Lightoj 1009. K-based Numbers(DP)-优快云博客

本文探讨了K进制有效数字的计数方法，详细介绍了如何计算包含N位的有效K进制数字的数量，考虑了K进制下有效数字的定义，并通过递归和动态规划两种方法给出了求解步骤。

1009. K-based Numbers

Time limit: 1.0 second
Memory limit: 64 MB

Let’s consider K-based numbers, containing exactlyN digits. We define a number to be valid if itsK-based notation doesn’t contain two successive zeros. For example:

1010230 is a valid 7-digit number;
1000198 is not a valid number;
0001235 is not a 7-digit number, it is a 4-digit number.

Given two numbers N and K, you are to calculate an amount of validK based numbers, containingN digits.

You may assume that 2 ≤ K ≤ 10; N ≥ 2;N +K ≤ 18.

Input

The numbers N and K in decimal notation separated by the line break.

Output

The result in decimal notation.

搜索一次就可以了，记录一下当前位置的数字还有长度

/***********************************************
 * Author: fisty
 * Created Time: 2015/6/10 10:26:53
 * File Name   : 1009.cpp
 *********************************************** */
#include <iostream>
#include <cstring>
#include <deque>
#include <cmath>
#include <queue>
#include <stack>
#include <list>
#include <map>
#include <set>
#include <string>
#include <vector>
#include <cstdio>
#include <bitset>
#include <algorithm>
using namespace std;
#define Debug(x) cout << #x << " " << x <<endl
#define Memset(x, a) memset(x, a, sizeof(x))
const int INF = 0x3f3f3f3f;
typedef long long LL;
typedef pair<int, int> P;
#define FOR(i, a, b) for(int i = a;i < b; i++)
int n, k;  
int ans;
void dfs(int cur, int m){
    //cur: 现在这一位 m:长度
    if(m == n) { ans++; return ;}
    for(int i = 0;i < k; i++){
        if(cur == 0 && i == 0) continue;  //当前一位与下一位不能同时为0
        dfs(i, m+1);
    }
}
int main() {
    //freopen("in.cpp", "r", stdin);
    cin.tie(0);
    ios::sync_with_stdio(false);
    cin >> n >> k;
    
    ans = 0;
    for(int i = 1;i < k; i++){
        dfs(i, 1);
    }

    cout << ans << endl;
    return 0;
}

还有就是写成递推的形式，第i位只有0和非0

定义dp[i][0]为第i位为0，dp[i][1] 为第i位非0

则 dp[i][0] = dp[i-1][1];

dp[i][1] = (k-1)*(dp[i][1-1] + dp[i-1][0]) (2 <= i <= n)

/***********************************************
 * Author: fisty
 * Created Time: 2015/6/10 10:26:53
 * File Name   : 1009.cpp
 *********************************************** */
#include <iostream>
#include <cstring>
#include <deque>
#include <cmath>
#include <queue>
#include <stack>
#include <list>
#include <map>
#include <set>
#include <string>
#include <vector>
#include <cstdio>
#include <bitset>
#include <algorithm>
using namespace std;
#define Debug(x) cout << #x << " " << x <<endl
#define Memset(x, a) memset(x, a, sizeof(x))
const int INF = 0x3f3f3f3f;
typedef long long LL;
typedef pair<int, int> P;
#define FOR(i, a, b) for(int i = a;i < b; i++)
int n, k;  
int ans;
LL dp[20][2];
int main() {
    //freopen("in.cpp", "r", stdin);
    cin.tie(0);
    ios::sync_with_stdio(false);
    cin >> n >> k;
    
    dp[1][0] = 0; dp[1][1] = k-1;
    for(int i = 2;i <= n; i++){
        dp[i][1] = (k-1) * (dp[i-1][1] + dp[i-1][0]);
        dp[i][0] = dp[i-1][1];
    }

    cout << dp[n][1] + dp[n][0] << endl;
    return 0;
}