「杂题记录」「CTSC2017」吉夫特

📅 Last Updated: 2021-01-20

📊 Word Count: 219

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给出一个长度为 $n$ 的数列 $A_i$ ，求有多少个长度 $k$ 的子序列 $A'$ ( $k\ge 2$ )满足： $\prod_{i=1}^{k-1}\dbinom{A'_i}{A'_{i+1}} > 0 \pmod{2}$ $n \le 211985, A_i \le 233333$ 。原题保证 $A_i$ 互不相同，但是不重要。

分析

根据 Lucas 定理，就是求有多少 $A$ 的子序列 $A'$ 满足: $\forall i \in [1, k-1]，S(A_i) \subseteq S(A_{i+1})$ $S(x)$ 表示二进制数 $x$ 表示的集合。

这东西可以直接 dp ，设 $f(i)$ 表示以 $i$ 结尾的合法子序列有多少： $f(n) \sum_{i=1}^{n-1} f(i)[A_i \operatorname{and}A_n=A_n]$ 直接暴力枚举是 $\mathcal{O}(n^2)$ 的。

考虑类似于分块一样的优化，考虑将 $A_i$ 拆开，设 $A_i$ 二进制下的前 9 位为 $x$ , 后 9 位为 $y$ 。 $g(x, y)$ 表示前九位恰好为 $x$ ，后九位是 $y$ 的子集的 $A_i$ 的对应 $f(i)$ 之和。

考虑维护这个东西，求出一个 $g(i)$ 后枚举子集更新。

考虑使用这个东西，在求一个 $g(i)$ 时，枚举子集求出。

成功均摊了复杂度。总复杂度 $\mathcal{O}(2^{9}n)$

const int _ = 241985;
const int MOD = 1e9 + 7;
int A[_], n, f[_];
int g[1 << 10][1 << 10]; // g[x][y]: 当前，所有满足 A_i 的前 9 位为 x ，后 9 位为 y 的超集。 
inline int & reduce(int &x) { if(x >= MOD) x-= MOD; if(x < 0) x += MOD; return x; }
int main(){
    ios::sync_with_stdio(false);
    cin >> n; rep(i, 1, n) cin >> A[i]; // 要求前面的数字为后面的超集。 
    register int LB = (1 << 9) - 1;
    register int All = ((1 << 9) - 1);
    f[1] = 1;
    g[A[1] >> 9][0] += 1;
    register int S0 = A[1] & LB; for(register int S = (S0); S; S = (S - 1) & (S0)) reduce(g[A[1] >> 9][S] += 1);
    register int $1;
    rep(i, 2, n) {
        int now = ((1 << 9) - 1) ^ (A[i] >> 9);
        int &ans = f[i] = 1 ;  
        $1 = A[i] >> 9;
        reduce(ans += g[$1][A[i] & LB]);
        for(int S = now; S; S = (S - 1) & (now)) reduce(ans += g[S | $1][A[i] & LB]);
        reduce(g[$1][0] += ans);
        for(int S = (A[i] & LB); S; S = (S - 1) & (A[i] & LB)) reduce(g[$1][S] += ans);
    }
    int Ans = 0;
    for(int i = 1; i <= n; i++) reduce(Ans += f[i]); reduce(Ans += MOD - n);
    cout << Ans << endl;
    return 0;
}

📚 Version History

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v1 📅 2021-01-20 15:35:06

🔗 Source Hash: 53811e666d0bf43b0c4e833e63a2957b9bf6a3737a7720e625785c972befe0f5

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    pdftitle={「杂题记录」「CTSC2017」吉夫特},
    pdfauthor={Jiayi Su (ShuYuMo)},
    hidelinks,
    pdfcreator={LaTeX via pandoc}}
  
  \title{「杂题记录」「CTSC2017」吉夫特}
  \author{Jiayi Su (ShuYuMo)}
  \date{2021-01-20 15:35:06}
  
  \begin{document}
  \maketitle
  
  \setstretch{1.3}
  给出一个长度为 \(n\) 的数列 \(A_i\) ，求有多少个长度 \(k\) 的子序列
  \(A'\) (\(k\ge 2\))满足： \[
  \prod_{i=1}^{k-1}\dbinom{A'_i}{A'_{i+1}} > 0 \pmod{2}
  \] \(n \le 211985, A_i \le 233333\)。原题保证 \(A_i\)
  互不相同，但是不重要。
  
  \subsection{分析}\label{ux5206ux6790}
  
  根据 Lucas 定理，就是求有多少 \(A\) 的子序列 \(A'\) 满足: \[
  \forall i \in [1, k-1]，S(A_i) \subseteq S(A_{i+1})
  \] \(S(x)\) 表示二进制数 \(x\) 表示的集合。
  
  这东西可以直接 dp ，设 \(f(i)\) 表示以 \(i\) 结尾的合法子序列有多少： \[
  f(n) \sum_{i=1}^{n-1} f(i)[A_i \operatorname{and}A_n=A_n]
  \] 直接暴力枚举是 \(\mathcal{O}(n^2)\) 的。
  
  考虑类似于分块一样的优化，考虑将 \(A_i\) 拆开，设 \(A_i\) 二进制下的前 9
  位为 \(x\), 后 9 位为 \(y\)。 \(g(x, y)\) 表示前九位恰好为 \(x\)
  ，后九位是 \(y\) 的子集的 \(A_i\) 的对应 \(f(i)\) 之和。
  
  考虑维护这个东西，求出一个 \(g(i)\) 后枚举子集更新。
  
  考虑使用这个东西，在求一个 \(g(i)\) 时，枚举子集求出。
  
  成功均摊了复杂度。 总复杂度 \(\mathcal{O}(2^{9}n)\)
  
  \begin{Shaded}
  \begin{Highlighting}[]
  \AttributeTok{const} \DataTypeTok{int}\NormalTok{ \_ }\OperatorTok{=} \DecValTok{241985}\OperatorTok{;}
  \AttributeTok{const} \DataTypeTok{int}\NormalTok{ MOD }\OperatorTok{=} \FloatTok{1e9} \OperatorTok{+} \DecValTok{7}\OperatorTok{;}
  \DataTypeTok{int}\NormalTok{ A}\OperatorTok{[}\NormalTok{\_}\OperatorTok{],}\NormalTok{ n}\OperatorTok{,}\NormalTok{ f}\OperatorTok{[}\NormalTok{\_}\OperatorTok{];}
  \DataTypeTok{int}\NormalTok{ g}\OperatorTok{[}\DecValTok{1} \OperatorTok{\textless{}\textless{}} \DecValTok{10}\OperatorTok{][}\DecValTok{1} \OperatorTok{\textless{}\textless{}} \DecValTok{10}\OperatorTok{];} \CommentTok{// g[x][y]: 当前，所有满足 A\_i 的前 9 位为 x ，后 9 位为 y 的超集。 }
  \KeywordTok{inline} \DataTypeTok{int} \OperatorTok{\&}\NormalTok{ reduce}\OperatorTok{(}\DataTypeTok{int} \OperatorTok{\&}\NormalTok{x}\OperatorTok{)} \OperatorTok{\{} \ControlFlowTok{if}\OperatorTok{(}\NormalTok{x }\OperatorTok{\textgreater{}=}\NormalTok{ MOD}\OperatorTok{)}\NormalTok{ x}\OperatorTok{{-}=}\NormalTok{ MOD}\OperatorTok{;} \ControlFlowTok{if}\OperatorTok{(}\NormalTok{x }\OperatorTok{\textless{}} \DecValTok{0}\OperatorTok{)}\NormalTok{ x }\OperatorTok{+=}\NormalTok{ MOD}\OperatorTok{;} \ControlFlowTok{return}\NormalTok{ x}\OperatorTok{;} \OperatorTok{\}}
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      \AttributeTok{register} \DataTypeTok{int}\NormalTok{ LB }\OperatorTok{=} \OperatorTok{(}\DecValTok{1} \OperatorTok{\textless{}\textless{}} \DecValTok{9}\OperatorTok{)} \OperatorTok{{-}} \DecValTok{1}\OperatorTok{;}
      \AttributeTok{register} \DataTypeTok{int}\NormalTok{ All }\OperatorTok{=} \OperatorTok{((}\DecValTok{1} \OperatorTok{\textless{}\textless{}} \DecValTok{9}\OperatorTok{)} \OperatorTok{{-}} \DecValTok{1}\OperatorTok{);}
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          \DataTypeTok{int} \OperatorTok{\&}\NormalTok{ans }\OperatorTok{=}\NormalTok{ f}\OperatorTok{[}\NormalTok{i}\OperatorTok{]} \OperatorTok{=} \DecValTok{1} \OperatorTok{;}  
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      \OperatorTok{\}}
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  \NormalTok{    cout }\OperatorTok{\textless{}\textless{}}\NormalTok{ Ans }\OperatorTok{\textless{}\textless{}}\NormalTok{ endl}\OperatorTok{;}
      \ControlFlowTok{return} \DecValTok{0}\OperatorTok{;}
  \OperatorTok{\}}
  \end{Highlighting}
  \end{Shaded}
  
  
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  LaTeX Font Warning: Font shape `TU/ARPLUKaiCN(0)/m/it' undefined
  (Font)              using `TU/ARPLUKaiCN(0)/m/n' instead on input line 145.
  
  [1] [2] (.../input.aux)
  
  LaTeX Font Warning: Some font shapes were not available, defaults substituted.
  
  
  LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
  
   )
  Output written on .../input.pdf (2 pages).
  Transcript written on .../input.log.
[INFO] [makePDF] Rerun needed
  Package hyperref Warning: Rerun to get /PageLabels entry.
  LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
[INFO] [makePDF] LaTeX run number 2
[INFO] [makePDF] LaTeX output
  This is XeTeX, Version 3.141592653-2.6-0.999995 (TeX Live 2023/Debian) (preloaded format=xelatex)
   restricted \write18 enabled.
  entering extended mode
  (.../input.tex
  LaTeX2e <2023-11-01> patch level 1
  L3 programming layer <2024-01-22>
  (.../article.cls
  Document Class: article 2023/05/17 v1.4n Standard LaTeX document class
  (.../[system]
  (.../xcolor.sty
  (.../color.cfg)
  (.../xetex.def)
  (.../[system]
  (.../geometry.sty
  (.../keyval.sty)
  (.../ifvtex.sty
  (.../iftex.sty)))
  (.../amsmath.sty
  For additional information on amsmath, use the `?' option.
  (.../amstext.sty
  (.../amsgen.sty))
  (.../amsbsy.sty)
  (.../amsopn.sty))
  (.../amssymb.sty
  (.../amsfonts.sty))
  (.../unicode-math.sty
  (.../expl3.sty
  (.../l3backend-xetex.def))
  (.../unicode-math-xetex.sty
  (.../xparse.sty)
  (.../l3keys2e.sty)
  (.../fontspec.sty
  (.../fontspec-xetex.sty
  (.../fontenc.sty)
  (.../fontspec.cfg)))
  (.../fix-cm.sty
  (.../ts1enc.def))
  (.../unicode-math-table.tex)))
  (.../lmodern.sty)
  (.../xeCJK.sty
  (.../ctexhook.sty)
  (.../xtemplate.sty)
  (.../xeCJK.cfg))
  (.../upquote.sty
  (.../textcomp.sty))
  (.../microtype.sty
  (.../etoolbox.sty)
  (.../microtype-xetex.def)
  (.../microtype.cfg))
  (.../setspace.sty)
  (.../parskip.sty
  (.../kvoptions.sty
  (.../ltxcmds.sty)
  (.../kvsetkeys.sty)))
  (.../fancyvrb.sty)
  (.../bookmark.sty
  (.../hyperref.sty
  (.../kvdefinekeys.sty)
  (.../pdfescape.sty
  (.../pdftexcmds.sty
  (.../infwarerr.sty)))
  (.../hycolor.sty)
  (.../auxhook.sty)
  (.../nameref.sty
  (.../refcount.sty)
  (.../gettitlestring.sty))
  (.../pd1enc.def)
  (.../intcalc.sty)
  (.../puenc.def)
  (.../url.sty)
  (.../bitset.sty
  (.../bigintcalc.sty))
  (.../atbegshi-ltx.sty))
  (.../hxetex.def
  (.../stringenc.sty)
  (.../rerunfilecheck.sty
  (.../atveryend-ltx.sty)
  (.../uniquecounter.sty)))
  (.../bkm-dvipdfm.def))
  (.../xurl.sty)
  (.../input.aux)
  *geometry* driver: auto-detecting
  *geometry* detected driver: xetex
  (.../mt-LatinModernRoman.cfg)
  (.../omllmm.fd)
  (.../umsa.fd)
  (.../mt-msa.cfg)
  (.../umsb.fd)
  (.../mt-msb.cfg)
  
  LaTeX Font Warning: Font shape `TU/ARPLUKaiCN(0)/b/n' undefined
  (Font)              using `TU/ARPLUKaiCN(0)/m/n' instead on input line 120.
  
  Missing character: There is no ， (U+FF0C) in font LatinModernMath-Regular/OT:sc
  ript=math;language=dflt;!
  
  Package xeCJK Warning: Unknown CJK family `\CJKttdefault' is being ignored.
  (xeCJK)                
  (xeCJK)                Try to use `\setCJKmonofont[<...>]{<...>}' to define
  (xeCJK)                it.
  
  
  LaTeX Font Warning: Font shape `TU/ARPLUKaiCN(0)/m/it' undefined
  (Font)              using `TU/ARPLUKaiCN(0)/m/n' instead on input line 145.
  
  [1] [2] (.../input.aux)
  
  LaTeX Font Warning: Some font shapes were not available, defaults substituted.
  
   )
  Output written on .../input.pdf (2 pages).
  Transcript written on .../input.log.
[WARNING] Missing character: There is no ， (U+FF0C) (U+FF0C) in font LatinModernMath-Regular/OT:sc