10077 - The Stern-Brocot Number System

Problem C

The Stern-Brocot Number System

Input: standard input

Output: standard output

 

The Stern-Brocot tree is a beautiful way for constructing the set ofall nonnegative fractionsm / n where m and n are relativelyprime. The idea is to start with two fractions and then repeat thefollowing operations as many times as desired:

Insert  between two adjacentfractions and.

For example, thefirst step gives us one new entry between  and,

and the nextgives two more:

The next givesfour more,

 

and then we willget 8, 16, and so on. The entire array can be regarded as an infinite binarytree structure whose top levels look like this:

 

 

The constructionpreserves order, and we couldn't possibly get the same fraction in twodifferent places.

We can, in fact,regard the Stern-Brocot tree as a number system for representing rationalnumbers, because each positive, reduced fraction occurs exactly once. Let's usethe lettersL and R to stand for going down to the left orright branch as we proceed from the root of the tree to a particular fraction;then a string ofL's and R's uniquely identifies a place in thetree. For example,LRRL means that wego left from  down to, then right to, then right to, then left to. We can considerLRRLto be a representation of . Every positive fraction gets represented in this way as aunique string ofL's and R's.

Well, actuallythere's a slight problem: The fraction  corresponds to theempty string, and we need a notation forthat. Let's agree to call it I,because that looks something like 1 and it stands for "identity".

In this problem,given a positive rational fraction, you are expected to represent it inStern-Brocot number system.

Input

The input file contains multipletest cases. Each test case consists of a line contains two positive integersm and n where m and n are relatively prime. The inputterminates with a test case containing two 1's form and n, and this casemust not be processed.

Output

For each test case in the inputfile output a line containing the representation of the given fraction in theStern-Brocot number system.

Sample Input

5 7
878 323
1 1

 

Sample Output

LRRL
RRLRRLRLLLLRLRRR

#include <stdio.h>struct {int x;int y;}tree[10];int main(void){int a,b;while(scanf("%d%d",&a,&b)==2){if(a==1&&b==1) break;tree[0].x=0,tree[0].y=1;tree[1].x=1,tree[1].y=1;tree[2].x=1,tree[2].y=0;while(tree[1].x!=a||tree[1].y!=b){if(a*tree[1].y>b*tree[1].x){printf("R");tree[0]=tree[1]; tree[1].x+=tree[2].x,tree[1].y+=tree[2].y;}else{printf("L");tree[2]=tree[1];tree[1].x+=tree[0].x,tree[1].y+=tree[0].y;}}printf("\n");}return 0;}