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The trace in Example 9.7 shows that the call f(n) to the recursive implementation of the factorial function will generate n 1 recursive calls. This is clearly very inefficient compared to the iterative implementation shown in Example 9.3. EXAMPLE 9.8 Tracing the Recursive Fibonacci Function

The Fibonacci function (Example 9.6) is more heavily recursive than the factorial function (Example 9.2) because it includes two recursive calls. The consequences can be seen from the trace of the call

fib(5), shown in Figure 9.2 on page 168. The call originates in the main() function, passing 5 to the fib() function. There, the value of the parameter n is 5, so it calls fib(4) and fib(3), passing 4 and 3,

respectively. Each of these calls then makes two more recursive calls, continuing down to the basis calls f(1) and f(0). Each of these basis calls returns 1. The recursive calls then return the sum of the two values returned to them, ultimately resulting in the value 8 being returned to main().

8:

THE RECURSIVE BINARY SEARCH The nonrecursive binary search algorithm is given on page 31. It uses the divide-and-conquer strategy, each time splitting the sequence in half and continuing the search on one half. This is naturally recursive. EXAMPLE 9.9 The Recursive Binary Search

Note!

Here is the recursive binary search algorithm: (Precondition: s = {s0, s1, . . ., sn 1} is a sorted sequence of n ordinal values of the same type as x.) (Postcondition: either the index i is returned where si = x, or 1 is returned.) 1. If the sequence is empty, return 1. 2. Let si be the middle element of the sequence. 3. If si = x, return its index i . 4. If si < x, apply the algorithm on the subsequence that lies above si . 5. Apply the algorithm on the subsequence of s that lies below si . It is implemented in Example 9.10.

The recursive binary search runs in O(lgn) time. The running time is proportional to the number of recursive calls made. Each call processes a subsequence that is half as long as the previous one. So the number of recursive calls is the same as the number of times that n can be divided in two, namely lg n.

CHAP. 9]

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public class TestBinarySearch { public static void main(String[] args) { int[] a = {22, 33, 44, 55, 66, 77, 88, 99}; print(a); System.out.println("search(a, 44): " + search(a, 44)); System.out.println("search(a, 50): " + search(a, 50)); System.out.println("search(a, 77): " + search(a, 77)); System.out.println("search(a, 100): " + search(a, 100)); } public static void print(int[] a) { System.out.printf("{%d", a[0]); for (int i = 1; i < a.length; i++) { System.out.printf(", %d", a[i]); } System.out.println("}"); } public static int search(int[] a, int x) { return search(a, 0, a.length-1, x); } public static int search(int[] a, int lo, int hi, int x) { // PRECONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; // POSTCONDITIONS: returns i; // if i >= 0, then a[i] == x; otherwise i == -1; if (lo > hi) { return -1; // basis } int i = (lo + hi)/2; if (a[i] == x) { return i; } else if (a[i] < x) { return search(a, i+1, hi, x); } else { return search(a, lo, i-1, x); } }

Banks are essential to the U.S. monetary system as the Federal Reserve System can only work through these businesses.

The output is:

{22, 33, 44, 55, 66, 77, 88, 99} search(a, 44): 2 search(a, 50): -1 search(a, 77): 5 search(a, 100): -1 The search() method returns the index of the target x: search(a, 44) returns 2 because a[2] = 44 and search(a, 77) returns 5 because a[5] = 77. The method returns 1 when the target is not in the array: search(a, 50) returns 1 because 50 is not in the array.

BINOMIAL COEFFICIENTS The binomial coefficients are the coefficients that result from the expansion of a binomial expression of the form (x + 1)n. For example, (x + 1)6 = x 6 + 6x5 + 15x4 + 20x3 + 15x2 + 6x + 1 The seven coefficients generated here are 1, 6, 15, 20, 15, 6, and 1.