right:=p[i]+i;
         k:=i;
       end;
     end;
     exit(w);
   end;

 procedure fft(var a:arr; ty:longint);
   var i,j,k,s:longint;
       w,p,u,v:point;
   begin
     for i:= to n- do
       if i<r[i] then
       begin
         w:=a[r[i]];
         a[r[i]]:=a[i];
         a[i]:=w;
       end;
     i:=;
     while i<n do
     begin
       p.x:=cos(pi/i);
       p.y:=ty*sin(pi/i);
       s:=i shl ;
       j:=;
       while j<n do
       begin
         w.x:=; w.y:=;
         for k:= to i- do
         begin
           u:=a[j+k];
           v:=w*a[j+k+i];
           a[j+k]:=u+v;
           a[j+k+i]:=u-v;
           w:=w*p;
         end;
         inc(j,s);
       end;
       i:=i shl ;
     end;

   end;
 begin
   readln(s);
   len:=length(s);
   d[]:=;
   for i:= to len do
     d[i]:=d[i-]* mod mo;
   n:=;
   l:=;
   while n<=(len-) shl  do
   begin
     n:=n*;
     inc(l);
   end;
   for i:= to n- do
     r[i]:=(r[i shr ] shr ) or ((i and ) shl (l-));
   for i:= to len do
     if s[i]='a' then a[i-].x:=;
   fft(a,);
   for i:= to n- do
   begin
     b[i]:=a[i]*a[i];
     a[i].x:=; a[i].y:=;
   end;
   for i:= to len do
     if s[i]='b' then a[i-].x:=;
   fft(a,);
   for i:= to n- do
     b[i]:=b[i]+a[i]*a[i];
   fft(b,-);
   for i:= to n- do
   begin
     x:=trunc(round(b[i].x/n));
     ans:=(ans+d[(x+) shr ]-) mod mo;
   end;
   writeln((ans-manacher+mo) mod mo);
 end.