牛客练习赛33题解

~~可能是最良心的一回练习赛了，之前的比赛是wannafly难度~~ ~~由于是比赛题解所以不会很详细，细节自己意会一下~~ ## T1 输出$$\frac{r}{x}-\frac{l-1}{x}$$ ```C #include<cstdio> #include<algorithm> using namespace std;typedef long long ll; ll l;ll r;int T;ll x; int main() { scanf("%d",&T); for(int z=1;z<=T;z++) { scanf("%lld%lld%lld",&l,&r,&x); printf("%lld\n",(r/x)-((l-1)/x)); }return 0; } ``` ## T2 记一下每种数字出现了多少次然后跑调和级数的暴力就行了 复杂度$O(n\ln n)$ ```C #include<cstdio> #include<algorithm> using namespace std;const int N=1e6+10; int cnt[N];int n;int k;int tot[N];int a=-1;int b=0; int main() { scanf("%d%d",&n,&k); for(int i=1,t;i<=n;i++)scanf("%d",&t),cnt[t]++; for(int i=1;i<=k;i++) for(int j=i;j<=k;j+=i)tot[j]+=cnt[i]; for(int i=1;i<=k;i++) if(tot[i]>a)a=tot[i],b=1;else if(tot[i]==a)b++; printf("%d %d\n",a,b);return 0; } ``` ## T3 因为1000就是8的倍数了因此一个数字能否被8整除仅仅由后三位决定 直接大力枚举后3位是什么然后开一个结构体表示每种数字出现了多少次，显然9越多越好如果9的个数一样8越多越好……以此类推，如果前面的数字都一样那么比较一下最后的3位 然后注意一些细节就是当数字的位数小于3位的时候直接枚举这个数字是什么就行了 输出的时候先输出所有的9然后输出所有的8....以此类推 ```C #include<cstdio> #include<algorithm> using namespace std;const int N=20;typedef long long ll; struct data { int val[N]; inline int& operator [](const int& x){return val[x];} friend bool operator <(data a,data b) { for(int i=9;i>=0;i--)if(a[i]!=b[i])return a[i]<b[i]; return a[10]<b[10]; } }ans,cur,bf;char mde[110];int n;int T; inline void solve() { scanf("%s",mde+1);for(n=1;mde[n+1]!='\0';n++); for(int i=1;i<=n;i++)bf[mde[i]-'0']++; int lim=1000; switch(n) { case 1:{lim=10;break;} case 2:{lim=100;break;} } for(int k=0;k<lim;k+=8) { cur=bf; if(n>=1)cur[k%10]--; if(n>=2)cur[(k/10)%10]--; if(n>=3)cur[(k/100)]--; cur[10]=k; for(int i=0;i<=9;i++)if(cur[i]<0)goto ed; ans=max(ans,cur);ed:; } for(int i=9;i>=1;i--)if(ans[i]!=0)goto pr; printf("%d\n",ans[10]);return; pr:; for(int i=9;i>=0;i--) for(int j=1;j<=ans[i];j++)printf("%c",'0'+i); printf("%c",'0'+(ans[10]/100)); printf("%c",'0'+((ans[10]/10)%10)); printf("%c",'0'+(ans[10]%10));printf("\n"); } inline void clear() {for(int i=10;i>=0;i--)ans[i]=0;for(int i=10;i>=0;i--)bf[i]=0;ans[10]=-1;} int main() { scanf("%d",&T); for(int z=1;z<=T;z++)clear(),solve();return 0; } ``` ## T4 ~~gb题,一开始没看出来~~ ~~老年选手退役预订~~ 发现操作的本质就是选择一段区间然后把这个区间的$k\mod(r-l+1)$个元素移到前面去 也就是把这个序列从abcd变成了abcd的形式,发现一个十分重要的事实是只有b和c之间的逆序对发生了改变，顺序对变成了逆序对逆序对变成了逆序对 因此如果$(r-l+1-k)k$是奇数的话答案奇偶性一定改变否则奇偶性一定不改变 初始的答案手懒了直接树状数组硬搞了 ```C #include<cstdio> #include<algorithm> using namespace std;const int N=1e5+10;typedef long long ll; ll ans;int a[N];int n;int m; struct treearray { int ta[N]; inline void c(int x){for(;x;x-=x&(-x))ta[x]++;} inline int q(int x){int r=0;for(;x<=n;x+=x&(-x))r+=ta[x];return r;} }ta; int main() { scanf("%d",&n); for(int i=1;i<=n;i++)scanf("%d",&a[i]); for(int i=1;i<=n;i++)ans+=ta.q(a[i]),ta.c(a[i]);ans&=1; scanf("%d",&m); for(int i=1,l,r,k;i<=m;i++) { scanf("%d%d%d",&l,&r,&k); k%=(r-l+1);ans^=(((r-l+1-k)*k)&1); printf("%lld\n",ans); }return 0; } ``` ## T5 ~~一开始以为是Space Issac那题的套路结果wa到怀疑人生~~ 发现答案只有26种因此我们大力把进行了0到25次变换的hash值全部搞出来就行了，然后题目中要求是最小变换，如果x变化b次到了y那么y变化26-i次就能到x，取个min就行了 ~~(一开始没考虑到有些字符会加过去有些字符加不过去然后疯狂把两个hahs值相减)~~ ```C #include<cstdio> #include<algorithm> using namespace std;const int N=1e5+10;typedef unsigned long long ll; const ll mod1=998244353;const ll mod2=1e9+7;const ll bas=233;int n;int m; inline ll po(ll a,ll p,ll mod){ll r=1;for(;p;p>>=1,a=a*a%mod)if(p&1)r=r*a%mod;return r;} char mde[N];ll hsh1[30][N];ll hsh2[30][N];ll imi1[N];ll imi2[N]; inline ll ghsh1(int p,int l,int r) {return (hsh1[p][r]+mod1-hsh1[p][l-1])*imi1[l-1]%mod1;} inline ll ghsh2(int p,int l,int r) {return (hsh2[p][r]+mod2-hsh2[p][l-1])*imi2[l-1]%mod2;} int main() { scanf("%d",&n);scanf("%s",mde+1); for(int j=0;j<=25;j++) { ll mi=1;for(int i=1;i<=n;i++,(mi*=bas)%=mod1)hsh1[j][i]=(hsh1[j][i-1]+mi*((mde[i]-'a'+j)%26+1))%mod1; } for(int j=0;j<=25;j++) { ll mi=1;for(int i=1;i<=n;i++,(mi*=bas)%=mod2)hsh2[j][i]=(hsh2[j][i-1]+mi*((mde[i]-'a'+j)%26+1))%mod2; } ll iv=po(bas,mod1-2,mod1);imi1[0]=1;for(int i=1;i<=n;i++)imi1[i]=imi1[i-1]*iv%mod1; iv=po(bas,mod2-2,mod2);imi2[0]=1;for(int i=1;i<=n;i++)imi2[i]=imi2[i-1]*iv%mod2; scanf("%d",&m); for(int i=1,x,y,le;i<=m;i++) { scanf("%d%d%d",&x,&y,&le); ll v1=ghsh1(0,y,y+le-1);ll v2=ghsh2(0,y,y+le-1); for(int i=0;i<=25;i++) if(ghsh1(i,x,x+le-1)==v1&&ghsh2(i,x,x+le-1)==v2){printf("%d\n",min(i,26-i));goto ed;} printf("-1\n");ed:; }return 0; } ``` ## T6 ~~noip的d2t2告诉我们打表是个好东西，脑子什么的可以扔掉了~~ 我是纯打表搞的这题 题意就是给定一个数组和一个多项式F，数组的第$i$项就是$F(i)$然后我们把这个数组做N次差分之后询问你这个数组中所有元素的和是多少其中$N$是最高次系数 首先让我们来手玩一下只有一个1的情况，你发现这个表长这样 ``` 1 1 -1 1 -2 1 1 -4 6 -4 1 1 -5 10 -10 5 -1 ``` 稍微有点脑子的人都知道这是杨辉三角，只不过加了点正负号 然后你发现除了第一行剩下部分的和都是0 那么我们接着玩一下别的幂次，比如1次，2次 你会发现这个数列会在第i次差分之后变成一个有限的数列 根据一点基本的常识：**数列之和的差分等于差分数列的和** 你会发现一个数列在第i次差分之后就变成0了 因此这题只有最高次是有用的 所以我们应该输出最高次系数乘上一个n次差分之后数组的和 让我们打个表手玩一下$F(x)=x^1,x^2,x^3,x^4...$的答案好了 发现是$1,2,6,24,120,720...$ 小学数奥知识告诉我们这东西叫阶乘 所以题目就是输出$n!$乘上最高次系数 好的$n!$模$10^9+7$ (多项式多点求值+任意模数fft,复杂度$O(\sqrt{P}log^{2}P)$) 显然这东西过于不科学了，我们考虑点接地气的方法 毕竟模数只有一个，我们还是分段打表吧…… 然后这题就做完了，你可以把块长取到$10^6$这样会代码短点 ```C #include<cstdio> #include<algorithm> using namespace std;typedef long long ll; const ll mod=1e9+7;const int N=1e3+10;const int P=1e6; ll pfac[N]={ 1,641102369,578095319,5832229,259081142,974067448,316220877,690120224,251368199, 980250487,682498929,134623568,95936601,933097914,167332441,598816162,336060741, 248744620,626497524,288843364,491101308,245341950,565768255,246899319,968999, 586350670,638587686,881746146,19426633,850500036,76479948,268124147,842267748, 886294336,485348706,463847391,544075857,898187927,798967520,82926604,723816384, 156530778,721996174,299085602,323604647,172827403,398699886,530389102,294587621, 813805606,67347853,497478507,196447201,722054885,228338256,407719831,762479457, 746536789,811667359,778773518,27368307,438371670,59469516,5974669,766196482, 606322308,86609485,889750731,340941507,371263376,625544428,788878910,808412394, 996952918,585237443,1669644,361786913,480748381,595143852,837229828,199888908, 526807168,579691190,145404005,459188207,534491822,439729802,840398449,899297830, 235861787,888050723,656116726,736550105,440902696,85990869,884343068,56305184, 973478770,168891766,804805577,927880474,876297919,934814019,676405347,567277637, 112249297,44930135,39417871,47401357,108819476,281863274,60168088,692636218, 432775082,14235602,770511792,400295761,697066277,421835306,220108638,661224977, 261799937,168203998,802214249,544064410,935080803,583967898,211768084,751231582, 972424306,623534362,335160196,243276029,554749550,60050552,797848181,395891998, 172428290,159554990,887420150,970055531,250388809,487998999,856259313,82104855, 232253360,513365505,244109365,1559745,695345956,261384175,849009131,323214113, 747664143,444090941,659224434,80729842,570033864,664989237,827348878,195888993, 576798521,457882808,731551699,212938473,509096183,827544702,678320208,677711203, 289752035,66404266,555972231,195290384,97136305,349551356,785113347,83489485, 66247239,52167191,307390891,547665832,143066173,350016754,917404120,296269301, 996122673,23015220,602139210,748566338,187348575,109838563,574053420,105574531, 304173654,542432219,34538816,325636655,437843114,630621321,26853683,933245637, 616368450,238971581,511371690,557301633,911398531,848952161,958992544,925152039, 914456118,724691727,636817583,238087006,946237212,910291942,114985663,492237273, 450387329,834860913,763017204,368925948,475812562,740594930,45060610,806047532, 464456846,172115341,75307702,116261993,562519302,268838846,173784895,243624360, 61570384,481661251,938269070,95182730,91068149,115435332,495022305,136026497, 506496856,710729672,113570024,366384665,564758715,270239666,277118392,79874094, 702807165,112390913,730341625,103056890,677948390,339464594,167240465,108312174, 839079953,479334442,271788964,135498044,277717575,591048681,811637561,353339603, 889410460,839849206,192345193,736265527,316439118,217544623,788132977,618898635, 183011467,380858207,996097969,898554793,335353644,54062950,611251733,419363534, 965429853,160398980,151319402,990918946,607730875,450718279,173539388,648991369, 970937898,500780548,780122909,39052406,276894233,460373282,651081062,461415770, 358700839,643638805,560006119,668123525,686692315,673464765,957633609,199866123, 563432246,841799766,385330357,504962686,954061253,128487469,685707545,299172297, 717975101,577786541,318951960,773206631,306832604,204355779,573592106,30977140, 450398100,363172638,258379324,472935553,93940075,587220627,776264326,793270300, 291733496,522049725,579995261,335416359,142946099,472012302,559947225,332139472, 499377092,464599136,164752359,309058615,86117128,580204973,563781682,954840109, 624577416,895609896,888287558,836813268,926036911,386027524,184419613,724205533, 403351886,715247054,716986954,830567832,383388563,68409439,6734065,189239124, 68322490,943653305,405755338,811056092,179518046,825132993,343807435,985084650, 868553027,148528617,160684257,882148737,591915968,701445829,529726489,302177126, 974886682,241107368,798830099,940567523,11633075,325334066,346091869,115312728, 473718967,218129285,878471898,180002392,699739374,917084264,856859395,435327356, 808651347,421623838,105419548,59883031,322487421,79716267,715317963,429277690, 398078032,316486674,384843585,940338439,937409008,940524812,947549662,833550543, 593524514,996164327,987314628,697611981,636177449,274192146,418537348,925347821, 952831975,893732627,1277567,358655417,141866945,581830879,987597705,347046911, 775305697,125354499,951540811,247662371,343043237,568392357,997474832,209244402, 380480118,149586983,392838702,309134554,990779998,263053337,325362513,780072518, 551028176,990826116,989944961,155569943,596737944,711553356,268844715,451373308, 379404150,462639908,961812918,654611901,382776490,41815820,843321396,675258797, 845583555,934281721,741114145,275105629,666247477,325912072,526131620,252551589, 432030917,554917439,818036959,754363835,795190182,909210595,278704903,719566487, 628514947,424989675,321685608,50590510,832069712,198768464,702004730,99199382, 707469729,747407118,302020341,497196934,5003231,726997875,382617671,296229203, 183888367,703397904,552133875,732868367,350095207,26031303,863250534,216665960, 561745549,352946234,784139777,733333339,503105966,459878625,803187381,16634739, 180898306,68718097,985594252,404206040,749724532,97830135,611751357,31131935, 662741752,864326453,864869025,167831173,559214642,718498895,91352335,608823837, 473379392,385388084,152267158,681756977,46819124,313132653,56547945,442795120, 796616594,256141983,152028387,636578562,385377759,553033642,491415383,919273670, 996049638,326686486,160150665,141827977,540818053,693305776,593938674,186576440, 688809790,565456578,749296077,519397500,551096742,696628828,775025061,370732451, 164246193,915265013,457469634,923043932,912368644,777901604,464118005,637939935, 956856710,490676632,453019482,462528877,502297454,798895521,100498586,699767918, 849974789,811575797,438952959,606870929,907720182,179111720,48053248,508038818, 811944661,752550134,401382061,848924691,764368449,34629406,529840945,435904287, 26011548,208184231,446477394,206330671,366033520,131772368,185646898,648711554, 472759660,523696723,271198437,25058942,859369491,817928963,330711333,724464507, 437605233,701453022,626663115,281230685,510650790,596949867,295726547,303076380, 465070856,272814771,538771609,48824684,951279549,939889684,564188856,48527183, 201307702,484458461,861754542,326159309,181594759,668422905,286273596,965656187, 44135644,359960756,936229527,407934361,267193060,456152084,459116722,124804049, 262322489,920251227,816929577,483924582,151834896,167087470,490222511,903466878, 361583925,368114731,339383292,388728584,218107212,249153339,909458706,322908524, 202649964,92255682,573074791,15570863,94331513,744158074,196345098,334326205, 9416035,98349682,882121662,769795511,231988936,888146074,137603545,582627184, 407518072,919419361,909433461,986708498,310317874,373745190,263645931,256853930, 876379959,702823274,147050765,308186532,175504139,180350107,797736554,606241871, 384547635,273712630,586444655,682189174,666493603,946867127,819114541,502371023, 261970285,825871994,126925175,701506133,314738056,341779962,561011609,815463367, 46765164,49187570,188054995,957939114,64814326,933376898,329837066,338121343, 765215899,869630152,978119194,632627667,975266085,435887178,282092463,129621197, 758245605,827722926,201339230,918513230,322096036,547838438,985546115,852304035, 593090119,689189630,555842733,567033437,469928208,212842957,117842065,404149413, 155133422,663307737,208761293,206282795,717946122,488906585,414236650,280700600, 962670136,534279149,214569244,375297772,811053196,922377372,289594327,219932130, 211487466,701050258,398782410,863002719,27236531,217598709,375472836,810551911, 178598958,247844667,676526196,812283640,863066876,857241854,113917835,624148346, 726089763,564827277,826300950,478982047,439411911,454039189,633292726,48562889, 802100365,671734977,945204804,508831870,398781902,897162044,644050694,892168027, 828883117,277714559,713448377,624500515,590098114,808691930,514359662,895205045, 715264908,628829100,484492064,919717789,513196123,748510389,403652653,574455974, 77123823,172096141,819801784,581418893,15655126,15391652,875641535,203191898, 264582598,880691101,907800444,986598821,340030191,264688936,369832433,785804644, 842065079,423951674,663560047,696623384,496709826,161960209,331910086,541120825, 951524114,841656666,162683802,629786193,190395535,269571439,832671304,76770272, 341080135,421943723,494210290,751040886,317076664,672850561,72482816,493689107, 135625240,100228913,684748812,639655136,906233141,929893103,277813439,814362881, 562608724,406024012,885537778,10065330,60625018,983737173,60517502,551060742, 804930491,823845496,727416538,946421040,678171399,842203531,175638827,894247956, 538609927,885362182,946464959,116667533,749816133,241427979,871117927,281804989, 163928347,563796647,640266394,774625892,59342705,256473217,674115061,918860977, 322633051,753513874,393556719,304644842,767372800,161362528,754787150,627655552, 677395736,799289297,846650652,816701166,687265514,787113234,358757251,701220427, 607715125,245795606,600624983,10475577,728620948,759404319,36292292,491466901, 22556579,114495791,647630109,586445753,482254337,718623833,763514207,66547751, 953634340,351472920,308474522,494166907,634359666,172114298,865440961,364380585, 921648059,965683742,260466949,117483873,962540888,237120480,620531822,193781724, 213092254,107141741,602742426,793307102,756154604,236455213,362928234,14162538, 753042874,778983779,25977209,49389215,698308420,859637374,49031023,713258160, 737331920,923333660,804861409,83868974,682873215,217298111,883278906,176966527, 954913,105359006,390019735,10430738,706334445,315103615,567473423,708233401, 48160594,946149627,346966053,281329488,462880311,31503476,185438078,965785236, 992656683,916291845,881482632,899946391,321900901,512634493,303338827,121000338, 967284733,492741665,152233223,165393390,680128316,917041303,532702135,741626808, 496442755,536841269,131384366,377329025,301196854,859917803,676511002,373451745, 847645126,823495900,576368335,73146164,954958912,847549272,241289571,646654592, 216046746,205951465,3258987,780882948,822439091,598245292,869544707,698611116 }; inline ll cfac(ll x) { ll ret=pfac[x/P];ll st=x-x%P; for(st=st+1;st<=x;st++)(ret*=st)%=mod; return ret; } int n;int m; int main() { scanf("%d%d",&n,&m);ll ans=cfac(n); for(int i=1,p,t;i<=m;i++) { scanf("%d%d",&p,&t); if(t==n){printf("%lld",ans*p%mod);return 0;} }return 0; } ```