Why is Binary Search a divide and conquer algorithm?

Question

I was asked if a Binary Search is a divide and conquer algorithm at an exam. My answer was yes, because you divided the problem into smaller subproblems, until you reached y

Answer 1

It isn't.

To complement @Kenci's post, DnC algorithms have a few general/common properties; they:

1. divide the original problem instance into a set of smaller sub-instances of itself;
2. independently solve each sub-instance;
3. combine smaller/independent sub-instance solutions to build a single solution for the larger/original instance

The problem with Binary Search is that it does not really even generate a set of independent sub-instances to be solved, as per step 1; it only simplifies the original problem by permanently discarding sections it's not interested in. In other words, it only reduces the problem's size and that's as far as it ever goes.

A DnC algorithm is supposed to not only identify/solve the smaller sub-instances of the original problem independently of each other, but also use that set of partial independent solutions to "build up" a single solution for the larger problem instance as a whole.

The book Fundamentals of Algorithmics, G. Brassard, P. Bratley says the following (bold my emphasis, italics in original):

It is probably the simplest application of divide-and-conquer, so simple in fact that strictly speaking this is an application of simplification rather than divide-and-conquer: the solution to any sufficiently large instance is reduced to that of a single smaller one, in this case of half size.

Section 7.3 Binary Search on p.226.

Answer 2

In a divide and conquer strategy :

1.Problem is divided into parts;

2.Each of these parts is attacked/solved independently, by applying the algorithm at hand (mostly recursion is used for this purpose) ;

3.And then the solutions of each partition/division and combined/merged together to arrive at the final solution to the problem as a whole (this comes under conquer)

Example, Quick sort, merge sort.

Basically, the binary search algorithm just divides its work space(input (ordered) array of size n) into half in each iteration. Therefore it is definitely deploying the divide strategy and as a result, the time complexity reduces down to O(lg n).So,this covers up the "divide" part of it.

As can be noticed, the final solution is obtained from the last comparison made, that is, when we are left with only one element for comparison. Binary search does not merge or combine solution.

In short, binary search divides the size of the problem (on which it has to work) into halves but doesn't find the solution in bits and pieces and hence no need of merging the solution occurs!

I know it's a bit too lengthy but i hope it helps :)

Also you can get some idea from : https://www.khanacademy.org/computing/computer-science/algorithms/binary-search/a/running-time-of-binary-search

Also i realised just now that this question was posted long back! My bad!

Answer 3

No, binary search is not divide and conquer. Yes, binary search is decrease and conquer. I believe divide and conquer algorithms have an efficiency of O(n log(n)) while decrease and conquer algorithms have an efficiency of O(log(n)). The difference being whether or not you need to evaluate both parts of the split in data or not.

Answer 4

A proper divide and conquer algorithm will require both parts to be processed.

Therefore, many people will not call binary-search a divide and conquer algorithm, it does divide the problem, but discards the other half.

But most likely, your examiners just wanted to see how you argue. (Good) exams aren't about the facts, but about how you react when the challenge goes beyond the original material.

So IMHO the proper answer would have been:

Well, technically, it consists only of a divide step, but needs to conquer only half of the original task then, the other half is trivially done already.

BTW: there is a nice variation of QuickSort, called QuickSelect, which actually exploits this difference to obtain an on average O(n) median search algorithm. It's like QuickSort - but descends only into the half it is interested in.