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UPC  13 Maker In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create EAN13 image in Visual Studio .NET applications. Generating GTIN  12 In Java Using Barcode generator for Java Control to generate, create GTIN  12 image in Java applications. Theorem 14.5 The insertion sort is correct. See the solution to Problem 14.23 on page 277 for a proof of this theorem. Theorem 14.6 The insertion sort runs in O(n2) time. See the solution to Problem 14.24 on page 277 for a proof of this theorem. Theorem 14.7 The insertion sort runs in O(n) time on a sorted sequence. See the solution to Problem 14.25 on page 283 for a proof of this theorem. THE SHELL SORT Theorem 14.7 suggests that if the sequence is nearly sorted, then the insertion sort will run nearly in O(n) time. That is true. The shell sort exploits that fact to obtain an algorithm that in general runs in better than O(n1.5) time. It applies the insertion sort repeatedly to skip subsequences such as {s0, s3, s6, s9, . . . , sn 2} and {s1, s4, s7, s10, . . . , sn 1}. These are two of the three skip3subsequences. Algorithm 14.4 The Shell Sort (Precondition: s = {s0 . . . sn 1} is a sequence of n ordinal values.) (Postcondition: The entire sequence s is sorted.) 1. Set d = 1. 2. Repeat step 3 until 9d > n. 3. Set d = 3d + 1. 4. Do steps 5 6 until d = 0. 5. Apply the insertion sort to each of the d skipdsubsequences of s. 6. Set d = d/3. Suppose that s has n = 200 elements. Then the loop at step 2 would iterate three times, increasing d from 1 to d = 4, 13, and 40. The first iteration of the loop at step 4 would apply the insertion sort to each of the 40 skip40subsequences {s0, s40, s80, s120, s160}, {s1, s41, s81, s121, s161}, {s2, s42, s82, s122, s162}, . . . , {s39, s79, s119, s159, s199}. Then step 6 would reduce d to 13, and then the second iteration of the loop at step 4 would apply the insertion sort to each of the thirteen skip13subsequences {s0, s13, s26, s39, s52, s65, . . . , s194}, {s1, s14, s27, s40, s53, s66, . . . , s195}, . . . , {s12, s25, s38, s51, s64, s77, . . . , s193}. Then step 6 would reduce d to 4, and the third iteration of the loop at step 4 would apply the insertion sort to each of the four skip4subsequences {s0, s4, s8, s12, . . ., s196}, {s1, s5, s9, s13, . . . , s197}, {s2, s6, s10, s14, . . . , s198}, and {s3, s7, s11, s15, . . . , s199}. Then step 6 would reduce d to 1, and, and the fourth Create Code128 In Java Using Barcode creation for Java Control to generate, create ANSI/AIM Code 128 image in Java applications. GTIN  12 Maker In Java Using Barcode creation for Java Control to generate, create UPCA image in Java applications. SORTING
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Code 128B Drawer In Java Using Barcode drawer for Android Control to generate, create Code128 image in Android applications. Paint UPCA In None Using Barcode creation for Microsoft Word Control to generate, create UCC  12 image in Word applications. iteration of the loop at step 4 would apply the insertion sort to the entire sequence. This entire process would apply the insertion sort 58 times: 40 times to subsequences of size n1 = 5, 13 times to subsequences of size n2 = 15, 4 times to subsequences of size n3 = 50, and once to the entire sequence of size n4 = n = 200. At first glance, the repeated use of the insertion sort within the shell sort would seem to take longer than simply applying the insertion sort directly just once to the entire sequence. Indeed, a direct calculation of the total number of comparisons, using the complexity function n2, yields 40(n12) + 13(n22) + 4(n32) + 1(n42) = 40(52) + 13(152) + 4(502) + 1(2002) = 53,925 which is quite a bit worse than the single n2 = 2002 = 40,000 But after the first iteration of step 4, the subsequent subsequences are nearly sorted. So the actual number of comparisons needed there is closer to n. Thus, the actual number of comparisons is more like 40(n12) + 13(n2) + 4(n3) + 1(n4) = 40(52) + 13(15) + 4(50) + 1(200) = 1,595 which is quite a bit better than 40,000. Theorem 14.8 The shell sort runs in O(n1.5) time. Note that, for n = 200, n1.5 = 2001.5 = 2,829, which is a lot better than n2 = 2002 = 40,000. EXAMPLE 14.7 The Shell Sort Encoding Barcode In VS .NET Using Barcode maker for .NET framework Control to generate, create bar code image in .NET framework applications. Code 3/9 Drawer In Java Using Barcode generator for Android Control to generate, create Code 39 Extended image in Android applications. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Scan UPCA Supplement 2 In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. UPCA Supplement 2 Encoder In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create GTIN  12 image in ASP.NET applications. public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length1]; int d = 1, j, n = a.length; // step 1 while (9*d < n) { // step 2 d = 3*d + 1; // step 3 } while (d > 0) { // step 4 for (int i = d; i < n; i++) { // step 5 int ai = a[i]; j = i; while (j >= d && a[jd] > ai) { a[j] = a[jd]; j = d; } a[j] = ai; } d /= 3; // step 6 } } THE MERGE SORT The merge sort applies the divideandconquer strategy to sort a sequence. First it subdivides the sequence into subsequences of singletons. Then it successively merges the subsequences pairwise until a single sequence is reformed. Each merge preserves order, so each merged subsequence is sorted. When the final merge is finished, the complete sequence is sorted. Although it can be implemented iteratively, the merge sort is naturally recursive: Split the sequence in two, sort each half, and then merge them back together preserving their order. The basis occurs when the subsequence contains only a single element. CHAP. 14]

