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Sec. 9.2 Self-Organizing Lists 309Example 9.2 Calculate the expected cost for searching a list ordered byfrequency when the probabilities are defined as{ 1/2iif 0 ≤ i ≤ n − 2p i =1/2 n if i = n − 1.Then,n−1∑C n ≈ (i + 1)/2 i+1 =i=0n∑(i/2 i ) ≈ 2.For this example, the expected number of accesses is a constant. This isbecause the probability for accessing the first record is high (one half), thesecond is much lower (one quarter) but still much higher than for the thirdrecord, and so on. This shows that for some probability distributions, orderingthe list by frequency can 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%of the records. The values of 80 and 20 are only estimates; every data access patternhas its own values. However, behavior of this nature occurs surprisingly oftenin practice (which explains the success of caching techniques widely used by webbrowsers for speeding access to web pages, and by disk drive and CPU manufacturersfor speeding access to data stored in slower memory; see the discussion onbuffer pools in Section 8.3). When the 80/20 rule applies, we can expect considerableimprovements to search performance from a list ordered by frequency ofaccess over standard sequential search in an unordered list.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=1When a frequency distribution follows the 80/20 rule, the average searchlooks at about 10-15% of the records in a list ordered by frequency.i=1