最近在洛谷刷题，遇到了几道能够使用“打表法”的题。之前我是完全不知道还有这种“旁门左道”的（doge），在打比赛的时候还是挺受用的，不过如果在平时练习，还是一板一眼来就好~
那我们话不多说，先简单了解一下“打表法”是个啥吧。
举个简单的例子：中学语文有一种题型，叫“古诗词默写”，给你上一句，我来下一句。做这种题，理论上有两种方法，一，是像“诗人”一样去推敲，去创造，凭借语感和文化底蕴去填；二，就是我们最常规的方法，提前背过就完事儿了，简单粗暴。而我们的“打表法”，就类似于后者。
再举一个例子，有句话，“充电2小时，通话5分钟”，但实际上，竞赛他只看你那“通话五分钟”，而‘充电2小时“可有可无。有则是”传统算法“，无则是”打表法“。
举完生活中的例子，我们现在来举2道例题。
例题1：回文素数
因为数据的规模到了1e9，如果用最常规方法（判断单个数字是否为回文和素数都需一重循环，再来一重a~b的循环，时间复杂度就来到了O（n*n），是无法通过全部测试点的。因此，打表法就是其中一个可行的解法。为什么说“其中一个”呢，因为还有优化后的传统算法，如优化素数判断的埃氏筛法和线性筛法。但今天主要说打表法，另外几种大家可以自行查阅。
那么，我们的思路如下：
首先：“提前充电2小时”，即把5~1e9中的所有回文素数找出来，并存在一个”txt“文件里，方便我们后续使用（复制粘贴~）本文以c/c++为实现语言。
代码如下：
#include<iostream>
#include<algorithm>
#include<cmath>
#include<map>
#include<queue>
#include<stack>
#include<stdlib.h>
#include<stdio.h>
#include<cstring>
#include<set>
using namespace std;
int f(int x)//先判断回文
{
int a=0;
int b=x;
int cnt=0;
while(x)
{
int y=x%10;
a=a*10+y;
x/=10;
cnt++;
}
return a==b?cnt:0;
}
bool g(int x)//再判断素数，因为回文比素数少
{
if(x==11)return 1;
int y=f(x);
if(y%2==0)return 0;
for(int i=2;i*i<=x;i++)
if(x%i==0)return 0;
return 1;
}
int main()
{
FILE* f;
f=fopen("hs.txt","w");
int cnt=0;
for(int i=5;i<=1000000000;i++)
{
if(g(i))
{
cout<<i<<endl;
cnt++;
fprintf(f,"%d ",i);
fputs(",",f);//写入“,"，因为数组初始化中用英文,隔开
}
}
fclose(f);
cout<<cnt;
return 0;
}
代码运行得到如下文件结果：
然后，”通话5分钟“，写我们的提交代码：
#include<iostream>
#include<algorithm>
#include<cmath>
#include<map>
#include<queue>
#include<stack>
#include<stdlib.h>
#include<stdio.h>
#include<cstring>
#include<set>
using namespace std;
int main()
{
int c[779]={5 ,7 ,11 ,101 ,131 ,151 ,181 ,191 ,313 ,353 ,373 ,383 ,727 ,757 ,787 ,797 ,919 ,929 ,10301 ,10501 ,10601 ,11311 ,11411 ,12421 ,12721 ,12821 ,13331 ,13831 ,13931 ,14341 ,14741 ,15451 ,15551 ,16061 ,16361 ,16561 ,16661 ,17471 ,17971 ,18181 ,18481 ,19391 ,19891 ,19991 ,30103 ,30203 ,30403 ,30703 ,30803 ,31013 ,31513 ,32323 ,32423 ,33533 ,34543 ,34843 ,35053 ,35153 ,35353 ,35753 ,36263 ,36563 ,37273 ,37573 ,38083 ,38183 ,38783 ,39293 ,70207 ,70507 ,70607 ,71317 ,71917 ,72227 ,72727 ,73037 ,73237 ,73637 ,74047 ,74747 ,75557 ,76367 ,76667 ,77377 ,77477 ,77977 ,78487 ,78787 ,78887 ,79397 ,79697 ,79997 ,90709 ,91019 ,93139 ,93239 ,93739 ,94049 ,94349 ,94649 ,94849 ,94949 ,95959 ,96269 ,96469 ,96769 ,97379 ,97579 ,97879 ,98389 ,98689 ,1003001 ,1008001 ,1022201 ,1028201 ,1035301 ,1043401 ,1055501 ,1062601 ,1065601 ,1074701 ,1082801 ,1085801 ,1092901 ,1093901 ,1114111 ,1117111 ,1120211 ,1123211 ,1126211 ,1129211 ,1134311 ,1145411 ,1150511 ,1153511 ,1160611 ,1163611 ,1175711 ,1177711 ,1178711 ,1180811 ,1183811 ,1186811 ,1190911 ,1193911 ,1196911 ,1201021 ,1208021 ,1212121 ,1215121 ,1218121 ,1221221 ,1235321 ,1242421 ,1243421 ,1245421 ,1250521 ,1253521 ,1257521 ,1262621 ,1268621 ,1273721 ,1276721 ,1278721 ,1280821 ,1281821 ,1286821 ,1287821 ,1300031 ,1303031 ,1311131 ,1317131 ,1327231 ,1328231 ,1333331 ,1335331 ,1338331 ,1343431 ,1360631 ,1362631 ,1363631 ,1371731 ,1374731 ,1390931 ,1407041 ,1409041 ,1411141 ,1412141 ,1422241 ,1437341 ,1444441 ,1447441 ,1452541 ,1456541 ,1461641 ,1463641 ,1464641 ,1469641 ,1486841 ,1489841 ,1490941 ,1496941 ,1508051 ,1513151 ,1520251 ,1532351 ,1535351 ,1542451 ,1548451 ,1550551 ,1551551 ,1556551 ,1557551 ,1565651 ,1572751 ,1579751 ,1580851 ,1583851 ,1589851 ,1594951 ,1597951 ,1598951 ,1600061 ,1609061 ,1611161 ,1616161 ,1628261 ,1630361 ,1633361 ,1640461 ,1643461 ,1646461 ,1654561 ,1657561 ,1658561 ,1660661 ,1670761 ,1684861 ,1685861 ,1688861 ,1695961 ,1703071 ,1707071 ,1712171 ,1714171 ,1730371 ,1734371 ,1737371 ,1748471 ,1755571 ,1761671 ,1764671 ,1777771 ,1793971 ,1802081 ,1805081 ,1820281 ,1823281 ,1824281 ,1826281 ,1829281 ,1831381 ,1832381 ,1842481 ,1851581 ,1853581 ,1856581 ,1865681 ,1876781 ,1878781 ,1879781 ,1880881 ,1881881 ,1883881 ,1884881 ,1895981 ,1903091 ,1908091 ,1909091 ,1917191 ,1924291 ,1930391 ,1936391 ,1941491 ,1951591 ,1952591 ,1957591 ,1958591 ,1963691 ,1968691 ,1969691 ,1970791 ,1976791 ,1981891 ,1982891 ,1984891 ,1987891 ,1988891 ,1993991 ,1995991 ,1998991 ,3001003 ,3002003 ,3007003 ,3016103 ,3026203 ,3064603 ,3065603 ,3072703 ,3073703 ,3075703 ,3083803 ,3089803 ,3091903 ,3095903 ,3103013 ,3106013 ,3127213 ,3135313 ,3140413 ,3155513 ,3158513 ,3160613 ,3166613 ,3181813 ,3187813 ,3193913 ,3196913 ,3198913 ,3211123 ,3212123 ,3218123 ,3222223 ,3223223 ,3228223 ,3233323 ,3236323 ,3241423 ,3245423 ,3252523 ,3256523 ,3258523 ,3260623 ,3267623 ,3272723 ,3283823 ,3285823 ,3286823 ,3288823 ,3291923 ,3293923 ,3304033 ,3305033 ,3307033 ,3310133 ,3315133 ,3319133 ,3321233 ,3329233 ,3331333 ,3337333 ,3343433 ,3353533 ,3362633 ,3364633 ,3365633 ,3368633 ,3380833 ,3391933 ,3392933 ,3400043 ,3411143 ,3417143 ,3424243 ,3425243 ,3427243 ,3439343 ,3441443 ,3443443 ,3444443 ,3447443 ,3449443 ,3452543 ,3460643 ,3466643 ,3470743 ,3479743 ,3485843 ,3487843 ,3503053 ,3515153 ,3517153 ,3528253 ,3541453 ,3553553 ,3558553 ,3563653 ,3569653 ,3586853 ,3589853 ,3590953 ,3591953 ,3594953 ,3601063 ,3607063 ,3618163 ,3621263 ,3627263 ,3635363 ,3643463 ,3646463 ,3670763 ,3673763 ,3680863 ,3689863 ,3698963 ,3708073 ,3709073 ,3716173 ,3717173 ,3721273 ,3722273 ,3728273 ,3732373 ,3743473 ,3746473 ,3762673 ,3763673 ,3765673 ,3768673 ,3769673 ,3773773 ,3774773 ,3781873 ,3784873 ,3792973 ,3793973 ,3799973 ,3804083 ,3806083 ,3812183 ,3814183 ,3826283 ,3829283 ,3836383 ,3842483 ,3853583 ,3858583 ,3863683 ,3864683 ,3867683 ,3869683 ,3871783 ,3878783 ,3893983 ,3899983 ,3913193 ,3916193 ,3918193 ,3924293 ,3927293 ,3931393 ,3938393 ,3942493 ,3946493 ,3948493 ,3964693 ,3970793 ,3983893 ,3991993 ,3994993 ,3997993 ,3998993 ,7014107 ,7035307 ,7036307 ,7041407 ,7046407 ,7057507 ,7065607 ,7069607 ,7073707 ,7079707 ,7082807 ,7084807 ,7087807 ,7093907 ,7096907 ,7100017 ,7114117 ,7115117 ,7118117 ,7129217 ,7134317 ,7136317 ,7141417 ,7145417 ,7155517 ,7156517 ,7158517 ,7159517 ,7177717 ,7190917 ,7194917 ,7215127 ,7226227 ,7246427 ,7249427 ,7250527 ,7256527 ,7257527 ,7261627 ,7267627 ,7276727 ,7278727 ,7291927 ,7300037 ,7302037 ,7310137 ,7314137 ,7324237 ,7327237 ,7347437 ,7352537 ,7354537 ,7362637 ,7365637 ,7381837 ,7388837 ,7392937 ,7401047 ,7403047 ,7409047 ,7415147 ,7434347 ,7436347 ,7439347 ,7452547 ,7461647 ,7466647 ,7472747 ,7475747 ,7485847 ,7486847 ,7489847 ,7493947 ,7507057 ,7508057 ,7518157 ,7519157 ,7521257 ,7527257 ,7540457 ,7562657 ,7564657 ,7576757 ,7586857 ,7592957 ,7594957 ,7600067 ,7611167 ,7619167 ,7622267 ,7630367 ,7632367 ,7644467 ,7654567 ,7662667 ,7665667 ,7666667 ,7668667 ,7669667 ,7674767 ,7681867 ,7690967 ,7693967 ,7696967 ,7715177 ,7718177 ,7722277 ,7729277 ,7733377 ,7742477 ,7747477 ,7750577 ,7758577 ,7764677 ,7772777 ,7774777 ,7778777 ,7782877 ,7783877 ,7791977 ,7794977 ,7807087 ,7819187 ,7820287 ,7821287 ,7831387 ,7832387 ,7838387 ,7843487 ,7850587 ,7856587 ,7865687 ,7867687 ,7868687 ,7873787 ,7884887 ,7891987 ,7897987 ,7913197 ,7916197 ,7930397 ,7933397 ,7935397 ,7938397 ,7941497 ,7943497 ,7949497 ,7957597 ,7958597 ,7960697 ,7977797 ,7984897 ,7985897 ,7987897 ,7996997 ,9002009 ,9015109 ,9024209 ,9037309 ,9042409 ,9043409 ,9045409 ,9046409 ,9049409 ,9067609 ,9073709 ,9076709 ,9078709 ,9091909 ,9095909 ,9103019 ,9109019 ,9110119 ,9127219 ,9128219 ,9136319 ,9149419 ,9169619 ,9173719 ,9174719 ,9179719 ,9185819 ,9196919 ,9199919 ,9200029 ,9209029 ,9212129 ,9217129 ,9222229 ,9223229 ,9230329 ,9231329 ,9255529 ,9269629 ,9271729 ,9277729 ,9280829 ,9286829 ,9289829 ,9318139 ,9320239 ,9324239 ,9329239 ,9332339 ,9338339 ,9351539 ,9357539 ,9375739 ,9384839 ,9397939 ,9400049 ,9414149 ,9419149 ,9433349 ,9439349 ,9440449 ,9446449 ,9451549 ,9470749 ,9477749 ,9492949 ,9493949 ,9495949 ,9504059 ,9514159 ,9526259 ,9529259 ,9547459 ,9556559 ,9558559 ,9561659 ,9577759 ,9583859 ,9585859 ,9586859 ,9601069 ,9602069 ,9604069 ,9610169 ,9620269 ,9624269 ,9626269 ,9632369 ,9634369 ,9645469 ,9650569 ,9657569 ,9670769 ,9686869 ,9700079 ,9709079 ,9711179 ,9714179 ,9724279 ,9727279 ,9732379 ,9733379 ,9743479 ,9749479 ,9752579 ,9754579 ,9758579 ,9762679 ,9770779 ,9776779 ,9779779 ,9781879 ,9782879 ,9787879 ,9788879 ,9795979 ,9801089 ,9807089 ,9809089 ,9817189 ,9818189 ,9820289 ,9822289 ,9836389 ,9837389 ,9845489 ,9852589 ,9871789 ,9888889 ,9889889 ,9896989 ,9902099 ,9907099 ,9908099 ,9916199 ,9918199 ,9919199 ,9921299 ,9923299 ,9926299 ,9927299 ,9931399 ,9932399 ,9935399 ,9938399 ,9957599 ,9965699 ,9978799 ,9980899 ,9981899 ,9989899};
int a,b;
cin>>a>>b;
for(int i=0;i<779;i++)//遍历所有回文素数
{
if(c[i]<a)continue;
if(c[i]>b)break;
else//满足a~b区间我们就输出，就是如此简单
{
cout<<c[i]<<endl;
}
}
return 0;
}
这就是第一道例题~
例题2：阶乘之和
首先看阶乘规模，到了50，一般做法一定会“空间爆炸”（MLE）。如果不用高精度做的话，就是打表法了。
当时做这道题的时候，我受第一道例题的启发，想到了可以用打表法。思路咱们就代码吧。
#include<iostream>
#include<algorithm>
#include<cmath>
#include<map>
#include<queue>
#include<stack>
#include<stdlib.h>
#include<stdio.h>
#include<cstring>
#include<set>
using namespace std;
string f(string a,string b)//计算两字符串的和，返回值仍然是字符串
{
int la=a.size(),lb=b.size();
string ans;
reverse(a.begin(),a.end()),reverse(b.begin(),b.end());//先反转，便于后续操作
if(la<lb)//平衡字符串长度
{
int cha=lb-la;
for(int i=0; i<cha; i++)a+='0';
}
else
{
int cha=la-lb;
for(int i=0; i<cha; i++)b+='0';
}
int len=max(la,lb);
for(int i=0; i<len; i++)//从低位开始，一位一位对应相加，相加的和逢十进一
{
int v1=a[i]-'0';
int v2=b[i]-'0';
int v3=v1+v2;
if(v3<10)
{
ans+='0'+v3;
}
else//逢十进一
{
ans+='0'+(v3-10);
if(i+1<len)
a[i+1]+=1;
else//处理越界
ans+='1';
}
}
reverse(ans.begin(),ans.end());
return ans;
}
int main()
{
string jc[51]= {"1", "1","2","6","24","120","720","5040","40320","362880","3628800","39916800","479001600","6227020800","87178291200","1307674368000","20922789888000","355687428096000","6402373705728000","121645100408832000","2432902008176640000","51090942171709440000","1124000727777607680000","25852016738884976640000","620448401733239439360000","15511210043330985984000000","403291461126605635584000000","10888869450418352160768000000","304888344611713860501504000000","8841761993739701954543616000000","265252859812191058636308480000000","8222838654177922817725562880000000","263130836933693530167218012160000000","8683317618811886495518194401280000000","295232799039604140847618609643520000000","10333147966386144929666651337523200000000","371993326789901217467999448150835200000000","13763753091226345046315979581580902400000000","523022617466601111760007224100074291200000000","20397882081197443358640281739902897356800000000","815915283247897734345611269596115894272000000000","33452526613163807108170062053440751665152000000000","1405006117752879898543142606244511569936384000000000","60415263063373835637355132068513997507264512000000000","2658271574788448768043625811014615890319638528000000000","119622220865480194561963161495657715064383733760000000000","5502622159812088949850305428800254892961651752960000000000","258623241511168180642964355153611979969197632389120000000000","12413915592536072670862289047373375038521486354677760000000000","608281864034267560872252163321295376887552831379210240000000000","30414093201713378043612608166064768844377641568960512000000000000"};//提前储存好1~50的阶乘，用字符串表示
int n;
cin>>n;
string s;
s="1";
for(int i=2; i<=n; i++)
{
s=f(s,jc[i]);//累加
}
cout<<s;
return 0;
}
我的思路是：
用字符串存储数据，这样不会超限，（这里省略了打表操作），然后，把传统的数据相加，转换成字符串相加，相加的逻辑和我们小学学的竖式加法，从低位开始，逢十进一。简而言之：“打表法”+“字符串加法“
当时写完代码提交上去，看到“AC”通过就很开心，觉得自己真聪明，然后，然后，就被打脸了哈哈。因为我看到了别人更nb的打表法，瞬间自己就觉得自己刚刚飘了。
附上他人代码：
所以，还是要保持谦逊的，人外有人啊~
