Binary Search
March 23, 2009
Children who are learning arithmetic sometimes play a number-guessing game: “I’m thinking of a number between 1 and 100. Can you guess it?” “Is the number less than 50?” “Yes.” “Is the number less than 25?” “No.” And so on, halving the interval at each step until only one number is left. This technique is known colloquially as the binary chop.
Your first task is to write a function that takes a target number and an array of numbers in non-decreasing order and returns either the position of the number in the array, or -1 to indicate the target number is not in the array. For instance, bsearch(32, [13 19 24 29 32 37 43]) should return 4, since 32 is the fourth element of the array (counting from zero).
Beware that this exercise is harder than it looks. Jon Bentley, in his book Programming Pearls, reports that 90% of professional programmers cannot write a proper implementation of binary search in two hours, and Donald Knuth, in the second volume of his book The Art of Computer Programming, reports that though the first binary search was published in 1946, the first bug-free binary search wasn’t published until 1962.
Thus, your second task is to write a suitable test program that shows the accuracy of your binary search function.
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])