Memoization is awesome!
I’ve already written one post on the subject in Python, but this time we’ll do the same in Racket. It’s particularly timely as without it, today’s post on determining the number of prime partitions of a number would take longer to run than I care to wait.
The basic macro isn’t actually too bad. Basically, we want to create a function with an associated hash table associating arguments with their result. It doesn’t work out so well if the function is non-deterministic or has state of its own, but for a purely functional solution, it’s exactly what we’re looking for.
; replace define with a memoized version (define-syntax define-memoized (syntax-rules () [(_ (f args ...) bodies ...) (define f ; store the cache as a hash of args => result (let ([results (make-hash)]) ; need to do this to capture both the names and the values (lambda (args ...) ((lambda vals ; if we haven't calculated it before, do so now (when (not (hash-has-key? results vals)) (hash-set! results vals (begin bodies ...))) ; return the cached result (hash-ref results vals)) args ...))))]))
The code gets a bit strange on the inner part as I’m actually creating two functions only to immediately apply the inner one. Why? Because it’s the only way that I could think of to capture both the names of the arguments and their values (which are the actual key to the hash). If you have a better solution, I’d love to hear it. 😄
But with that, it’s really straight forward to turn a normal function into a memoized one. Here’s an example using that particularly poor way of writing Fibonacci:
; example, fibonacci without memoization (define (fib n) (cond [(< n 1) 1] [else (+ (fib (- n 1)) (fib (- n 2)))])) ; example, fibonacci with memoization (define-memoized (mfib n) (cond [(< n 1) 1] [else (+ (mfib (- n 1)) (mfib (- n 2)))]))
So, moment of truth. Does it work?
> (time (fib 35)) cpu time: 5022 real time: 5038 gc time: 37 24157817 > (time (mfib 35)) cpu time: 0 real time: 0 gc time: 0 24157817
I’d say that’s a pretty definitive yes. With this, we can calculate values we wouldn’t even have considered with the non-memoized version.
> (time (mfib 1000)) cpu time: 2 real time: 1 gc time: 0 113796925398360272257523782552224175572745930353730513145086634176691092536145 985470146129334641866902783673042322088625863396052888690096969577173696370562 180400527049497109023054114771394568040040412172632376
And that’s all there is too it. Memoization in Racket. Bam!