URAL 1081 Binary Lexicographic Sequence （递推 + 递归）

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大体题意：

定义一个合法的二进制序列为序列中没有两个1是相邻的，对于所有长度为n的合法序列按照字典序排序后（保留前导0），求第K大的串是多少？

思路：

我们先得求出一个n位的合法二进制的个数是多少！

f[1] = 2;

f[2] = 3

f[3] = 5

f[4] = 8

这几个很容易算出来，然后我们就很明显的发现规律了，这是一个斐波那契数列！

其实想一想确实这个样！

你构造第n位时（高位），要么填0，要么填1，填0的话，直接是f(n-1),填1的话，要填10，结果是f(n-2)

因此 f(n) = f(n-1) + f(n-2);

然后在考虑第K大！

f(n-2)的数第一个数是1，f(n-1) 第一个数是0，显然f(n-1)字典序要小！

因此如果k <= f(n-1) ，那么这一位必然填0，继续递归！

相反如果k > f(n-1) 这一位必然是10，继续递归！

注意 k 不合法时，要输出-1，没看到wa了一次！

详细见代码：

#include <cstdio>#include <cstring>#include <algorithm>#include <iostream>#include <string>using namespace std;typedef long long ll;int n, k;ll f[50];void init(){    f[0] = 1;    f[1] = 2;    for (int i = 2; i <= 45; ++i){        f[i] = f[i-1] + f[i-2];    }}string ans;void dfs(int c,int o){    if (c == 0){        return;    }    if (c == 1){        if (o == 1) {            ans += "0";            return;        }        ans += "1";        return;    }    if (f[c-1] >= o){        ans += "0";        dfs(c-1,o);    }    else {        ans += "10";        dfs(c-2,o-f[c-1]);    }}int main(){    init();    while(scanf("%d %d",&n, &k)!=EOF){        ans = "";        if (k > f[n]){            puts("-1");            continue;        }        dfs(n,k);        printf("%s\n",ans.c_str());    }    return 0;}

1081. Binary Lexicographic Sequence

Time limit: 0.5 second
Memory limit: 64 MB

Consider all the sequences with length (0 < N < 44), containing only the elements 0 and 1, and no two ones are adjacent (110 is not a valid sequence of length 3, 0101 is a valid sequence of length 4). Write a program which finds the sequence, which is on K-th place (0 < K < 109) in the lexicographically sorted in ascending order collection of the described sequences.

Input

The first line of input contains two positive integers N and K.

Output

Write the found sequence or −1 if the number K is larger then the number of valid sequences.

Sample

inputoutput

3 1

Problem Author: Emil Kelevedzhiev
Problem Source: Winter Mathematical Festival Varna '2001 Informatics Tournament

Tags: dynamic programming ()

Difficulty: 255 Printable version Submit solution Discussion (24)
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