A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 9.1 Searching Unsorted and Sorted Arrays 323It is not always practical to reduce an algorithm’s growth rate. There is a practicalitywindow for every problem, in that we have a practical limit to how big aninput we wish to solve for. If our problem size never grows too big, it might notmatter if we can reduce the cost by an extra log factor, because the constant factorsin the two algorithms might overwhelm the savings.For our two algorithms, let us look further and check the actual number ofcomparisons used. For binary search, we need about log n − 1 total comparisons.Quadratic binary search requires about 2.4 lg lg n comparisons. If we incorporatethis observation into our table, we get a different picture about the relative differences.n lg n − 1 2.4 lg lg n Difference16 3 4.8 worse256 7 7.2 ≈ same64K 15 9.6 1.62 32 31 12 2.6But we still are not done. This is only a count of comparisons! Binary searchis inherently much simpler than QBS, because binary search only needs to calculatethe midpoint position of the array before each comparison, while quadraticbinary search must calculate an interpolation point which is more expensive. Sothe constant factors for QBS are even higher.Not only are the constant factors worse on average, but QBS is far more dependentthan binary search on good data distribution to perform well. For example,imagine that you are searching a telephone directory for the name “Young.” Normallyyou would look near the back of the book. If you found a name beginningwith ‘Z,’ you might look just a little ways toward the front. If the next name youfind also begins with ’Z,‘ you would look a little further toward the front. If thisparticular telephone directory were unusual in that half of the entries begin with ‘Z,’then you would need to move toward the front many times, each time eliminatingrelatively few records from the search. In the extreme, the performance of interpolationsearch might not be much better than sequential search if the distribution ofkey values is badly calculated.While it turns out that QBS is not a practical algorithm, this is not a typicalsituation. Fortunately, algorithm growth rates are usually well behaved, so that asymptoticalgorithm analysis nearly always gives us a practical indication for whichof two algorithms is better.