Frozen_Guardian

2019ICPC(上海) - Counting Sequences I(dfs打表)

题目链接：点击查看

题目大意：对于n>1，求 $\prod_{i=1}^{n}a_{i}=\sum_{i=1}^{n}a_{i}$ 的长度为n的数组个数

题目分析：因为长度为n的数组中，组成数字相同但排列组合不同的数组，不算同一种情况，所以我们在求出一种满足条件的数组后，需要考虑所有排列组合的情况，然后枚举的时候也有技巧，如果盲目枚举3000位，那得枚举到何年何月。。我们需要分析一下这个式子，可以得出一个简单地结论，就是当n很大的时候，绝大部分都是1，为什么这么说呢，现在假设n>14，并且假设现在有14个大于1的数，然后其余的数都为1，因为2的14次方就已经大于3000了，所以那14个大于1的数累乘起来必定是大于3000的，所以将这些元素累加起来的结果是绝对不可能满足上述公式的。

现在已经剪枝了绝大部分，因为大于1的元素最多有13个，所以当n小于13的时候枚举n就行，当n大于13的时候，枚举一下13个不同的元素，然后剩下的用1补齐，然后计算全排列即可，因为我数论贼菜，所以全排列公式是我现学现卖的。。举个很简单的例子，比如3000个数中，有5个2,3个4，其余的都是1，那么排列组合就是(3000!)/(2992!*5!*3!)，因为涉及到了除法，所以我们可以预处理一下每个全排列的逆元即可。

首先先说一下几个小性质，也是挺简单地性质，虽然我不会证明。。

每个枚举的 $a_{i}$ 都是小于等于n的
所有 $a_{i}$ 之和小于等于2*n

在dfs中还需要剪枝的地方就是乘积了，因为在枚举的时候，我选择枚举 $a_{i}$ 是不严格升序枚举的，所以当累乘大于2*n时就可以直接返回了，并且枚举每一位的时候，下限是上一次枚举的数值，上限是2*n/mul (mul为累乘到当前的值)，因为要保证累乘的结果总是小于2*n，所以这样枚举也可以达到大幅降低时间复杂度的效果，下面挂上打表代码，因为跑了不到半分钟，然后如果是直接跑的话大概时间是3秒钟，但是题目给的是1秒钟，只能把3000个数值跑出来然后打表了。。

打表代码：

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
using namespace std;
 
typedef long long LL;

const int inf=0x3f3f3f3f;

const int mod=1e9+7;

const int limit=13;
 
const int N=3e3+100;

LL q_pow(LL a,LL b)
{
	LL ans=1;
	while(b)
	{
		if(b&1)
			ans=ans*a%mod;
		a=a*a%mod;
		b>>=1;
	}
	return ans;
}

LL fac[N],ifac[N],cnt[N];

LL ans;

int n;

void dfs(int pos=0,int pre=1,LL plus=0,LL mul=1)//pos：当前枚举了多少位，pre：上一位是多少，plus：累加，mul：累乘
{
	if(pos==min(n,limit))
	{
		int t=n>limit?n-limit:0;//计算有在至多13个不同的数值中有多少个1
		if(plus+t!=mul)
			return;
		LL temp=fac[n];//按照公式实现
		temp=temp*ifac[cnt[1]+t]%mod;//实现。。
		for(int i=2;i2*n)//剪枝
		return;
	for(int i=pre;i<=2*n/mul;i++)//枚举
	{
		cnt[i]++;
		dfs(pos+1,i,plus+i,mul*i);
		cnt[i]--;
	}
}

int main()
{
	freopen("output.txt","w",stdout);
	fac[0]=ifac[0]=1;
	for(int i=1;i

 
  AC代码： 
  #include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
using namespace std;
 
typedef long long LL;

const int inf=0x3f3f3f3f;
 
const int N=3e3+100;

int ans[N]={0,0,1,6,12,40,30,84,224,144,135,715,627,1326,1274,420,480,2720,5202,16530,3800,2590,924,16951,552,9300,68575,82134,21168,24360,165300,3720,45632,199584,2244,22015,2520,29304,77330,57798,398320,1285760,4255062,81270,41624,48510,97290,1479889,106032,11760,180075,132600,3254004,429936,1267866,38115,2293060,20948298,40667106,299425,208860,2325320,344162,7394058,7636608,82875520,8580,305118,4013836,171258,173880,11850965,70320672,1866318,14396330,4872900,17100,895356,468468,499122,252800,20509200,7012845,1391827,578676,25229190,26418340,211474989,784847184,497121123,15339150,49140,393484,1972158,821748,2924575,53598240,55872,45638306,15086610,753390000,50040450,417498852,18788230,1649648,1168440,1202040,63736369,2461428,68719050,162308630,564673856,24864,5916680,12882,83753235,89751520,869785371,124424274,827222264,916501173,142157979,2679303,3661464,39416252,40718500,31500,4096512,135201152,132178560,5399940,191928100,764103895,303683688,755446880,403450390,878166801,7620488,63013698,181404174,188003060,1490370,40044,7198477,2965248,73807320,222581380,897305600,3241644,486312756,838638473,668099857,87240552,277345965,3652110,279481895,569777520,909958004,17438618,102598248,6080160,331917600,704201656,12886128,6522608,11108130,856576600,6916639,773139192,393768648,453484704,5029110,10088316,597113652,30102,2755725,479740800,6824611,490800468,6727676,453731610,810770040,174453052,127995139,76742980,579462920,6400260,197469008,621276832,63131059,3519180,655161515,315686842,28570176,246855106,708291090,7567560,468395047,15329358,7998606,803840600,57548093,624161197,809432154,689858065,197982428,630455180,357065367,320910720,936082576,87780,963476640,74767476,379431305,9754548,380663735,10031040,156955682,25639852,20815512,5347980,198166963,886901487,225570529,242056473,501024386,533868153,603037815,920524624,245275990,718210895,212520,12540528,543375195,461026027,32361615,540843679,392427831,168324999,208713599,700359503,28225920,380401421,67909312,81646278,630637420,1293931,889031838,949548624,259817649,549967486,898626104,686911972,56583450,24419560,817851652,16842240,891525490,17040642,862442760,43568980,561311662,17984466,437032941,809577120,18959160,388056851,262408964,560202110,38713404,657023276,821728799,818922105,190585256,635328541,948130242,157183696,704797050,309867908,277830386,51523419,134870012,749747645,184738215,448245518,318027562,275886879,333883532,395070958,48524256,475894014,49114980,236370917,784569639,979573108,921821882,773970379,110038107,105145128,235499832,26910000,64267522,208632348,110356236,214584532,461333294,560051640,903997718,590609103,218012655,780213045,692419287,439786580,279209594,383153779,93074310,814843017,988535918,205584668,145543973,54218796,420841210,489026136,101531951,476789328,74482200,664404815,556029323,35179968,106293320,217140,72747180,127035152,215692694,111253064,388964221,61191507,848268878,659514248,487518216,265388569,771127213,748161940,681164077,541338475,70311036,85182422,882943718,804827001,59357303,21376250,43611750,43737408,807625711,44111586,713948361,860057995,984432333,137008032,883198273,305220824,581376307,283279826,758860975,670467540,619538990,874650824,656833322,504482780,851303417,791600048,321615528,207289111,969710687,224897578,969569932,975009846,12704182,147213128,181988425,550724752,138047730,536076394,19423120,474768550,534347422,277774654,873715625,305094536,448946783,855709629,60538530,209522824,693786590,927479183,18760111,123571800,467930686,548061617,339961478,377197825,929122116,510292238,260505648,898560496,606820688,73823363,167968493,753754165,643514715,617757502,468134774,139190904,429596203,499872283,804316856,158784988,500436898,330820304,833779290,309908865,798032191,19144628,330236216,227597688,791835097,548987726,78035958,933622580,464458901,606535622,336637759,837971159,466100351,501035131,722499131,998404247,40098260,401772714,106748090,42495200,86735880,877149,383556205,196692,263769300,896584125,250541410,706926755,322185268,663522473,595974710,915622764,342038221,55675413,585447394,719165182,809889789,569918083,97217522,72682038,663560186,103370673,909754905,634643969,791101815,701424289,688651692,481800017,873272360,331481464,944159323,607048176,450140201,307580852,633067279,507471325,680972531,932698257,847108034,329935200,841861733,931230222,86653572,298186557,317306433,382511484,941206421,998651093,269587042,223414638,901347339,381257681,299192826,716507553,865165128,88498570,843173027,268662409,919135809,671588778,222222296,816798687,794783532,382042584,329002916,363573101,606409032,111669003,820596994,868832953,599417016,293953140,806616576,764889370,63761101,853583648,276375792,65522872,998575523,883199315,676000997,149580165,348663191,678678481,255020512,485229036,109282577,448477583,149990544,76877264,304513527,150568236,655124776,151988148,534097455,270093859,221569978,619992276,544538574,150812922,389192157,202124556,637466796,904314865,928886810,154958903,986123802,493231088,174281319,670239135,620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int main()
{
//	freopen("output.txt","w",stdout);
	int w;
	for(cin>>w;w;w--)
	{
		int n;
		scanf("%d",&n);
		printf("%d\n",ans[n]);
	}
	
 
 
 
 
 
 
 
 
 
	return 0;
}

nosql数据库技术与应用知识点皆过客，揽星河 NoSQL nosql 数据库大数据数据分析数据结构非关系型数据库
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浅谈MapReduce Android路上的人 Hadoop 分布式计算 mapreduce 分布式框架 hadoop
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Hadoop 傲雪凌霜，松柏长青后端大数据 hadoop 大数据分布式
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