A Memoization Macro for
Common Lisp
Description of the problem
Imagine you have a recursive pure (side-effect free) function. As the
function calls itself with smaller and smaller input arguments as we
move down on the recursion tree, there is a big possibility that we
calculate and re-calculate the function with the same input over and
over again which is wasteful in terms of time, energy and memory.
One good option is to store the results of our calculations in a
cache, and then recall them as we traverse the recursion tree. This is
done in two different ways:
In dynamic-programming
we build our computations ground-up, that is, we start with the
base cases and work our way up gradually increasing the input and
storing the results of our calculations in a cache. Then we call our
recursive function.
In memoization we work
our way down. Given an input we check the cache if our function is
called with that input. If that is the case we return the cached result.
Otherwise we go on making our recursive calculation.
I am a great fan of the second approach. In the past, as you might
recall, I employed
it successfully in calculating several difficult functions. And, as
it happens, it is not that difficult to implement.
A Macro
I will define a macro mem-defun
which you can use to
replace a function definition with a memoized version of the same
function. On the syntactic level, there is no difference. Just replace
defun
with mem-defun
and it is done.
(defmacro mem-defun (name args body)
(let ((hash-name (gensym)))
`(let ((,hash-name (make-hash-table :test 'equal)))
(defun ,name ,args
(or (gethash (list ,@args) ,hash-name)
(setf (gethash (list ,@args) ,hash-name)
,body))))))
MEM-DEFUN
A Nice Encapsulation Trick
In the definition I employed a nice trick I learned from someone else
a long long time ago, but I fortunately I forgot when and where, about
encapsulating a state within a function:
(let ((n 0))
(defun counter () (incf n)))
COUNTER
Now, if you call counter
back to back, it will do the
right increment. Plus, the variable n
is completely
encapsulated within counter
.
It is not very functional of me, but, hey it gets the job
done. Neat, isn’t it?
A Test
I will use the following function which calculates the number of
unordered -partitions of an
integer :
(defun par (n k)
(cond ((< n k) 0)
((= n k) 1)
((= k 1) 1)
((= k 2) (if (evenp n) (/ n 2) (/ (1- n) 2)))
(t (+ (par (1- n) (1- k)) (par (- n k) k)))))
PAR
(time (par 125 25))
Evaluation took:
4.039 seconds of real time
4.036252 seconds of total run time (3.980249 user, 0.056003 system)
[ Run times consist of 0.024 seconds GC time, and 4.013 seconds non-GC time. ]
99.93% CPU
14,104,722,285 processor cycles
1,604,091,864 bytes consed
139620591
(mem-defun par (n k)
(cond ((< n k) 0)
((= n k) 1)
((= k 1) 1)
((= k 2) (if (evenp n) (/ n 2) (/ (1- n) 2)))
(t (+ (par (1- n) (1- k)) (par (- n k) k)))))
PAR
(time (par 125 25))
Evaluation took:
0.000 seconds of real time
0.000000 seconds of total run time (0.000000 user, 0.000000 system)
100.00% CPU
1,186,483 processor cycles
170,440 bytes consed
139620591
Addendum
Reiner Joswig pointed
out that, if a return value for our function to be memoized is `NIL`,
the macro I wrote above will fail, and the function will be recomputed.
So, if your function has a possibility of returning a NIL, use the
version below:
(defmacro mem-defun (name args body)
(let ((hash-name (gensym)))
`(let ((,hash-name (make-hash-table :test 'equal)))
(defun ,name ,args
(multiple-value-bind (val found-p)
(gethash (list ,@args) ,hash-name)
(if found-p
val
(setf (gethash (list ,@args) ,hash-name)
,body)))))))
He also pointed me to Norvig’s PAIP, specifically his version a
macro doing the same thing and much more [here]. I knew the code,
and there are few things I didn’t like about the way he designed the
memoization macro. But Norvig’s code is very well-structured and fun to
read. I definitely recommend to go through it and the PAIP book for
one’s edification.
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