**Thinking inside the box**, and kindly contributed to R-bloggers)

There was another question recently on StackOverflow that I had meant to

discuss in a follow-up post here. User deltanovember asked

about slow recursive functions

and used the very classic

Fibonacci number

as an example. To recap, Fibonacci number are defined with two initial

values F(0) = 0 and F(1) = 1; thereafter the Fibonacci number F(n) is defined

as the sum of the two preceding numbers: F(n) = F(n-2) + F(n-1).

This leads to very straightforward implementations using recursion:

## R implementation of recursive Fibonacci sequence fibR <- function(n) { if (n == 0) return(0) if (n == 1) return(1) return (fibR(n - 1) + fibR(n - 2)) }

Unfortunately, this elegant implementation which remain close to the abtract

formulation of the recurrence algorithm performs *very* poorly in R as

there is noticeable overhead in function calls which becomes dominant in a

recursion. This lead to the original question on StackOverflow, and the

accepted answer uses a trick presented by Pat Burns in his lovely

R Inferno:

rewrite the solution using a computer science trick called memoization:

fibonacci <- local({ memo <- c(1, 1, rep(NA, 100)) f <- function(x) { if(x == 0) return(0) if(x < 0) return(NA) if(x > length(memo)) stop("'x' too big for implementation") if(!is.na(memo[x])) return(memo[x]) ans <- f(x-2) + f(x-1) memo[x] <<- ans ans } })

That is a fair answer, and even more was suggested with a link to a terrific

analysis calling the Fibonacci recurrence

the worst algorithm in the world.

That is also fair, but all the basic research into better algorithms

exploiting some structure of the problem to advance performance (and of

course understanding) is overlooking one crucial part: *algorithm analysis
is essentially independent of the language*. So whatever improvements we

obtain by thinking

**really hard**about a problem are then

available for other implementations too.

So with a tip of the hat to the old Larry Wall quote about

Lazyness, Impatience and Hubris, I would like to suggest what I consider a

much simpler route to much better performance: recode it in C++ using

both Rcpp (for the R/C++ integration) and

inline for the on-the-fly compilation, linking and loading of C++ code into R.

## inline to compile, load and link the C++ code require(inline) ## we need a pure C/C++ function as the generated function ## will have a random identifier at the C++ level preventing ## us from direct recursive calls incltxt <- ' int fibonacci(const int x) { if (x == 0) return(0); if (x == 1) return(1); return (fibonacci(x - 1)) + fibonacci(x - 2); }' ## now use the snippet above as well as one argument conversion ## in as well as out to provide Fibonacci numbers via C++ fibRcpp <- cxxfunction(signature(xs="int"), plugin="Rcpp", incl=incltxt, body =' int x = Rcpp::as<int>(xs); return Rcpp::wrap( fibonacci(x) ); ')

This single R function call `cxxfunction()`

takes the code

embedded in the arguments to the `body`

variable (for the core

function) and the `incltxt`

variable for the helper function we

need to call. This helper function is needed for the recursion as

`cxxfunction()`

will use an randomized internal identifier for the

function called from R preventing us from calling this (unknown) indentifier.

But the rest of the algorithm is simple, and as beautiful as the initial

recurrence. Three lines, three statements, and three cases for F(0), F(1) and

the general case F(n) solved by recursive calls. This also illustrates how

easy it is to get an `integer`

from R to C++ and back: the

`as`

and `wrap`

simply *do the right thing*

converting to and from the `SEXP`

types used internally by the C

API of R.

A performance comparison of the basic R version `fibR`

, its

byte-compiled variant `fibRC`

and

and the C++ version `fibRcpp`

shown above is very compelling.

We have added a file `fibonacci.r`

to the large and still growing

set of examples included with Rcpp, and we can just

execute that script with `Rscript`

or (as here) `r`

from the littler package:

edd@max:~/svn/rcpp/pkg/Rcpp/inst/examples/Misc$ r fibonacci.r Loading required package: inline Loading required package: methods Loading required package: compiler test replications elapsed relative user.self sys.self 3 fibRcpp(N) 1 0.092 1.0000 0.09 0.00 2 fibRC(N) 1 61.480 668.2609 61.47 0.00 1 fibR(N) 1 61.877 672.5761 61.83 0.02 edd@max:~/svn/rcpp/pkg/Rcpp/inst/examples/Misc$

So the recursion for the original argument of N=35 takes just over a

minute at about 61.5 and 61.9 seconds, respectively, for the R version and

its byte-compiled variant (as per the column titled *elapsed*). So

byte-compilation essentially offers no help for the bottleneck of slow function calls.

The C++ versions relying on Rcpp which created in a few lines of code and a

single call to `cxxfunction`

however takes just 92

**milli**seconds—or a relative gain of well over six-hundred times.

That provides another nice demonstration of what Rcpp can do. Improved

algorithms for well-understood problems are surely one way to accelerate solutions.

But there are (many ?) times when we do not have the luxury of being able to

think through to a new and improved approach. Or worse, such an approach may

even introduce new errors or inaccurracies if we get it wrong on a first try.

With Rcpp, we are able to the express the problem as written in its original statement: a simple

recursion. The gain relative to a slow R implementation is noteworthy—and

could of course be improved further if we really needed to by relying on

better algorithms like memoization. But for day to day tasks, I gladly take

speedups of (up to) a few hundred times thanks to

Rcpp without having to do *hard*

algorithmic work.

Before closing, a quick reminder that I will be giving two classes on

Rcpp in a few weeks. These will be in New York on September 24, and San Franciso on October 8, see

this blog post as well as

this page at Revolution Analytics (who are a

co-organiser of the classes) for details and registration information.

**leave a comment**for the author, please follow the link and comment on their blog:

**Thinking inside the box**.

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