Commit 1224346a authored by Philipp Marek's avatar Philipp Marek
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Merge branch 'master' into 'master'

Re-implement median function using the quick-select algorithm

I wanted to submit a re-implementation of alexandria:median. Currently the function works by fully sorting the sequence, which has a complexity of O(n log n). This commit implements it as a special case of the quick select algorithm, which brings its complexity down to O(n).

The new implementation is more complicated, though I think it's still simpler than some other algorithms in alexandria. Anyways, the performance gains are dramatic enough that I thought I'd at least submit the patch. I did some testing via the timing code below, and observed speedups of 5-10x even on sequences with just 100 elements.

I could also provide the quick select algorithm as its own function if there's any interest, though it seems like this would have to go into the alexandria-2 branch.

;; Speed-testing function
(defun test-median-speed (mag)
  (labels ((get-vec-median (vec)
             (if (oddp (length vec))
                 (aref vec (ash (length vec) -1))
                 (* 1/2 (+ (aref vec (ash (length vec) -1))
                           (aref vec (1- (ash (length vec) -1))))))))
    (let ((arr (loop for i from 0 upto (expt 10 mag) collect (random 1000000))))
      (assert (=
               (time (get-vec-median (sort (alexandria:copy-sequence 'vector arr) #'<)))
               (time (alexandria:median arr)))))))

See merge request !17
parents 3b849bc0 46d8b0ad
......@@ -104,15 +104,50 @@ interpolation coefficient V."
"Returns the mean of SAMPLE. SAMPLE must be a sequence of numbers."
(/ (reduce #'+ sample) (length sample)))
(declaim (inline median))
(defun median (sample)
"Returns median of SAMPLE. SAMPLE must be a sequence of real numbers."
(let* ((vector (sort (copy-sequence 'vector sample) #'<))
(length (length vector))
(middle (truncate length 2)))
(if (oddp length)
(aref vector middle)
(/ (+ (aref vector middle) (aref vector (1- middle))) 2))))
;; Implements and uses the quick-select algorithm to find the median
(labels ((randint-in-range (start-int end-int)
"Returns a random integer in the specified range, inclusive"
(+ start-int (random (1+ (- end-int start-int)))))
(partition (vec start-i end-i)
"Implements the partition function, which performs a partial
sort of vec around the (randomly) chosen pivot.
Returns the index where the pivot element would be located
in a correctly-sorted array"
(if (= start-i end-i)
(let ((pivot-i (randint-in-range start-i end-i)))
(rotatef (aref vec start-i) (aref vec pivot-i))
(let ((swap-i end-i))
(loop for i from swap-i downto (1+ start-i) do
(when (>= (aref vec i) (aref vec start-i))
(rotatef (aref vec i) (aref vec swap-i))
(decf swap-i)))
(rotatef (aref vec swap-i) (aref vec start-i))
(let* ((vector (copy-sequence 'vector sample))
(len (length vector))
(mid-i (ash len -1))
(i 0)
(j (1- len)))
(loop for correct-pos = (partition vector i j)
while (/= correct-pos mid-i) do
(if (< correct-pos mid-i)
(setf i (1+ correct-pos))
(setf j (1- correct-pos))))
(if (oddp len)
(aref vector mid-i)
(* 1/2
(+ (aref vector mid-i)
(reduce #'max (make-array
:displaced-to vector))))))))
(declaim (inline variance))
(defun variance (sample &key (biased t))
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