1009. K-based Numbers

https://acm.timus.ru/problem.aspx?space=1&num=1009

思路

典型dp问题

对于n位k位底的数， 求不存在连续0出现的数目。

设f(i) 为 i位，最后一位为不为0的数目

设g(i)为i位，最后一位为0的数目

g(1) = 0 --- 1位数，且最低位为0， 最高位不为0， 不存在这种情况，顾值为0

f(1) = k-1 ， 一位数， 最低位不为0，最高位也不为0

转移公式

g(i) = f(i-1)  ---- 对于i位数，最低位为0， 其总数目等于前i-1位最低位为1的总数目

f(i) = (f(i-1) + g(i-1)) * (k-1)

下图中的1表示不为0， 为1-》k-1

https://whiteboard-online.org/boards/DFIdh9VqHW8BhPbbyWgNTu8zNJNByPrAkSwP2NOX1ak-#70,0,0.9

Code

#include <iomanip>
#include <bits/stdc++.h>
#include <iostream>
using namespace std;
#include <limits.h>
#include <math.h>
#include <vector>
#include <set>
#include <map>
#include <stack>
#include <queue>

typedef long long LL;
typedef pair<int, int> pii;
typedef pair<LL, LL> pll;
typedef pair<string, string> pss;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<pii> vii;
typedef vector<LL> vl;
typedef vector<vl> vvl;

double EPS = 1e-9;
int INF = 1000000005;
long long INFF = 1000000000000000005LL;
double PI = acos(-1);
int dirx[8] = { -1, 0, 0, 1, -1, -1, 1, 1 };
int diry[8] = { 0, 1, -1, 0, -1, 1, -1, 1 };

#ifdef TESTING
#define DEBUG fprintf(stderr, "====TESTING====\n")
#define VALUE(x) cerr << "The value of " << #x << " is " << x << endl
#define debug(...) fprintf(stderr, __VA_ARGS__)
#else
#define DEBUG
#define VALUE(x)
#define debug(...)
#endif

#define FOR(a, b, c) for (int(a) = (b); (a) < (c); ++(a))
#define FORN(a, b, c) for (int(a) = (b); (a) <= (c); ++(a))
#define FORD(a, b, c) for (int(a) = (b); (a) >= (c); --(a))
#define FORSQ(a, b, c) for (int(a) = (b); (a) * (a) <= (c); ++(a))
#define FORC(a, b, c) for (char(a) = (b); (a) <= (c); ++(a))
#define FOREACH(a, b) for (auto&(a) : (b))
#define REP(i, n) FOR(i, 0, n)
#define REPN(i, n) FORN(i, 1, n)
#define MAX(a, b) a = max(a, b)
#define MIN(a, b) a = min(a, b)
#define SQR(x) ((LL)(x) * (x))
#define RESET(a, b) memset(a, b, sizeof(a))
#define fi first
#define se second
#define mp make_pair
#define pb push_back
#define ALL(v) v.begin(), v.end()
#define ALLA(arr, sz) arr, arr + sz
#define SIZE(v) (int)v.size()
#define SORT(v) sort(ALL(v))
#define REVERSE(v) reverse(ALL(v))
#define SORTA(arr, sz) sort(ALLA(arr, sz))
#define REVERSEA(arr, sz) reverse(ALLA(arr, sz))
#define PERMUTE next_permutation
#define TC(t) while (t--)

/******************************** COMMON FUNC START ***************************************/

LL quick_pow(LL x,LL n,LL m){
    LL res = 1;
    while(n > 0){
        if(n & 1)    res = res * x % m;
        x = x * x % m;
        n >>= 1;//相当于n=n/2.详情请参考位移运算符。
    }

    return res;
}

inline string IntToString(LL a)
{
    char x[100];
    sprintf(x, "%lld", a);
    string s = x;
    return s;
}

inline LL StringToInt(string a)
{
    char x[100];
    LL res;
    strcpy(x, a.c_str());
    sscanf(x, "%lld", &res);
    return res;
}

inline string GetString(void)
{
    char x[1000005];
    scanf("%s", x);
    string s = x;
    return s;
}

inline string uppercase(string s)
{
    int n = SIZE(s);
    REP(i, n)
    if (s[i] >= 'a' && s[i] <= 'z')
        s[i] = s[i] - 'a' + 'A';
    return s;
}

inline string lowercase(string s)
{
    int n = SIZE(s);
    REP(i, n)
    if (s[i] >= 'A' && s[i] <= 'Z')
        s[i] = s[i] - 'A' + 'a';
    return s;
}

inline void OPEN(string s)
{
#ifndef TESTING
    freopen((s + ".in").c_str(), "r", stdin);
    freopen((s + ".out").c_str(), "w", stdout);
#endif
}

/******************************** COMMON FUNC END ***************************************/

/******************************** GRAPH START ***************************************/
// Graph class represents a directed graph
// using adjacency list representation
class Graph {
public:
    map<int, bool> visited;
    map<int, list<int> > adj;

    // function to add an edge to graph
    void addEdge(int v, int w);

    // DFS traversal of the vertices
    // reachable from v
    void DFS(int v);
};

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v’s list.
}

void Graph::DFS(int v)
{
    // Mark the current node as visited and
    // print it
    visited[v] = true;
    cout << v << " ";

    // Recur for all the vertices adjacent
    // to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFS(*i);
}

/******************************** GRAPH END ***************************************/

/*
https://atcoder.jp/contests/abcxxx/tasks/abcxxx_d
*/

int n, k;

int f[20], g[20];

int main()
{
    cin >> n >> k;

    g[1] = 0;
    f[1] = k-1;
    
    for(int i=2; i<=n; i++){
        f[i] = (f[i-1] + g[i-1]) * (k-1);
        g[i] = f[i-1];
    }

    cout << (f[n] + g[n]) << endl;

    return 0;
}