P2657 [SCOI2009]windy數

題目描述

windy定義了一種windy數。不含前導零且相鄰兩個數字之差至少為2的正整數被稱為windy數。 windy想知道，

在A和B之間，包括A和B，總共有多少個windy數？

輸入輸出格式

輸入格式：

包含兩個整數，A B。

輸出格式：

一個整數

輸入輸出樣例

輸入樣例#1： 

1 10

輸出樣例#1： 

9

輸入樣例#2： 

25 50

輸出樣例#2： 

20

說明

100%的數據，滿足 1 <= A <= B <= 2000000000 。

 

Solution：

　　本題顯然數位$DP$，暫時不會（留着填坑～）。

　　提供打表的思路，先線下每$10^6$個處理一次，統計出$2000$個答案（前綴和$sum[i]$表示$1$到$i*10^6$中滿足條件的個數）。

　　那么查詢時就直接瞎搞模擬，最多計算$10^6$次。

打表代碼：

 

/**************************************************************
    Problem: 1026
    User: five20
    Language: C++
    Result: Accepted
    Time:256 ms
    Memory:1296 kb
****************************************************************/
 
#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;

 

 

 

 

過來填坑啦～：2018-05-26

　　本題套上一個數位$dp$的板子。定義狀態$f[i][j]$表示第$i$位為數字$j$時的合法個數。

　　由於本題的約數條件是相鄰兩位數字之差不小於$2$（且所有個位數均視為滿足條件），直接套板子肯定有問題，比如$[45,4500]$，枚舉時當前幾位連續為$0$時必須往后判斷，否則會出兮兮，那么就必須去處理前導$0$的問題。

　　於是在普通的約束上界$limit1$的條件下（即每次枚舉數字時是否受限），再加一個$limit2$來判斷前導$0$的情況。

　　那么每次枚舉當前位數字時，判斷一下上一位是否受$limit2$的限制且當前位為$0$，是的話就說明這時存在前導$0$，往后搜索時將$lst$賦值為$-2$（這樣$0-(-2)\geq 2$）。

　　然后所有的不受限制的情況都記憶化一下，最后前綴和思想相減就是答案了。

$DP$代碼：

#include<bits/stdc++.h>
#define il inline
#define ll long long
#define For(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
using namespace std;
ll a,b,f[20][10],cnt,p[20];
il ll dfs(int pos,int lst,bool limit1,bool limit2){
    if(!pos)return 1;
    if(!limit2&&!limit1&&f[pos][lst]!=-1)return f[pos][lst];
    int op,up=limit1?p[pos]:9;
    ll tmp=0;
    For(i,0,up){
        if(abs(i-lst)<2)continue;
        op=i;
        if(limit2&&!i)op=-2;
        tmp+=dfs(pos-1,op,(limit1)&&(i==up),(op==-2));
    }
    if(!limit1&&!limit2)f[pos][lst]=tmp;
    return tmp;
}
il ll solve(ll x){
    cnt=0;
    while(x)p[++cnt]=x%10,x/=10;
    return dfs(cnt,-2,1,1);
}
int main(){
    cin>>a>>b;
    memset(f,-1,sizeof(f));
    printf("%lld\n",solve(b)-solve(a-1));
    return 0;
}