什么是Calkin Wilf序列?
Calkin-Wilf树(或序列)是一种特殊的二叉树,它是通过从分数1/1开始并在每个分数a / b之下迭代地添加a /(a + b)和(a + b)/ b来获得的。该树生成每个有理数。按顺序写出术语将给出1 / 1、1 / 2、2 / 1、1 / 3、3 / 2、2 / 3、3 / 1、1 / 4、4 / 3、3 / 5、5 / 2,2/5,5/3,3/4,4/1,…序列具有以下特性:每个分母是下一个分子。
上图是列出所有有理数的Calkin-Wilf树。节点a / b的子代计算为a /(a + b)和(a + b)/ b。
任务是在此树的广度优先遍历中找到第n个有理数。
例子:
Input : 13
Output : [5, 3]
Input : 5
Output : [3, 2]
说明:这棵树是一个完美的二叉搜索树,我们需要floor(log(n))步骤来计算第n个有理数。该概念类似于在二叉搜索树中进行搜索。给定n,我们将其一直除以2,直到得到0。我们按以下方式在每个阶段返回分数:
if n%2 == 0
update frac[1]+=frac[0]
else
update frac[0]+=frac[1]
下面是在Calkin Wilf序列中找到第n个数字的程序:
C++
// C++ program to find the
// nth number in Calkin
// Wilf sequence:
# include
using namespace std;
int frac[] = {0, 1};
// returns 1x2 int array
// which contains the nth
// rational number
int nthRational(int n)
{
if (n > 0)
nthRational(n / 2);
// ~n&1 is equivalent to
// !n%2?1:0 and n&1 is
// equivalent to n%2
frac[~n & 1] += frac[n & 1];
}
// Driver Code
int main()
{
int n = 13; // testing for n=13
// converting array
// to string format
nthRational(n);
cout << "[" << frac[0] << ","
<< frac[1] << "]" << endl;
return 0;
}
// This code is contributed
// by Harshit Saini
Java
// Java program to find the nth number
// in Calkin Wilf sequence:
import java.util.*;
public class GFG {
static int[] frac = { 0, 1 };
public static void main(String args[])
{
int n = 13; // testing for n=13
// converting array to string format
System.out.println(Arrays.toString(nthRational(n)));
}
// returns 1x2 int array which
// contains the nth rational number
static int[] nthRational(int n)
{
if (n > 0)
nthRational(n / 2);
// ~n&1 is equivalent to !n%2?1:0
// and n&1 is equivalent to n%2
frac[~n & 1] += frac[n & 1];
return frac;
}
}
Python3
# Python program to find
# the nth number in Calkin
# Wilf sequence:
frac = [0, 1]
# returns 1x2 int array
# which contains the nth
# rational number
def nthRational(n):
if n > 0:
nthRational(int(n / 2))
# ~n&1 is equivalent to
# !n%2?1:0 and n&1 is
# equivalent to n%2
frac[~n & 1] += frac[n & 1]
return frac
# Driver code
if __name__ == "__main__":
n = 13 # testing for n=13
# converting array
# to string format
print(nthRational(n))
# This code is contributed
# by Harshit Saini
C#
// C# program to find the nth number
// in Calkin Wilf sequence:
using System;
class GFG
{
static int[] frac = { 0, 1 };
public static void Main(String []args)
{
int n = 13; // testing for n=13
// converting array to string format
Console.WriteLine("[" + String.Join(",",
nthRational(n)) + "]");
}
// returns 1x2 int array which
// contains the nth rational number
static int[] nthRational(int n)
{
if (n > 0)
nthRational(n / 2);
// ~n&1 is equivalent to !n%2?1:0
// and n&1 is equivalent to n%2
frac[~n & 1] += frac[n & 1];
return frac;
}
}
// This code is contributed by 29AjayKumar
输出:
[5, 3]
解释:
对于n = 13
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