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Figure 13.10 Trace of heap insertions into an array
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Theorem. Every subtree of a heap is also a heap. Proof: Let T be a heap, and let S be a subtree of T. (See Figure 13.11.) By definition, T is a complete binary tree with the heap property. Thus, by the theorem in the solution, S is also a complete binary tree. Let x be the root of S, and let p be any root-to-leaf path in S. Then x is an element of T since S is a subtree of T, and there is a unique path q in T from x to the root of T. Also, p is a path in T that connects x to a leaf of T since S is a subtree Figure 13.11 Subtree of a heap of T. Let q 1 represent the reverse of the path 1p represent the concatenation of q, and let q q 1 with p in T. Then q 1p is a root-to-leaf path in T. Hence the elements along q 1p must be descending because T has the heap property. Therefore the elements along p are descending. Thus S also has the heap property.
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75, 45, 35 into a heap.
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Figure 13.12 shows a trace of the insertion of the keys 50, 95, 70, 30, 90, 25, 35, 80, 60, 40, 20, 10,
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Figure 13.12 Insertion into a heap
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[CHAP. 13
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Figure 13.12 (continued) Insertion into a heap
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We saw in 2 that searching through an array is far more efficient if its elements are sorted first. That fact is obvious to anyone who has ever looked up a number in a phone book or a word in a dictionary. This chapter outlines nine of the most common algorithms for sorting a linear data structure such as an array or a list. CODE PRELIMINARIES All of the sorting algorithms are implemented in this chapter using the same Java signature
public static void sort(int[] a)
for sorting an array of integers. This can easily be modified to sort an array of any other primitive type or an array of objects that implement the Comparable interface. The pseudocode and the Java code include preconditions, postconditions, and loop invariants. These are used to prove that the algorithms are correct and to analyze their complexity. In the pseudocode we use the notation s = {s0, s1, . . . , sn 1} for a sequence of n elements. The notation {sp . . . sq 1} denotes the subsequence {sp, sp+1, . . . , sq 1} of elements from sp. to sq 1 . In Java code comments, we represent the subsequence by s[p..q). For example, {s3 . . . s7} and s[3..8) would both denote the subsequence {s3, s4, s5, s6, s7}. Unless otherwise noted, sorted will always mean that the elements of the sequence are in ascending order: s0 s1 s2 sn 1. All the exchange sorts use this swap() method: EXAMPLE 14.1 A swap() Method for Arrays
1 2 3 4 5 6 7 8 9 10
private static void swap(int[] a, int i, int j) { // PRECONDITIONS: 0 <= i < a.length; 0 <= j < a.length; // POSTCONDITION: a[i] and a[j] are interchanged; if (i == j) { return; } int temp=a[j]; a[j] = a[i]; a[i] = temp; }
This method simply swaps the ith and jth elements of the array.
SORTING
[CHAP. 14
Note the use of preconditions and postconditions, included as comments at lines 2 and 3 of Example 14.1. A precondition is a condition that is assumed to be true before the method is invoked. A postcondition is a condition that is guaranteed to be true after the method has been invoked, provided that the preconditions were true. Preconditions and postconditions define the contract for a method: the consumer guarantee that defines the method. They can be used to prove, logically, that the method will always work as expected. (For example, see Theorem 14.10 on page 262.) All the array examples use this print() method: EXAMPLE 14.2 A print() Method for Arrays
private static void print(int[] a) { for (int ai : a) { 3 System.out.printf("%s ", ai); 4 } 5 System.out.println(); 6 } Note the use of Java s enhanced for loop construct at line 2. The variable ai represents array elements a[i] as the index i traverses its entire range from 0 to a.length 1.
THE JAVA Arrays.sort() METHOD The standard Java class library defines a sort() method in the java.util.Arrays class. It actually includes twenty overloaded versions of the method: four for arrays of objects, two for arrays of each primitive type except boolean, and two for generic types. (See page 95.) The signatures for the two sort() methods for arrays of ints are
public static void sort(int[] a) public static void sort(int[] a, int p, int q)
The first of these sorts the entire array. The second sorts the subarray a[p..q). EXAMPLE 14.3 Using the Arrays.sort() Method
1 2 3 4 5 6
public static void main(String[] args) { int[] a = { 77, 44, 99, 66, 33, 55, 88, 22 }; print(a); java.util.Arrays.sort(a); print(a); }
The output is:
77 44 99 66 33 55 88 22 22 33 44 55 66 77 88 99
For arrays of elements of a primitive type, the Arrays.sort() method implements the quick sort. (See Algorithm 14.6 on page 263.). For arrays of elements of a reference type, it implements the merge sort. (See Algorithm 14.5 on page 261.) THE BUBBLE SORT The bubble sort makes n 1 passes through a sequence of n elements. Each pass moves through the array from left to right, comparing adjacent elements and swapping each pair that is out of order. This gradually moves the larger elements to the right. It is called the bubble sort
CHAP. 14]
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