A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 9.2 Self-Organizing Lists 325A geometric probability distribution can yield quite different results.Example 9.2 Calculate the expected cost for searching a list ordered byfrequency when the probabilities are defined as{ 1/2iif 1 ≤ i ≤ n − 1p i =1/2 n−1 if i = n.Then,C n ≈n∑(i/2 i ) ≈ 2.i=1For this example, the expected number of accesses is a constant. This isbecause the probability for accessing the first record is high, the second ismuch lower but still much higher than for record three, and so on. Thisshows that for some probability distributions, ordering the list by frequencycan yield an efficient search technique.In many search applications, real access patterns follow a rule of thumb calledthe 80/20 rule. The 80/20 rule says that 80% of the record accesses are to 20% ofthe records. The values of 80 and 20 are only estimates; every application has itsown values. However, behavior of this nature occurs surprisingly often in practice(which explains the success of caching techniques widely used by disk drive andCPU manufacturers for speeding access to data stored in slower memory; see thediscussion on buffer pools in Section 8.3). When the 80/20 rule applies, we canexpect reasonable search performance from a list ordered by frequency of access.Example 9.3 The 80/20 rule is an example of a Zipf distribution. Naturallyoccurring distributions often follow a Zipf distribution. Examplesinclude the observed frequency for the use of words in a natural languagesuch as English, and the size of the population for cities (i.e., view therelative proportions for the populations as equivalent to the “frequency ofuse”). Zipf distributions are related to the Harmonic Series defined in Equation2.10. Define the Zipf frequency for item i in the distribution for nrecords as 1/(iH n ) (see Exercise 9.4). The expected cost for the serieswhose members follow this Zipf distribution will ben∑C n = i/iH n = n/H n ≈ n/ log e n.i=1