JMU CS280 SP18 WK06


2018-02-13 21:30 CET

JMU CS280 SP18 WK06


2018-02-17 05:59 CET
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Problem D
A Rational Sequence (Take 3)

A sequence of positive rational numbers is defined as follows:

An infinite full binary tree labeled by positive rational numbers is defined by:

  • The label of the root is $1/1$.

  • The left child of label $p/q$ is $p/(p+q)$.

  • The right child of label $p/q$ is $(p+q)/q$.

The top of the tree is shown in the following figure:


The sequence is defined by doing a level order (breadth first) traversal of the tree (indicated by the light dashed line). So that:

\[ F(1) = 1/1, F(2) = 1/2, F(3) = 2/1, F(4) = 1/3, F(5) = 3/2, F(6) = 2/3, \ldots \]

Write a program to compute the $n^{\text {th}}$ element of the sequence, $F(n)$. Does this problem sound familiar? Well it should! We had variations of this problem at the $2014$ and $2015$ Greater NY ACM ICPC Regionals.


The first line of input contains a single integer $P$, ($1 \le P \le 1\, 000$), which is the number of data sets that follow. Each data set should be processed identically and independently. Each data set consists of a single line of input. It contains the data set number, $K$, and the index, $N$, of the sequence element to compute ($1 \le N \le 2\, 147\, 483\, 647$).


For each data set there is a single line of output. It contains the data set number, $K$, followed by a single space which is then followed by the numerator of the fraction, followed immediately by a forward slash (‘/’) followed immediately by the denominator of the fraction. Inputs will be chosen so neither the numerator nor the denominator will overflow an $32$-bit unsigned integer.

Sample Input 1 Sample Output 1
1 1
2 4
3 11
4 1431655765
1 1/1
2 1/3
3 5/2
4 2178309/1346269