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14 At level 0 of recursion, merge_sort now has result_A and result_B merge_sort calls merge to put the two lists of numbers in order, and then it returns the entire set of four numbers in sorted order Aside from being an interesting exercise in recursion, the merge_sort provides attractive performance The merge sort has a theta of n(lg n), which for large problems is much better than the theta of n2 for the insertion sort The recursion in merge_sort divides the problem into many subproblems by repeatedly halving the size of the list to be sorted The number of times the list must be divided by two in order to create lists of length one is equal to the logarithm to the base 2 of the number of elements in the list.

Property 9 allows the existence of arrays of arrays. Technically, Java allows multidimensional arrays only with primitive types. But for objects, an array of arrays is essentially the same thing. Since arrays themselves are objects, an array of arrays is an array of objects, and some of those component objects could also be nonarrays. (See Example 2.1.) Note that a consequence of Property 13 is that changing a reference component value to null has no effect upon the length of the array; null is still a valid value of a reference component. EXAMPLE 2.1 Some Array Definitions

Here are some valid array definitions:

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In the case of our 4-element example, the logarithm to the base 2 of 4 is 2, because 22 = 4 This can be written as log2 n, but in computer science, because of the ubiquity of binary math, this is usually written as lg n, meaning logarithm to the base 2 of n The total running time T of the merge sort consists of the time to recursively solve two problems of half the size, and then to combine the results One way of expressing the time required is this: T = 2T(n/2) + merge Since merge runs in (nA + nB), and since nA + nB = n, we will restate this: T = 2T(n/2) + (n) A recursion tree is a way to visualize the time required.

public class ArrayDefs { public static void main(String[] args) { 3 float x[]; 4 x = new float[100]; 5 args = new String[10]; 6 boolean[] isPrime = new boolean[1000]; 7 int fib[] = {0, 1, 1, 2, 3, 5, 8, 13}; 8 short[][][] b = new short[4][10][5]; 9 double a[][] = {{1.1,2.2}, {3.3,4.4}, null, {5.5,6.6}, null}; 10 a[4] = new double[66]; 11 a[4][65] = 3.14; 12 Object[] objects = {x, args, isPrime, fib, b, a}; 13 } 14 } Line 3 declares x[] to be an array of floats but does not allocate any storage for the array. Line 4 defines x[] to have 100 float components.

At the top level, we have the time required for merge (n), plus the time required for the two subproblems: (n) T(n/2) T(n/2) At the next level, we have the time required for the two merges of the two subproblems, and for the further subdivision of the two subproblems: (n) (n/2) (n/2) T(n/4) T(n/4) T(n/4) T(n/4).

Line 5 declares args[] to be an array of 10 String objects. Note the two different (equivalent) ways to declare an array: The brackets may be a suffix on the type identifier or on the array identifier. Line 5 defines args[] to have 10 String components. Line 6 defines isPrime[] to be an array of 1000 boolean variables. Line 7 defines fib[] to be an array of 8 ints, initializing them to the 8 values listed. So for example, fib[4] has the value 3, and fib[7] has the value 13. Line 8 defines b[][][] to be a three-dimensional array of 4 components, each of which is a twodimensional array of 10 components, each of which is a one-dimensional array of 5 component elements of type short. Line 9 defines a[][] to be an array of five components, each of which is an array of elements of type double. Only three of the five component arrays are allocated. Then line 10 allocates a 66-element array of doubles to a[4], and line 11 assigns 3.14 to its last element. Line 12 defines the array objects to consist of six components, each of which is itself an array. The components of the first four component arrays are elements (nonarrays). But the components of the components b and a are not elements because they are also arrays. The actual elements of the objects array include 2, 5, and 13 (components of the component fib), null (components of the component a), and 2.2 and 3.14 (components of the components of the component a).

The array a[][] defined in Example 2.1 is called a ragged array because it is a two-dimensional array with rows of different lengths. The element type of an array in Java can be a primitive type, a reference type, or an array type. The simplest, of course, are arrays of primitive type elements, such as x[], isPrime[], and fib[] in Example 2.1. These are arrays that can be sorted.

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