Question

write a function to generate test data.  The common-lisp function (random n) will return a random number in the range zero to n. there is a problem of number partitioning.  Given a set of  integerts, the task is to divide them into two mutually exclusive and  collectively exhaustive subsets, so that the sums of the numbers in each subset  are as nearly equal as possible.  For example, given the set (1 4 9 16 25), an  optimal partition divides the numbers into the sets (1 25) and (4 9 16).  Since  the sum of the numbers in the first set is 26, and the sum of the numbers in  the second set is 29, the difference of the sums is 3, and this is the smallest  difference that can be achieved in this case. the number partitioning problems generally become more difficult with increasing magnitude of the numbers, so be sure to test your program against integers at least as large as a million.  Note that it is feasible to optimally solve instances with very large numbers of integers if the integers are small.

Answer 1

;;;; -*- Mode: Lisp; indent-tabs-mode: nil -*-

(in-package :it.bese.fiveam)

;;;; ** Random (QuickCheck-ish) testing

;;;; FiveAM provides the ability to automatically generate a
;;;; collection of random input data for a specific test and run a
;;;; test multiple times.

;;;; Specification testing is done through the FOR-ALL macro. This
;;;; macro will bind variables to random data and run a test body a
;;;; certain number of times. Should the test body ever signal a
;;;; failure we stop running and report what values of the variables
;;;; caused the code to fail.

;;;; The generation of the random data is done using "generator
;;;; functions" (see below for details). A generator function is a
;;;; function which creates, based on user supplied parameters, a
;;;; function which returns random data. In order to facilitate
;;;; generating good random data the FOR-ALL macro also supports guard
;;;; conditions and creating one random input based on the values of
;;;; another (see the FOR-ALL macro for details).

;;;; *** Public Interface to the Random Tester

(defparameter *num-trials* 100
"Number of times we attempt to run the body of the FOR-ALL test.")

(defparameter *max-trials* 10000
"Number of total times we attempt to run the body of the
FOR-ALL test including when the body is skipped due to failed
guard conditions.
Since we have guard conditions we may get into infinite loops
where the test code is never run due to the guards never
returning true. This second run limit prevents that.")

(defmacro for-all (bindings &body body)
"Bind BINDINGS to random variables and test BODY *num-trials* times.
BINDINGS is a list of binding forms, each element is a list
of (BINDING VALUE &optional GUARD). Value, which is evaluated
once when the for-all is evaluated, must return a generator which
be called each time BODY is evaluated. BINDING is either a symbol
or a list which will be passed to destructuring-bind. GUARD is a
form which, if present, stops BODY from executing when IT returns
NIL. The GUARDS are evaluated after all the random data has been
generated and they can refer to the current value of any
binding. NB: Generator forms, unlike guard forms, can not contain
references to the bound variables.
Examples:
(for-all ((a (gen-integer)))
    (is (integerp a)))
(for-all ((a (gen-integer) (plusp a)))
    (is (integerp a))
    (is (plusp a)))
(for-all ((less (gen-integer))
            (more (gen-integer) (< less more)))
    (is (<= less more)))
(for-all (((a b) (gen-two-integers)))
    (is (integerp a))
    (is (integerp b)))"
(with-gensyms (test-lambda-args)
    `(perform-random-testing
      (list ,@(mapcar #'second bindings))
      (lambda (,test-lambda-args)
        (destructuring-bind ,(mapcar #'first bindings)
            ,test-lambda-args
          (if (and ,@(delete-if #'null (mapcar #'third bindings)))
              (progn ,@body)
              (throw 'run-once
                (list :guard-conditions-failed))))))))

;;;; *** Implementation

;;;; We could just make FOR-ALL a monster macro, but having FOR-ALL be
;;;; a preproccessor for the perform-random-testing function is
;;;; actually much easier.

(defun perform-random-testing (generators body)
(loop
     with random-state = *random-state*
     with total-counter = *max-trials*
     with counter = *num-trials*
     with run-at-least-once = nil
     until (or (zerop total-counter)
               (zerop counter))
     do (let ((result (perform-random-testing/run-once generators body)))
          (ecase (first result)
            (:pass
             (decf counter)
             (decf total-counter)
             (setf run-at-least-once t))
            (:no-tests
             (add-result 'for-all-test-no-tests
                         :reason "No tests"
                         :random-state random-state)
             (return-from perform-random-testing nil))
            (:guard-conditions-failed
             (decf total-counter))
            (:fail
             (add-result 'for-all-test-failed
                         :reason "Found failing test data"
                         :random-state random-state
                         :failure-values (second result)
                         :result-list (third result))
             (return-from perform-random-testing nil))))
     finally (if run-at-least-once
                 (add-result 'for-all-test-passed)
                 (add-result 'for-all-test-never-run
                             :reason "Guard conditions never passed"))))

(defun perform-random-testing/run-once (generators body)
(catch 'run-once
    (bind-run-state ((result-list '()))
      (let ((values (mapcar #'funcall generators)))
        (funcall body values)
        (cond
          ((null result-list)
           (throw 'run-once (list :no-tests)))
          ((every #'test-passed-p result-list)
           (throw 'run-once (list :pass)))
          ((notevery #'test-passed-p result-list)
           (throw 'run-once (list :fail values result-list))))))))

(defclass for-all-test-result ()
((random-state :initarg :random-state)))

(defclass for-all-test-passed (test-passed for-all-test-result)
())

(defclass for-all-test-failed (test-failure for-all-test-result)
((failure-values :initarg :failure-values)
   (result-list :initarg :result-list)))

(defgeneric for-all-test-failed-p (object)
(:method ((object for-all-test-failed)) t)
(:method ((object t)) nil))

(defmethod reason ((result for-all-test-failed))
(format nil "Falsifiable with ~S" (slot-value result 'failure-values)))

(defclass for-all-test-no-tests (test-failure for-all-test-result)
())

(defclass for-all-test-never-run (test-failure for-all-test-result)
())

;;;; *** Generators

;;;; Since this is random testing we need some way of creating random
;;;; data to feed to our code. Generators are regular functions which
;;;; create this random data.

;;;; We provide a set of built-in generators.

(defun gen-integer (&key (max (1+ most-positive-fixnum))
                         (min (1- most-negative-fixnum)))
"Returns a generator which produces random integers greater
than or equal to MIN and less than or equal to MAX."
(lambda ()
    (+ min (random (1+ (- max min))))))

(defun gen-float (&key bound (type 'short-float))
"Returns a generator which produces floats of type TYPE. BOUND,
if specified, constrains the results to be in the range (-BOUND,
BOUND)."
(lambda ()
    (let* ((most-negative (ecase type
                            (short-float most-negative-short-float)
                            (single-float most-negative-single-float)
                            (double-float most-negative-double-float)
                            (long-float most-negative-long-float)))
           (most-positive (ecase type
                            (short-float most-positive-short-float)
                            (single-float most-positive-single-float)
                            (double-float most-positive-double-float)
                            (long-float most-positive-long-float)))
           (bound (or bound (max most-positive (- most-negative)))))
      (coerce
       (ecase (random 2)
         (0 ;; generate a positive number
          (random (min most-positive bound)))
         (1 ;; generate a negative number
          (- (random (min (- most-negative) bound)))))
       type))))

(defun gen-character (&key (code-limit char-code-limit)
                           (code (gen-integer :min 0 :max (1- code-limit)))
                           (alphanumericp nil))
"Returns a generator of characters.
CODE must be a generator of random integers. ALPHANUMERICP, if
non-NIL, limits the returned chars to those which pass
alphanumericp."
(lambda ()
    (loop
       for count upfrom 0
       for char = (code-char (funcall code))
       until (and char
                  (or (not alphanumericp)
                      (alphanumericp char)))
       when (= 1000 count)
       do (error "After 1000 iterations ~S has still not generated ~:[a valid~;an alphanumeric~] character :(."
                 code alphanumericp)
       finally (return char))))

(defun gen-string (&key (length (gen-integer :min 0 :max 80))
                        (elements (gen-character))
                        (element-type 'character))
"Returns a generator which produces random strings. LENGTH must
be a generator which produces integers, ELEMENTS must be a
generator which produces characters of type ELEMENT-TYPE."
(lambda ()
    (loop
       with length = (funcall length)
       with string = (make-string length :element-type element-type)
       for index below length
       do (setf (aref string index) (funcall elements))
       finally (return string))))

(defun gen-list (&key (length (gen-integer :min 0 :max 10))
                      (elements (gen-integer :min -10 :max 10)))
"Returns a generator which produces random lists. LENGTH must be
an integer generator and ELEMENTS must be a generator which
produces objects."
(lambda ()
    (loop
       repeat (funcall length)
       collect (funcall elements))))

(defun gen-tree (&key (size 20)
                      (elements (gen-integer :min -10 :max 10)))
"Returns a generator which produces random trees. SIZE controls
the approximate size of the tree, but don't try anything above
30, you have been warned. ELEMENTS must be a generator which
will produce the elements."
(labels ((rec (&optional (current-depth 0))
             (let ((key (random (+ 3 (- size current-depth)))))
               (cond ((> key 2)
                      (list (rec (+ current-depth 1))
                            (rec (+ current-depth 1))))
                     (t (funcall elements))))))
    (lambda ()
      (rec))))

(defun gen-buffer (&key (length (gen-integer :min 0 :max 50))
                        (element-type '(unsigned-byte 8))
                        (elements (gen-integer :min 0 :max (1- (expt 2 8)))))
(lambda ()
    (let ((buffer (make-array (funcall length) :element-type element-type)))
      (map-into buffer elements))))

(defun gen-one-element (&rest elements)
(lambda ()
    (nth (random (length elements)) elements)))

;;;; The trivial always-produce-the-same-thing generator is done using
;;;; cl:constantly.