min-max 容斥学习笔记
Posted qq62c30ac77b2a7
tags:
篇首语：本文由小常识网(cha138.com)小编为大家整理，主要介绍了min-max 容斥学习笔记相关的知识，希望对你有一定的参考价值。
关于 %22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6D%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-69%22%20x%3D%22878%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%20x%3D%221224%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2212%22%20x%3D%222046%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6D%22%20x%3D%223047%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%223925%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%224455%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
容斥，我们希望构造一个系数使得%3D%5Csum_%7BT%5Csubseteq%20S%7Df_%7B%7CT%7C%7Dmin(T)%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-6D%22%20d%3D%22M21%20287Q22%20293%2024%20303T36%20341T56%20388T88%20425T132%20442T175%20435T205%20417T221%20395T229%20376L231%20369Q231%20367%20232%20367L243%20378Q303%20442%20384%20442Q401%20442%20415%20440T441%20433T460%20423T475%20411T485%20398T493%20385T497%20373T500%20364T502%20357L510%20367Q573%20442%20659%20442Q713%20442%20746%20415T780%20336Q780%20285%20742%20178T704%2050Q705%2036%20709%2031T724%2026Q752%2026%20776%2056T815%20138Q818%20149%20821%20151T837%20153Q857%20153%20857%20145Q857%20144%20853%20130Q845%20101%20831%2073T785%2017T716%20-10Q669%20-10%20648%2017T627%2073Q627%2092%20663%20193T700%20345Q700%20404%20656%20404H651Q565%20404%20506%20303L499%20291L466%20157Q433%2026%20428%2016Q415%20-11%20385%20-11Q372%20-11%20364%20-4T353%208T350%2018Q350%2029%20384%20161L420%20307Q423%20322%20423%20345Q423%20404%20379%20404H374Q288%20404%20229%20303L222%20291L189%20157Q156%2026%20151%2016Q138%20-11%20108%20-11Q95%20-11%2087%20-5T76%207T74%2017Q74%2030%20112%20181Q151%20335%20151%20342Q154%20357%20154%20369Q154%20405%20129%20405Q107%20405%2092%20377T69%20316T57%20280Q55%20278%2041%20278H27Q21%20284%2021%20287Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-61%22%20d%3D%22M33%20157Q33%20258%20109%20349T280%20441Q331%20441%20370%20392Q386%20422%20416%20422Q429%20422%20439%20414T449%20394Q449%20381%20412%20234T374%2068Q374%2043%20381%2035T402%2026Q411%2027%20422%2035Q443%2055%20463%20131Q469%20151%20473%20152Q475%20153%20483%20153H487Q506%20153%20506%20144Q506%20138%20501%20117T481%2063T449%2013Q436%200%20417%20-8Q409%20-10%20393%20-10Q359%20-10%20336%205T306%2036L300%2051Q299%2052%20296%2050Q294%2048%20292%2046Q233%20-10%20172%20-10Q117%20-10%2075%2030T33%20157ZM351%20328Q351%20334%20346%20350T323%20385T277%20405Q242%20405%20210%20374T160%20293Q131%20214%20119%20129Q119%20126%20119%20118T118%20106Q118%2061%20136%2044T179%2026Q217%2026%20254%2059T298%20110Q300%20114%20325%20217T351%20328Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-78%22%20d%3D%22M52%20289Q59%20331%20106%20386T222%20442Q257%20442%20286%20424T329%20379Q371%20442%20430%20442Q467%20442%20494%20420T522%20361Q522%20332%20508%20314T481%20292T458%20288Q439%20288%20427%20299T415%20328Q415%20374%20465%20391Q454%20404%20425%20404Q412%20404%20406%20402Q368%20386%20350%20336Q290%20115%20290%2078Q290%2050%20306%2038T341%2026Q378%2026%20414%2059T463%20140Q466%20150%20469%20151T485%20153H489Q504%20153%20504%20145Q504%20144%20502%20134Q486%2077%20440%2033T333%20-11Q263%20-11%20227%2052Q186%20-10%20133%20-10H127Q78%20-10%2057%2016T35%2071Q35%20103%2054%20123T99%20143Q142%20143%20142%20101Q142%2081%20130%2066T107%2046T94%2041L91%2040Q91%2039%2097%2036T113%2029T132%2026Q168%2026%20194%2071Q203%2087%20217%20139T245%20247T261%20313Q266%20340%20266%20352Q266%20380%20251%20392T217%20404Q177%20404%20142%20372T93%20290Q91%20281%2088%20280T72%20278H58Q52%20284%2052%20289Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-73%22%20d%3D%22M131%20289Q131%20321%20147%20354T203%20415T300%20442Q362%20442%20390%20415T419%20355Q419%20323%20402%20308T364%20292Q351%20292%20340%20300T328%20326Q328%20342%20337%20354T354%20372T367%20378Q368%20378%20368%20379Q368%20382%20361%20388T336%20399T297%20405Q249%20405%20227%20379T204%20326Q204%20301%20223%20291T278%20274T330%20259Q396%20230%20396%20163Q396%20135%20385%20107T352%2051T289%207T195%20-10Q118%20-10%2086%2019T53%2087Q53%20126%2074%20143T118%20160Q133%20160%20146%20151T160%20120Q160%2094%20142%2076T111%2058Q109%2057%20108%2057T107%2055Q108%2052%20115%2047T146%2034T201%2027Q237%2027%20263%2038T301%2066T318%2097T323%20122Q323%20150%20302%20164T254%20181T195%20196T148%20231Q131%20256%20131%20289Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-3D%22%20d%3D%22M56%20347Q56%20360%2070%20367H707Q722%20359%20722%20347Q722%20336%20708%20328L390%20327H72Q56%20332%2056%20347ZM56%20153Q56%20168%2072%20173H708Q722%20163%20722%20153Q722%20140%20707%20133H70Q56%20140%2056%20153Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJSZ2-2211%22%20d%3D%22M60%20948Q63%20950%20665%20950H1267L1325%20815Q1384%20677%201388%20669H1348L1341%20683Q1320%20724%201285%20761Q1235%20809%201174%20838T1033%20881T882%20898T699%20902H574H543H251L259%20891Q722%20258%20724%20252Q725%20250%20724%20246Q721%20243%20460%20-56L196%20-356Q196%20-357%20407%20-357Q459%20-357%20548%20-357T676%20-358Q812%20-358%20896%20-353T1063%20-332T1204%20-283T1307%20-196Q1328%20-170%201348%20-124H1388Q1388%20-125%201381%20-145T1356%20-210T1325%20-294L1267%20-449L666%20-450Q64%20-450%2061%20-448Q55%20-446%2055%20-439Q55%20-437%2057%20-433L590%20177Q590%20178%20557%20222T452%20366T322%20544L56%20909L55%20924Q55%20945%2060%20948Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-54%22%20d%3D%22M40%20437Q21%20437%2021%20445Q21%20450%2037%20501T71%20602L88%20651Q93%20669%20101%20677H569H659Q691%20677%20697%20676T704%20667Q704%20661%20687%20553T668%20444Q668%20437%20649%20437Q640%20437%20637%20437T631%20442L629%20445Q629%20451%20635%20490T641%20551Q641%20586%20628%20604T573%20629Q568%20630%20515%20631Q469%20631%20457%20630T439%20622Q438%20621%20368%20343T298%2060Q298%2048%20386%2046Q418%2046%20427%2045T436%2036Q436%2031%20433%2022Q429%204%20424%201L422%200Q419%200%20415%200Q410%200%20363%201T228%202Q99%202%2064%200H49Q43%206%2043%209T45%2027Q49%2040%2055%2046H83H94Q174%2046%20189%2055Q190%2056%20191%2056Q196%2059%20201%2076T241%20233Q258%20301%20269%20344Q339%20619%20339%20625Q339%20630%20310%20630H279Q212%20630%20191%20624Q146%20614%20121%20583T67%20467Q60%20445%2057%20441T43%20437H40Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2286%22%20d%3D%22M84%20346Q84%20468%20166%20546T360%20635Q361%20635%20370%20635T395%20635T430%20636T475%20636T524%20636H679Q694%20628%20694%20616Q694%20607%20681%20597L522%20596H470H441Q366%20596%20338%20592T266%20568Q244%20557%20224%20542T179%20500T139%20433T124%20346V341Q124%20253%20185%20185Q244%20121%20328%20103Q348%2098%20366%2098T522%2096H681Q694%2086%20694%2076Q694%2064%20679%2056H526Q510%2056%20480%2056T434%2055Q350%2055%20289%2071T172%20141Q84%20223%2084%20346ZM104%20-131T104%20-118T118%20-98H679Q694%20-106%20694%20-118T679%20-138H118Q104%20-131%20104%20-118Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-53%22%20d%3D%22M308%2024Q367%2024%20416%2076T466%20197Q466%20260%20414%20284Q308%20311%20278%20321T236%20341Q176%20383%20176%20462Q176%20523%20208%20573T273%20648Q302%20673%20343%20688T407%20704H418H425Q521%20704%20564%20640Q565%20640%20577%20653T603%20682T623%20704Q624%20704%20627%20704T632%20705Q645%20705%20645%20698T617%20577T585%20459T569%20456Q549%20456%20549%20465Q549%20471%20550%20475Q550%20478%20551%20494T553%20520Q553%20554%20544%20579T526%20616T501%20641Q465%20662%20419%20662Q362%20662%20313%20616T263%20510Q263%20480%20278%20458T319%20427Q323%20425%20389%20408T456%20390Q490%20379%20522%20342T554%20242Q554%20216%20546%20186Q541%20164%20528%20137T492%2078T426%2018T332%20-20Q320%20-22%20298%20-22Q199%20-22%20144%2033L134%2044L106%2013Q83%20-14%2078%20-18T65%20-22Q52%20-22%2052%20-14Q52%20-11%20110%20221Q112%20227%20130%20227H143Q149%20221%20149%20216Q149%20214%20148%20207T144%20186T142%20153Q144%20114%20160%2087T203%2047T255%2029T308%2024Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-66%22%20d%3D%22M118%20-162Q120%20-162%20124%20-164T135%20-167T147%20-168Q160%20-168%20171%20-155T187%20-126Q197%20-99%20221%2027T267%20267T289%20382V385H242Q195%20385%20192%20387Q188%20390%20188%20397L195%20425Q197%20430%20203%20430T250%20431Q298%20431%20298%20432Q298%20434%20307%20482T319%20540Q356%20705%20465%20705Q502%20703%20526%20683T550%20630Q550%20594%20529%20578T487%20561Q443%20561%20443%20603Q443%20622%20454%20636T478%20657L487%20662Q471%20668%20457%20668Q445%20668%20434%20658T419%20630Q412%20601%20403%20552T387%20469T380%20433Q380%20431%20435%20431Q480%20431%20487%20430T498%20424Q499%20420%20496%20407T491%20391Q489%20386%20482%20386T428%20385H372L349%20263Q301%2015%20282%20-47Q255%20-132%20212%20-173Q175%20-205%20139%20-205Q107%20-205%2081%20-186T55%20-132Q55%20-95%2076%20-78T118%20-61Q162%20-61%20162%20-103Q162%20-122%20151%20-136T127%20-157L118%20-162Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-7C%22%20d%3D%22M139%20-249H137Q125%20-249%20119%20-235V251L120%20737Q130%20750%20139%20750Q152%20750%20159%20735V-235Q151%20-249%20141%20-249H139Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-69%22%20d%3D%22M184%20600Q184%20624%20203%20642T247%20661Q265%20661%20277%20649T290%20619Q290%20596%20270%20577T226%20557Q211%20557%20198%20567T184%20600ZM21%20287Q21%20295%2030%20318T54%20369T98%20420T158%20442Q197%20442%20223%20419T250%20357Q250%20340%20236%20301T196%20196T154%2083Q149%2061%20149%2051Q149%2026%20166%2026Q175%2026%20185%2029T208%2043T235%2078T260%20137Q263%20149%20265%20151T282%20153Q302%20153%20302%20143Q302%20135%20293%20112T268%2061T223%2011T161%20-11Q129%20-11%20102%2010T74%2074Q74%2091%2079%20106T122%20220Q160%20321%20166%20341T173%20380Q173%20404%20156%20404H154Q124%20404%2099%20371T61%20287Q60%20286%2059%20284T58%20281T56%20279T53%20278T49%20278T41%20278H27Q21%20284%2021%20287Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-6E%22%20d%3D%22M21%20287Q22%20293%2024%20303T36%20341T56%20388T89%20425T135%20442Q171%20442%20195%20424T225%20390T231%20369Q231%20367%20232%20367L243%20378Q304%20442%20382%20442Q436%20442%20469%20415T503%20336T465%20179T427%2052Q427%2026%20444%2026Q450%2026%20453%2027Q482%2032%20505%2065T540%20145Q542%20153%20560%20153Q580%20153%20580%20145Q580%20144%20576%20130Q568%20101%20554%2073T508%2017T439%20-10Q392%20-10%20371%2017T350%2073Q350%2092%20386%20193T423%20345Q423%20404%20379%20404H374Q288%20404%20229%20303L222%20291L189%20157Q156%2026%20151%2016Q138%20-11%20108%20-11Q95%20-11%2087%20-5T76%207T74%2017Q74%2030%20112%20180T152%20343Q153%20348%20153%20366Q153%20405%20129%20405Q91%20405%2066%20305Q60%20285%2060%20284Q58%20278%2041%20278H27Q21%20284%2021%20287Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6D%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%22878%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%221408%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%221980%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-73%22%20x%3D%222370%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%222839%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%20x%3D%223506%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(4563%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ2-2211%22%20x%3D%2230%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(0%2C-1117)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-54%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-2286%22%20x%3D%22704%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-53%22%20x%3D%221483%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(6234%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(490%2C-187)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-7C%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-54%22%20x%3D%22278%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-7C%22%20x%3D%22983%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6D%22%20x%3D%227717%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-69%22%20x%3D%228595%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%20x%3D%228941%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%229541%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-54%22%20x%3D%229931%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%2210635%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
考虑 %22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6B%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2B%22%20x%3D%22743%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%20x%3D%221744%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
大的贡献
[Luogu#4707] 重返现世（min-max容斥+背包dp）
bzoj 4036 按位或 —— min-max容斥+FMT
Posted qq62c30ac77b2a7
tags:
篇首语：本文由小常识网(cha138.com)小编为大家整理，主要介绍了min-max 容斥学习笔记相关的知识，希望对你有一定的参考价值。
关于 容斥，我们希望构造一个系数使得
考虑 大的贡献
[Luogu#4707] 重返现世（min-max容斥+背包dp）
bzoj 4036 按位或 —— min-max容斥+FMT