Binary Search
March 23, 2009
Our binary search sets the variables lo and hi to the two ends of the vector, and maintains the invariant that if the target number is present, it must be between lo and hi; if lo ever exceeds hi, we report that the target number is not present. Otherwise, we calculate the mid-point of the vector and make a three-way decision: if the number at the mid-point is the target number, we report success, otherwise we loop on the appropriate sub-vector:
(define (bsearch t vs)
(let loop ((lo 0) (hi (- (vector-length vs) 1)))
(if (< hi lo) -1
(let ((mid (quotient (+ lo hi) 2)))
(cond ((< t (vector-ref vs mid))
(loop lo (- mid 1)))
((< (vector-ref vs mid) t)
(loop (+ mid 1) hi))
(else mid))))))
Our test generates two types of vectors and tests each for all sizes from zero to a user-requested limit. The first vector has the integers from 0 to limit-1, and tests each number from 0 to limit-1 for a successful search, plus the two fractions 1/2 above and below each number from 0 to limit-1 for unsuccessful search. The first vector has all elements set to 1, and searches for 1 for a successful search and 1/2 and 3/2 for an unsuccessful search.
Assert is a macro that compares its two arguments and reports if they are not equal; the message includes the text of the expression, making it easy to see what failed. Range returns a list of non-negative integers less than n. Both appear in the Standard Prelude.
(define (test-search limit)
(do ((n 0 (+ n 1))) ((= n limit))
(let ((in-order (list->vector (range n)))
(all-equal (make-vector n 1)))
(do ((i 0 (+ i 1))) ((= i n))
(assert (bsearch i in-order) i)
(assert (bsearch (+ i 1/2) in-order) -1)
(assert (bsearch (- i 1/2) in-order) -1))
(assert (bsearch 1/2 all-equal) -1)
(assert (bsearch 3/2 all-equal) -1)
(if (positive? n) (assert (< -1 (bsearch1 all-equal) n) #t)))))
Performing the test shows that all is well:
> (test-search 12)
>
This solution is available at http://programmingpraxis.codepad.org/UxAnLcJF.
Programming Praxis 
Hm, it seems fairly trivial, but given the description I’m undoubtedly missing something. Anyway, here’s my attempt in Haskell:
import Data.List import qualified Data.Sequence as S import Test.QuickCheck main = quickCheck prop_bsearch bsearch :: Int -> S.Seq Int -> Maybe Int bsearch n s | s == S.empty = Nothing | i == n = Just x | i > n = bsearch n $ S.take x s | otherwise = fmap (x + 1 +) (bsearch n $ S.drop (x + 1) s) where i = S.index s x x = div (S.length s) 2 prop_bsearch n [] = bsearch n (S.fromList []) == Nothing prop_bsearch n s = maybe (notElem n s) ((== n) . (sort s !!)) . bsearch n . S.fromList $ sort sPlease be more specific: since the array is nondecreasing, there might be more than one element having the same value. Which index should the function return? Is the index of any element having the target value acceptable, or should it be the index of the first, or the last, or something else?
You may assume all elements of the array are distinct.
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Sorry long post as I included fairly exhaustive test conditions using schemeunit. There are several edge cases with a binary search so I thought I’d include tests for all the ones I thought of.
(define (bsearch target items) (let scan ([lower 0] [upper (sub1 (vector-length items))]) (if (> lower upper) -1 (let* ([mid (quotient (+ upper lower) 2)] [mid-item (vector-ref items mid)]) (cond [(< target mid-item) (scan lower (sub1 mid))] [(> target mid-item) (scan (add1 mid) upper)] [else mid]))))) (require (planet schematics/schemeunit:3:4)) (define empty-set #()) (define single-item-set #(1)) (define odd-size-set #(11 12 23 34 45 56 67)) (define even-size-set #(11 12 23 34 45 56 67 78)) (define all-equal-set #(11 11 11 11 11)) (define *not-found* -1) (define bsearch-tests (test-suite "Tests for bsearch.ss" (test-case "Base sets" (check-equal? (bsearch 3 empty-set) *not-found*) ; empty set fails (check-equal? (bsearch 3 single-item-set) *not-found*) ; single element fail (check-equal? (bsearch 1 single-item-set) 0) ; find single element ) (test-case "Odd number of elements" (check-equal? (bsearch 11 odd-size-set) 0) ; find first item (check-equal? (bsearch 45 odd-size-set) 4) ; find some item (check-equal? (bsearch 34 odd-size-set) 3) ; find exact mid item (check-equal? (bsearch 67 odd-size-set) 6) ; find last item (check-equal? (bsearch 1 odd-size-set) *not-found*) ; fail low (check-equal? (bsearch 102 odd-size-set) *not-found*) ; fail high (check-equal? (bsearch 36 odd-size-set) *not-found*) ; fail mid-way ) (test-case "Even number of elements" (check-equal? (bsearch 11 even-size-set) 0) ; find first item (check-equal? (bsearch 45 even-size-set) 4) ; find some item (check-equal? (bsearch 34 even-size-set) 3) ; find exact mid item (check-equal? (bsearch 78 even-size-set) 7) ; find last item (check-equal? (bsearch 10 even-size-set) *not-found*) ; fail low (check-equal? (bsearch 102 even-size-set) *not-found*) ; fail high (check-equal? (bsearch 36 even-size-set) *not-found*) ; fail mid-way ) (test-case "Identical elements set" (check-not-equal? (bsearch 11 even-size-set) *not-found*) ; finds first item (check-equal? (bsearch 10 all-equal-set) *not-found*) ; fail low (check-equal? (bsearch 12 all-equal-set) *not-found*) ; fail high ) )) (require (planet schematics/schemeunit:3/text-ui)) (run-tests bsearch-tests 'normal)barryallison,
What test cases would you use to test the following algorithms
1.) insertion sort
2.) mergesort
3.) heapsort
4.) binary trees methods (e.g. insert, delete, etc.)
In Python,
my c solution
int binary_search(int arr[], int n, int key)
{
int left, right, mid;
left = 0;
right = n – 1;
while(right > left)
{
mid = (left + right)/2;
if(key arr[mid])
{
left = mid + 1;
}
else
return mid;
}
return -1;
}
[\sourcecode]
My c solution
int binary_search(int arr[], int n, int key) { int left, right, mid; left = 0; right = n - 1; while(right > left) { mid = (left + right)/2; if(key < arr[mid]) { right = mid - 1; } else if(key > arr[mid]) { left = mid + 1; } else return mid; } return -1; }ruby solution (http://codepad.org/jUgepFQa) Doesn’t exactly use the binary chop *shameface*
Another interesting binary search algorithm that creates the binary representation of the position at which the targeted value can be found:
https://github.com/ftt/programming-praxis/blob/master/20090323-binary-search/binary-search.py
#! /usr/bin/env python def binsearch(l, x): lower, upper = 0, len(l)-1 if l[upper] == x: return upper while True: mid = (lower + upper) / 2 if l[mid] == x: return mid elif l[mid] < x: lower = mid else: upper = mid if __name__ == "__main__": from random import sample for x in xrange(100): l = sample(xrange(100), 25) l.sort() assert binsearch(l, l[x/4]) == l.index(l[x/4])