{"id":72835,"date":"2026-02-06T17:25:46","date_gmt":"2026-02-06T09:25:46","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/72835.html"},"modified":"2026-02-06T17:25:46","modified_gmt":"2026-02-06T09:25:46","slug":"p4717-%e3%80%90%e6%a8%a1%e6%9d%bf%e3%80%91%e5%bf%ab%e9%80%9f%e8%8e%ab%e6%af%94%e4%b9%8c%e6%96%af-%e6%b2%83%e5%b0%94%e4%bb%80%e5%8f%98%e6%8d%a2-fmt-fwt%e9%a2%98%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/72835.html","title":{"rendered":"P4717 \u3010\u6a21\u677f\u3011\u5feb\u901f\u83ab\u6bd4\u4e4c\u65af \/ \u6c83\u5c14\u4ec0\u53d8\u6362 (FMT \/ FWT)\u9898\u89e3"},"content":{"rendered":"

P4717 \u3010\u6a21\u677f\u3011\u5feb\u901f\u83ab\u6bd4\u4e4c\u65af \/ \u6c83\u5c14\u4ec0\u53d8\u6362 (FMT \/ FWT)<\/h2>\n

\u9898\u76ee\u63cf\u8ff0<\/h3>\n

\u7ed9\u5b9a\u957f\u5ea6\u4e3a 2n2^n<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e24\u4e2a\u5e8f\u5217 A,BA,B<\/span><\/span>A<\/span>,<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span>，\u8bbe<\/p>\n

Ci=\u2211j\u2295k=iAj\u00d7BkC_i=\\\\sum_{j\\\\oplus k = i}A_j \\\\times B_k<\/span><\/span>C<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>j<\/span>\u2295<\/span>k<\/span>=<\/span>i<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>B<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5206\u522b\u5f53 \u2295\\\\oplus<\/span><\/span>\u2295<\/span><\/span><\/span><\/span><\/span> \u662f or, and, xor \u65f6\u6c42\u51fa CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8f93\u5165\u683c\u5f0f<\/h3>\n

\u7b2c\u4e00\u884c，\u4e00\u4e2a\u6574\u6570 nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u3002
\n\u7b2c\u4e8c\u884c，2n2^n<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u6570 A0,A1,\u2026,A2n\u22121A_0, A_1, \\\\ldots, A_{2^n-1}<\/span><\/span>A<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>A<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>A<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002
\n\u7b2c\u4e09\u884c，2n2^n<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u6570 B0,B1,\u2026,B2n\u22121B_0, B_1, \\\\ldots, B_{2^n-1}<\/span><\/span>B<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>B<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>B<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8f93\u51fa\u683c\u5f0f<\/h3>\n

\u4e09\u884c，\u6bcf\u884c 2n2^n<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u6570，\u5206\u522b\u4ee3\u8868 \u2295\\\\oplus<\/span><\/span>\u2295<\/span><\/span><\/span><\/span><\/span> \u662f or, and, xor \u65f6 C0,C1,\u2026,C2n\u22121C_0, C_1, \\\\ldots, C_{2^n-1}<\/span><\/span>C<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>C<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>C<\/span><\/span>2<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u503c \u200amod\u200a\u00a0998244353\\\\bmod\\\\ 998244353<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span>\u00a0<\/span>998244353<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8f93\u5165\u8f93\u51fa\u6837\u4f8b #1<\/h3>\n

\u8f93\u5165 #1<\/h4>\n

2
\n2 4 6 8
\n1 3 5 7<\/p>\n

\u8f93\u51fa #1<\/h4>\n

2 22 46 250
\n88 64 112 56
\n100 92 68 60<\/p>\n

\u8bf4\u660e\/\u63d0\u793a<\/h3>\n

1\u2264n\u2264171 \\\\le n \\\\le 17<\/span><\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>17<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u601d\u8def<\/h3>\n

\u6a21\u677f\u9898，\u76f4\u63a5\u5199\u5373\u53ef\u3002<\/p>\n

\u4ee3\u7801\u89c1\u4e0b<\/h3>\n

#<\/span>include<\/span><bits\/stdc++.h><\/span> <\/span>
\nusing<\/span> namespace<\/span> std;<\/span>
\nlong<\/span> long<\/span> n,<\/span>m,<\/span>aa[<\/span>200005<\/span>]<\/span>,<\/span>bb[<\/span>200005<\/span>]<\/span>,<\/span>a[<\/span>200005<\/span>]<\/span>,<\/span>b[<\/span>200005<\/span>]<\/span>;<\/span>
\nconst<\/span> long<\/span> long<\/span> mod=<\/span>998244353<\/span>;<\/span>
\nvoid<\/span> or1<\/span>(<\/span>long<\/span> long<\/span> *<\/span>f,<\/span>long<\/span> long<\/span> x)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> o=<\/span>2<\/span>,<\/span>k=<\/span>1<\/span>;<\/span>o<=<\/span>n;<\/span>o*=<\/span>2<\/span>,<\/span>k*=<\/span>2<\/span>)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i+=<\/span>o)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> j=<\/span>0<\/span>;<\/span>j<=<\/span>k–<\/span>1<\/span>;<\/span>j++<\/span>)<\/span>{<\/span>
\n f[<\/span>i+<\/span>j+<\/span>k]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>+<\/span>f[<\/span>i+<\/span>j]<\/span>*<\/span>x)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n }<\/span>
\n }<\/span>
\n return<\/span> ;<\/span>
\n}<\/span>
\nvoid<\/span> and1<\/span>(<\/span>long<\/span> long<\/span> *<\/span>f,<\/span>long<\/span> long<\/span> x)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> o=<\/span>2<\/span>,<\/span>k=<\/span>1<\/span>;<\/span>o<=<\/span>n;<\/span>o*=<\/span>2<\/span>,<\/span>k*=<\/span>2<\/span>)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i+=<\/span>o)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> j=<\/span>0<\/span>;<\/span>j<=<\/span>k–<\/span>1<\/span>;<\/span>j++<\/span>)<\/span>{<\/span>
\n f[<\/span>i+<\/span>j]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j]<\/span>+<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>*<\/span>x)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n }<\/span>
\n }<\/span>
\n return<\/span> ;<\/span>
\n}<\/span>
\nvoid<\/span> xor1<\/span>(<\/span>long<\/span> long<\/span> *<\/span>f,<\/span>long<\/span> long<\/span> x)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> o=<\/span>2<\/span>,<\/span>k=<\/span>1<\/span>;<\/span>o<=<\/span>n;<\/span>o*=<\/span>2<\/span>,<\/span>k*=<\/span>2<\/span>)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i+=<\/span>o)<\/span>{<\/span>
\n for<\/span>(<\/span>int<\/span> j=<\/span>0<\/span>;<\/span>j<=<\/span>k–<\/span>1<\/span>;<\/span>j++<\/span>)<\/span>{<\/span>
\n f[<\/span>i+<\/span>j]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j]<\/span>+<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>)<\/span>%<\/span>mod;<\/span>
\n f[<\/span>i+<\/span>j+<\/span>k]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j]<\/span>–<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>–<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>+<\/span>mod*<\/span>2<\/span>)<\/span>%<\/span>mod;<\/span>
\n f[<\/span>i+<\/span>j]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j]<\/span>*<\/span>x)<\/span>%<\/span>mod;<\/span>
\n f[<\/span>i+<\/span>j+<\/span>k]<\/span>=<\/span>(<\/span>f[<\/span>i+<\/span>j+<\/span>k]<\/span>*<\/span>x)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n }<\/span>
\n }<\/span>
\n return<\/span> ;<\/span>
\n}<\/span>
\nint<\/span> main<\/span>(<\/span>)<\/span>{<\/span>
\n cin>><\/span>m;<\/span>
\n n=<\/span>pow<\/span>(<\/span>2<\/span>,<\/span>m)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n cin>><\/span>aa[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n cin>><\/span>bb[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>aa[<\/span>i]<\/span>;<\/span>
\n b[<\/span>i]<\/span>=<\/span>bb[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n or1<\/span>(<\/span>a,<\/span>1<\/span>)<\/span>;<\/span>
\n or1<\/span>(<\/span>b,<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>(<\/span>a[<\/span>i]<\/span>*<\/span>b[<\/span>i]<\/span>)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n or1<\/span>(<\/span>a,<\/span>mod–<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n cout<<<\/span>a[<\/span>i]<\/span><<<\/span>" "<\/span>;<\/span>
\n }<\/span>
\n cout<<<\/span>endl;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>aa[<\/span>i]<\/span>;<\/span>
\n b[<\/span>i]<\/span>=<\/span>bb[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n and1<\/span>(<\/span>a,<\/span>1<\/span>)<\/span>;<\/span>
\n and1<\/span>(<\/span>b,<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>(<\/span>a[<\/span>i]<\/span>*<\/span>b[<\/span>i]<\/span>)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n and1<\/span>(<\/span>a,<\/span>mod–<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n cout<<<\/span>a[<\/span>i]<\/span><<<\/span>" "<\/span>;<\/span>
\n }<\/span>
\n cout<<<\/span>endl;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>aa[<\/span>i]<\/span>;<\/span>
\n b[<\/span>i]<\/span>=<\/span>bb[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n xor1<\/span>(<\/span>a,<\/span>1<\/span>)<\/span>;<\/span>
\n xor1<\/span>(<\/span>b,<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n a[<\/span>i]<\/span>=<\/span>(<\/span>a[<\/span>i]<\/span>*<\/span>b[<\/span>i]<\/span>)<\/span>%<\/span>mod;<\/span>
\n }<\/span>
\n xor1<\/span>(<\/span>a,<\/span>mod\/<\/span>2<\/span>+<\/span>1<\/span>)<\/span>;<\/span>
\n for<\/span>(<\/span>int<\/span> i=<\/span>0<\/span>;<\/span>i<=<\/span>n–<\/span>1<\/span>;<\/span>i++<\/span>)<\/span>{<\/span>
\n cout<<<\/span>a[<\/span>i]<\/span><<<\/span>" "<\/span>;<\/span>
\n }<\/span>
\n cout<<<\/span>endl;<\/span>
\nreturn<\/span> 0<\/span>;<\/span>
\n}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

P4717 \u3010\u6a21\u677f\u3011\u5feb\u901f\u83ab\u6bd4\u4e4c\u65af \/ \u6c83\u5c14\u4ec0\u53d8\u6362 (FMT \/ FWT)
\n\u9898\u76ee\u63cf\u8ff0
\n\u7ed9\u5b9a\u957f\u5ea6\u4e3a 2n2^n2n \u4e24\u4e2a\u5e8f\u5217 A,BA,BA,B，\u8bbe
\nCi\u2211j\u2295kiAj\u00d7BkC_i\\\\sum_{j\\\\oplus k i}A_j \\\\times B_kCi\u200bj\u2295ki\u2211\u200bAj\u200b\u00d7Bk\u200b
\n\u5206\u522b\u5f53 \u2295\\\\oplus\u2295 \u662f or, and, xor \u65f6\u6c42\u51fa CCC\u3002
\n\u8f93\u5165\u683c\u5f0f
\n\u7b2c\u4e00\u884c，\u4e00\u4e2a\u6574\u6570 nnn\u3002
\n\u7b2c\u4e8c\u884c，2n2^n2n \u4e2a\u6570 A0,A1,\u2026,A2<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,1813,427],"topic":[],"class_list":["post-72835","post","type-post","status-publish","format-standard","hentry","category-server","tag-c","tag-1813","tag-427"],"yoast_head":"\nP4717 \u3010\u6a21\u677f\u3011\u5feb\u901f\u83ab\u6bd4\u4e4c\u65af \/ \u6c83\u5c14\u4ec0\u53d8\u6362 (FMT \/ FWT)\u9898\u89e3 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/72835.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"P4717 \u3010\u6a21\u677f\u3011\u5feb\u901f\u83ab\u6bd4\u4e4c\u65af \/ \u6c83\u5c14\u4ec0\u53d8\u6362 (FMT \/ FWT)\u9898\u89e3 - 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