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

nonrecursive part of each call. Moreover, we can also see that the memory space
used by the algorithm (in addition to the array A) is also roughly proportional to n,
since we need a constant amount of memory space for each of the n boxes in the
trace at the time we make the final recursive call (for n = 1).
Reversing an Array by Recursion
Next, let us consider the problem of reversing the n elements of an array, A, so
that the first element becomes the last, the second element becomes second to the
last, and so on. We can solve this problem using linear recursion, by observing
that the reversal of an array can be achieved by swapping the first and last
elements and then recursively reversing the remaining elements in the array. We
describe the details of this algorithm in Code Fragment 3.32, using the convention
that the first time we call this algorithm we do so as ReverseArray(A,0,n − 1).
Code Fragment 3.32: Reversing the elements of an
array using linear recursion.
Note that, in this algorithm, we actually have two base cases, namely, when i = j
and when i > j. Moreover, in either case, we simply terminate the algorithm, since
a sequence with zero elements or one element is trivially equal to its reversal.
Furthermore, note that in the recursive step we are guaranteed to make progress
towards one of these two base cases. If n is odd, we will eventually reach the i = j
case, and if n is even, we will eventually reach the i > j case. The above argument
immediately implies that the recursive algorithm of Code Fragment 3.32 is
guaranteed to terminate.
Defining Problems in Ways That Facilitate Recursion
To design a recursive algorithm for a given problem, it is useful to think of the
different ways we can subdivide this problem to define problems that have the
same general structure as the original problem. This process sometimes means we
need to redefine the original problem to facilitate similar-looking subproblems.
For example, with the ReverseArray algorithm, we added the parameters i and
j so that a recursive call to reverse the inner part of the array A would have the
same structure (and same syntax) as the call to reverse all of A. Then, rather than
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