Chapter 9.1.1 Recursive Binary Search

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

 

9.1.1 Recursive Binary Search

 

Let’s start with an example that you’ve seen before: the binary search algorithm from Subsection 7.4.1. Binary search is used to find a specified value in a sorted list of items (or, if it does not occur in the list, to determine that fact). The idea is to test the element in the middle of the list. If that element is equal to the specified value, you are done. If the specified value is less than the middle element of the list, then you should search for the value in the first half of the list.

 

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

 

Otherwise, you should search for the value in the second half of the list. The method used to search for the value in the first or second half of the list is binary search. That is, you look at the middle element in the half of the list that is still under consideration, and either you’ve found the value you are looking for, or you have to apply binary search to one half of the remaining elements. And so on! This is a recursive description, and we can write a recursive subroutine to implement it.

Before we can do that, though, there are two considerations that we need to take into account. Each of these illustrates an important general fact about recursive subroutines. First of all, the binary search algorithm begins by looking at the “middle element of the list.” But what if the list is empty? If there are no elements in the list, then it is impossible to look at the middle element.

In the terminology of Subsection 8.2.1, having a non-empty list is a “precondition” for looking at the middle element, and this is a clue that we have to modify the algorithm to take this precondition into account. What should we do if we find ourselves searching for a specified value in an empty list? The answer is easy: If the list is empty, we can be sure that the value does not occur in the list, so we can give the answer without any further work.

 

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

 

An empty list is a base case for the binary search algorithm. A base case for a recursive algorithm is a case that is handled directly, rather than by applying the algorithm recursively. The binary search algorithm actually has another type of base case: If we find the element we are looking for in the middle of the list, we are done. There is no need for further recursion.

The second consideration has to do with the parameters to the subroutine. The problem is phrased in terms of searching for a value in a list. In the original, non-recursive binary search subroutine, the list was given as an array. However, in the recursive approach, we have to able to apply the subroutine recursively to just a part of the original list.

Where the original subroutine was designed to search an entire array, the recursive subroutine must be able to search part of an array. The parameters to the subroutine must tell it what part of the array to search. This illustrates a general fact that in order to solve a problem recursively, it is often necessary to generalize the problem slightly.

Here is a recursive binary search algorithm that searches for a given value in part of an array of integers:

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

In this routine, the parameters loIndex and hiIndex specify the part of the array that is to be searched. To search an entire array, it is only necessary to call binarySearch(A, 0, A.length – 1, value). In the two base cases—when there are no elements in the specified range of indices and when the value is found in the middle of the range—the subroutine can return an answer immediately, without using recursion. In the other cases, it uses a recursive call to compute the answer and returns that answer.

Most people find it difficult at first to convince themselves that recursion actually works. The key is to note two things that must be true for recursion to work properly: There must be one or more base cases, which can be handled without using recursion. And when recursion is applied during the solution of a problem, it must be applied to a problem that is in some sense smaller—that is, closer to the base cases—than the original problem.

The idea is that if you can solve small problems and if you can reduce big problems to smaller problems, then you can solve problems of any size. Ultimately, of course, the big problems have to be reduced, possibly in many, many steps, to the very smallest problems (the base cases).

Doing so might involve an immense amount of detailed bookkeeping. But the computer does that bookkeeping, not you! As a programmer, you lay out the big picture: the base cases and the reduction of big problems to smaller problems. The computer takes care of the details involved in reducing a big problem, in many steps, all the way down to base cases.

 

Chapter 9.1.1 Recursive Binary Search | Introduction to Programming Using Java

 

Trying to think through this reduction in detail is likely to drive you crazy, and will probably make you think that recursion is hard. Whereas in fact, recursion is an elegant and powerful method that is often the simplest approach to solving a complex problem.

A common error in writing recursive subroutines is to violate one of the two rules: There must be one or more base cases, and when the subroutine is applied recursively, it must be applied to a problem that is smaller than the original problem.

If these rules are violated, the result can be an infinite recursion, where the subroutine keeps calling itself over and over, without ever reaching a base case. Infinite recursion is similar to an infinite loop. However, since each recursive call to the subroutine uses up some of the computer’s memory, a program that is stuck in an infinite recursion will run out of memory and crash before long. (In Java, the program will crash with an exception of type StackOverflowError.)

 

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