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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

WHAT IS COMPUTER SCIENCE Computer Science is defined in different ways by different authors. Wikipedia (http://en.wikipedia.org/ wiki/Computer_science) defines computer science as the collection of a variety of disciplines related to computing, both theoretical and practical: theoretical foundations of information and computation, language theory, algorithm analysis and development, implementation of computing systems, computer graphics, databases, data communications, etc. The US National Coordination Office for Networking and Information Technology Research and Development (NITRD) defines computer science in a similarly broad way:

Slope = _

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Vl R3 R2 Slope = _ R2 + R3 R1 (b)

the systematic study of computing systems and computation. The body of knowledge resulting from this discipline contains theories for understanding computing systems and methods; design methodology, algorithms, and tools; methods for the testing of concepts; methods of analysis and verification; and knowledge representation and implementation. (http://www.nitrd.gov/pubs/bluebooks/1995/section.5.html)

public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; 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[j-d] > ai) { a[j] = a[j-d]; j -= d; } a[j] = ai; } d /= 3; // step 6 } }

Fig. 9-18

path through R3 . Assume an ideal op amp and a diode characterized by Fig. 2-2(a). (b) Determine the relationship between vo and vS , and sketch the transfer characteristic.

THE MERGE SORT The merge sort applies the divide-and-conquer 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 re-formed. 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.

Another broad definition comes from the Association for Computing Machinery (ACM) Model Curriculum. It says that computer science is the study of computers and algorithmic processes, including their principles, their hardware and software design, their applications, and their impact on society. A famous definition of computer science by Gibbs and Tucker (Gibbs and Tucker, A Model Curriculum for a Liberal Arts Degree in Computer Science, Comm. of the ACM, vol. 29, no. 3, March 1986) emphasizes algorithm development and analysis as the central focus of computer science. It s also a fair question to ask, How is computer science a science In contrast to physics, biology, and chemistry, computer science is not based on the study of the natural world. In that sense, computer science is more like mathematics than science. Some argue that computer science is really computer art (where art means practice). On the other hand, computer scientists do use the scientific method to propose and test hypotheses, and some very nonobvious discoveries in computer science have important real-world implications. An example, which we will discuss later, is the discovery that some important problems simply cannot be solved by computation. Despite many variations, essentially all definitions of computer science emphasize the study of algorithms. Algorithms, in one form or another, are central to computer science. Computer science combines the theoretical concepts of algorithm design and analysis with the practical considerations of how to implement algorithms on a computer and solve practical problems.

(a) The diode voltage vD is found by writing a loop equation. Since the inverting input is a virtual ground, vo appears across R2 and vD vo i3 R3 vo When vD 0; vo V , and (1) gives V R3 V R4 2 V vo R R3 R4 3 1

CHAP. 14]

(b) For vo > V , the diode blocks and R2 constitutes the only feedback path. Since i1 i2 , vS v o R1 R2 3