A Practical Introduction to Data Structures and Algorithm Analysis

86 Chap. 3 Algorithm Analysisof its correct position once it has been placed there, and each swap operation placesat least one integer in its correct position. Thus, this code fragment has cost Θ(n).However, it requires more time to run than the first code fragment. On my computerthe second version takes nearly twice as long to run as the first, but it only requireshalf the space.A second principle for the relationship between a program’s space and timerequirements applies to programs that process information stored on disk, as discussedin Chapter 8 and thereafter. Strangely enough, the disk-based space/timetradeoff principle is almost the reverse of the space/time tradeoff principle for programsusing main memory.The disk-based space/time tradeoff principle states that the smaller you canmake your disk storage requirements, the faster your program will run. This is becausethe time to read information from disk is enormous compared to computationtime, so almost any amount of additional computation needed to unpack the data isgoing to be less than the disk-reading time saved by reducing the storage requirements.Naturally this principle does not hold true in all cases, but it is good to keepin mind when designing programs that process information stored on disk.3.10 Speeding Up Your ProgramsIn practice, there is not such a big difference in running time between an algorithmwhose growth rate is Θ(n) and another whose growth rate is Θ(n log n). There is,however, an enormous difference in running time between algorithms with growthrates of Θ(n log n) and Θ(n 2 ). As you shall see during the course of your studyof common data structures and algorithms, it is not unusual that a problem whoseobvious solution requires Θ(n 2 ) time also has a solution that requires Θ(n log n)time. Examples include sorting and searching, two of the most important computerproblems.Example 3.18 The following is a true story. A few years ago, one ofmy graduate students had a big problem. His thesis work involved severalintricate operations on a large database. He was now working on the finalstep. “Dr. Shaffer,” he said, “I am running this program and it seems tobe taking a long time.” After examining the algorithm we realized that itsrunning time was Θ(n 2 ), and that it would likely take one to two weeksto complete. Even if we could keep the computer running uninterruptedfor that long, he was hoping to complete his thesis and graduate beforethen. Fortunately, we realized that there was a fairly easy way to convertthe algorithm so that its running time was Θ(n log n). By the next day he