$$x_{k} = x_{k-1} modulo x_{k-2}.$$
'modulo' binary operation returns the remainder from a given division. Stopping criterion for the recurrence relation is reached when $x_{k-2}=0$ and the result of GCD will be the current value of $x_{k-1}$. This process can be visualised as successive divisions. Let's implement this in R in a naive way.
# Naive Euclid algorithm by msuzen gcd <- function(a, b) { rk_1 <- a; rk_2 <- b; # Recurrence Formula: r_k = r_k-1 modulo r_k-2 # Increment k until r_k-2 == 0 while(rk_2 != 0) { rk <- rk_1%%rk_2; # remainder rk_1 <- rk_2; # proceed in recurrence rk_2 <- rk; } return(rk_1) }
This is a straight forward task. Let's make the problem little more generic. What happends if we would like to know GCD of $n$ natural numbers, $x_{1},..., x_{n}$? Than, a solution is to apply GCD operation pairwise, for example if $n=3$:
$$GCD(x_{1}, GCD(x_{2}, x_{3})) = GCD(GCD(x_{1}, x_{2}), x_{3})$$
How can we implement this for a vector of non-negative integers?
Tree Approach
The simplest way to reach GCD of $n$ numbers is probably thinking of this process as a binary tree, formed by pairing elements of set of integers as we obtain GCDs. It is relatively easy to implement this because ordering of pairs is not important. We can start from the beginning and pair up as we obtain the results. Here is the naive implementation.
1 2 3 4 5 6 7 8 | gcdN <- function(X) { n <- length(X) gcdEnd <- X[1] for(i in 2:n) { gcdEnd <- gcd(gcdEnd, X[i]) } return(gcdEnd) } |
'Reduce' operation
So called tree approach we have given above is actually noting but a Reduce operation in the context of MapReduce. The function gcdN can be replaced with a single line.
Reduce("gcd", X)