打表打表

算法

你总会遇见很多题目，看着有点逻辑，但是数学推导非常难受。看着有点规律，但是样例太少无法确定。或者你很肯定的能给出暴力做法，但是卡你时间复杂度。

这些种种情况，我们都可以用到打表！打表一般用到的就是暴力循环，暴力判断，dfs bfs。（这里将不定时更新作者遇到的有趣的打表题）

什么是打表？

我们先说说狭义的打表（仅代表个人理解）

打表就是将所有输入情况的答案保存在代码中，输入数据后直接输出就可以了

打表法具有快速，易行（可以写暴力枚举程序）的特点，缺点是代码可能太大，或者情况覆盖不完

对于不会超时，数据规模适合打表，为了简洁你也可以打表。

比如这道题

这道题n<=20完全可以打表

只要你得出每个数字需要多少个火柴，问题就简单了。而不管几位的数字都是

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int a[10]={6,2,5,5,4,5,6,3,7,6}

这就是0-9需要的火柴数量。往后以此类推，10需要一个1和一个0。11需要两个1，也就是说我们一位位拆分即可。

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#include<bits/stdc++.h>
using namespace std;
int main(){
	freopen("ans.txt","w",stdout);
	int a[10]={6,2,5,5,4,5,6,3,7,6};
	int i,j,temp=0,num=0,k,in[2020],n;
	in[0]=6;
	for(i=1;i<=2000;i++){
		k=i;
		temp=0;
		while(k){
			temp+=a[k%10];
			k/=10;
		}
		in[i]=temp;
	}
	n=0;
	Again:
	num=0;
	for(i=0;i<=999;i++){
		for(j=0;j<=999;j++){
			if(n==in[i]+in[j]+in[i+j]+4) num++;
		}
	}
	printf("%d,",num);
	if(n<24){n++;goto Again;}
	return 0;
}

然后得到答案你就可以这样，面向答案编程

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#include<iostream>
using namespace std;
int ans[]={0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,8,9,6,9,29,39,38,65,88,128};
int n;
int main(){
	cin>>n;
	cout<<ans[n]<<'\n';
	return 0;
}

也许这道题你还没有领略到打表的方便之处。

试试这道题

数位dp题，你也许会觉得难。但是没关系，我们提供更暴力更简单的方法。

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#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cmath>
#define For(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
using namespace std;
const int N=1e6+5;
int ans,a,b,sum[2005]={0, 202174, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155315, 136131, 138503, 138214, 138252, 138252, 138214, 138503, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 138503, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 0, 0, 0, 138214, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 0, 0, 0, 138252, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 0, 0, 0, 138252, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 0, 0, 0, 138214, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 0, 0, 0, 138503, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 0, 0, 0, 136131, 155315, 155315, 136131, 138503, 138214, 138252, 138252, 0, 0, 0, 155315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
inline void check(int x){
    if(x<=9){ans++;return;}
    int a=x,b=-5;
    while(a){
        if(abs(a%10-b)<2)return;
        b=a%10;a=a/10;
    }
    ans++;
}
int main(){
    cin>>a>>b;
    if(b-a<=1000000){
        For(i,a,b)check(i);
        cout<<ans;
        return 0;
    }
    int p=ceil(a*1.0/1000000),q=floor(b*1.0/1000000);
    For(i,1,2000)sum[i]+=sum[i-1];
    ans+=sum[q]-sum[p];
    if(!ans)ans++;
    p=p*1000000,q=q*1000000;
    if(a!=p)
    For(i,a,p-1)check(i);
    if(b!=q)
    For(i,q+1,b)check(i);
    cout<<ans;
    return 0;

这里引用了five20的代码。由于过于暴力我们就不多解释了。这一长串数组就是打表打出来的。然后直接暴力筛选。

打表真的有用吗？

有的朋友。也许我们不能遇见上面这些及其粗暴的方法能写的题目。但是往往我们可以找到很多答案来寻找规律。这就是我理解的广义的打表。

可以看看这道题

题意：构造一个矩阵，使相邻两个数之和两两不重复。

刚拿到这道题目的你是不是有点摸不着头脑？

这很明显你不能很暴力的来遍历拿到一个矩阵，所以我们需要找到某种特殊的方式，让我们按照这个方式构造出的矩阵一定相邻两个数之和不重复。

这时候我们不妨打表。注释写在代码上了。

原理就是尝试填充一个 n x m的矩阵，使得矩阵中相邻元素的和都是唯一的。如果找到这样的填充方式，它会输出该矩阵。

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#include<bits/stdc++.h>
using namespace std;
int a[1005][1005];
int n,m;
int tot=0;
bool vis[10000];
bool check()
{
    set<int>s;
    for(int i=1;i<=n-1;i++)
    {
        for(int j=1;j<=m-1;j++)
        {
            if(s.count(a[i][j]+a[i+1][j]))  // 检查当前元素和下方元素的和
            {
                return false;
            }
            if(s.count(a[i][j]+a[i][j+1]))  // 检查当前元素和右方元素的和
            {
                return false;
            }
            s.insert(a[i][j]+a[i][j+1]);    // 插入当前元素和右方元素的和
            s.insert(a[i][j]+a[i+1][j]);    // 插入当前元素和下方元素的和
        }
    }
    // 检查最后一列的元素和下方元素的和
    for(int i=1;i<=n-1;i++)
    {
        if(s.count(a[i][m]+a[i+1][m]))return false;
        s.insert(a[i][m]+a[i+1][m]);
    }
    // 检查最后一行的元素和右方元素的和
    for(int i=1;i<=m-1;i++)
    {
        if(s.count(a[n][i]+a[n][i+1]))return false;
        s.insert(a[n][i]+a[n][i+1]);
    }
    return true;
}
void dfs(int dep)
{
    tot++;
    if(dep==n*m+1)
    {
        if(check())
        {
            for(int i=1;i<=n;i++)
            {
                for(int j=1;j<=m;j++)
                {
                    cout <<a[i][j]<<" ";
                }
                cout<<endl;
            }
        }
        return;
    }
    for(int i=1;i<=n*m;i++)
    {
        if(vis[i]) continue;  // 如果数字i已经被使用，跳过
        vis[i]=1;             // 标记数字i为已使用
        int hang;
        if(tot%m==0)hang=tot/m;
        else hang=tot/m+1;
        int lie=tot%m;
        if(lie==0)lie=m;
        a[hang][lie]=i;       // 将数字i放入矩阵的当前位置
        dfs(dep+1);           // 递归填充下一个位置
        a[hang][lie]=-1;      // 回溯：恢复矩阵的当前位置
        tot--;                // 回溯：恢复tot
        vis[i]=0;             // 回溯：恢复数字i的标记
    }
}

}
int main()
{
    int t;
    cin >>t;
    while(t--)
    {
        cin >>n>>m;
        dfs(1);
    }
}

打表出来我们发现一个规律

如果是奇数行那不动，偶数行循环右移一位即可（不要问我循环右移是什么）

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#include<bits/stdc++.h>
using namespace std;
#define int long long 
#define asd(i,a,b) for(int i=a;i<=b;i++)
#define des(i,a,b) for(int i=a;i>=b;i--)
const int N=2e5+7,inf=2e18,mod=998244353;

void solve(){
    int n,m;cin>>n>>m;
    cout<<"Yes"<<endl;
    if(m&1){
        int cnt=0;
        asd(i,1,n){
            if(i&1){
                asd(j,1,m)cout<<++cnt<<" ";
            }else{
                int tmp=++cnt;

                asd(j,2,m)cout<<++cnt<<" ";
                cout<<tmp<<" ";
            }
            cout<<endl;
        }
    }else{
        int cnt=0;
        asd(i,1,n){
            asd(j,1,m)cout<<++cnt<<" ";
            cout<<endl;
        }
    }
    
}
signed main(){
    ios::sync_with_stdio(0),cin.tie(0),cout.tie(0);
    int t=1;
    cin>>t;
    while(t--){
        solve();
    }
    return 0;
}

附上代码。