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t-subset-sum.clisp
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(defun subset-sum (set sum &optional subset)
(when set
(destructuring-bind (head . tail) set
(or (and (= head sum) (cons head subset))
(subset-sum tail sum subset)
(subset-sum tail (- sum head) (cons head subset))))))
(setq city-pops '(18897109
12828837
9461105
6371773
5965343
5946800
5582170
5564635
5268860
4552402
4335391
4296250
4224851
4192887
3439809
3279833
3095313
2812896
2783243
2710489
2543482
2356285
2226009
2149127
2142508
2134411))
(subset-sum city-pops 100000000)
(2134411 2142508 2226009 2543482 2812896 3095313 3279833 4224851 4296250 4335391 4552402 5268860 5582170 5946800 6371773 9461105 12828837 18897109)
; (apply '+ '(2134411 2142508 2226009 2543482 2812896 3095313 3279833 4224851 4296250 4335391 4552402 5268860 5582170 5946800 6371773 9461105 12828837 18897109))
; 100000000
(setq numbers #(18897109
12828837
9461105
6371773
5965343
5946800
5582170
5564635
5268860
4552402
4335391
4296250
4224851
4192887
3439809
3279833
3095313
2812896
2783243
2710489
2543482
2356285
2226009
2149127
2142508
2134411))
(defun found-solution()
"Called whenever the algorithm has found a solution"
(let ((total 0))
(format t " ")
(dotimes (i (length numbers))
(when (aref flags i)
(incf total (aref numbers i))
(format t "~A " (aref numbers i)))
)
(format t " => ~A~%" total)
(incf solutions)))
(defun find-solutions(k target-sum callback)
"Core backtracking algorithm"
(when (zerop target-sum)
(funcall callback)
(return-from find-solutions))
(unless (= k (length numbers))
(let ((nk (aref numbers k)))
(when (>= target-sum nk)
;; try subtracting numbers[k] from target-sum
(setf (aref flags k) t)
(find-solutions (+ 1 k) (- target-sum nk) callback)
(setf (aref flags k) nil)))
;; recurse without subtracting first
(find-solutions (+ 1 k) target-sum callback)))
(defun find-subset-sum(target-sum)
"Set up and run backtracking algorithm based on 'numbers' array"
(setf flags (make-array (list (length numbers))))
(setf solutions 0)
(find-solutions 0 target-sum #'found-solution)
(format t "Found ~A different solutions.~%" solutions))
(defun subset-sum-test(size)
"Test subset sum algorithm using random numbers"
(let* ((total 0) target-sum)
;; init numbers array with random values up to 1000
(setf numbers (make-array (list size)))
(dotimes (i size)
(setf (aref numbers i) (random 1000))
(incf total (aref numbers i)))
(setf target-sum (floor (/ total 2))) ;; random target sum
(format t "Now listing all subsets which sum up to ~A:~%" target-sum)
(find-subset-sum target-sum)))