Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

The simplest form of recursion is linear recursion, where a method is defined so
that it makes at most one recursive call each time it is invoked. This type of
recursion is useful when we view an algorithmic problem in terms of a first or last
element plus a remaining set that has the same structure as the original set.
Summing the Elements of an Array Recursively
Suppose, for example, we are given an array, A, of n integers that we wish to sum
together. We can solve this summation problem using linear recursion by
observing that the sum of all n integers in A is equal to A[0], if n = 1, or the sum
of the first n − 1 integers in A plus the last element in A. In particular, we can
solve this summation problem using the recursive algorithm described in Code
Fragment 3.31.
Code Fragment 3.31: Summing the elements in an
array using linear recursion.
This example also illustrates an important property that a recursive method should
always possess—the method terminates. We ensure this by writing a nonrecursive
statement for the case n = 1. In addition, we always perform the recursive call on
a smaller value of the parameter (n − 1) than that which we are given (n), so that,
at some point (at the "bottom" of the recursion), we will perform the nonrecursive
part of the computation (returning A[0]). In general, an algorithm that uses linear
recursion typically has the following form:
• Test for base cases. We begin by testing for a set of base cases (there
should be at least one). These base cases should be defined so that every
possible chain of recursive calls will eventually reach a base case, and the
handling of each base case should not use recursion.
• Recur. After testing for base cases, we then perform a single recursive
call. This recursive step may involve a test that decides which of several
possible recursive calls to make, but it should ultimately choose to make just
one of these calls each time we perform this step. Moreover, we should define
each possible recursive call so that it makes progress towards a base case.
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