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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); } } }

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().

in order to make h(0) = 1.

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

The output is:

with no conjugate reciprocal zeros, i.e., a # 1 /a; for any k and 1. Show that x ( n ) is uniquely defined k to within a constant by the phase of its discrete-time Fourier transform.

Let xl (n) be a finite-length sequence that has a z-transform with no conjugate reciprocal zeros and a DTFT with the same phase as x(n). Then XI^) = G(z)X(z) where G(z) is the system function of a filter that has zero phase. Thus, G(z) must have zeros in conjugate reciprocal

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.

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