A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 9.1 Searching Unsorted and Sorted Arrays 321looking for. The search generally does not start at the middle of the dictionary. Aperson looking for a word starting with ‘S’ generally assumes that entries beginningwith ‘S’ start about three quarters of the way through the dictionary. Thus, he orshe will first open the dictionary about three quarters of the way through and thenmake a decision based on what they find as to where to look next. In other words,people typically use some knowledge about the expected distribution of key valuesto “compute” where to look next. This form of “computed” binary search is calleda dictionary search or interpolation search. In a dictionary search, we search Lat a position p that is appropriate to the value of K as follows.p =K − L[1]L[n] − L[1]The location of a particular key within the key range is translated into theexpected position for the corresponding record in the table, and this position ischecked first. As with binary search, the value of the key found eliminates allrecords either above or below that position. The actual value of the key found canthen be used to compute a new position within the remaining range of the table.The next check is made based on the new computation. This proceeds until eitherthe desired record is found, or the table is narrowed until no records are left.A variation on dictionary search is known as Quadratic Binary Search (QBS),and we will analyze this in detail because its analysis is easier than that of thegeneral dictionary search. QBS will first compute p and t hen examine L[⌈pn⌉]. IfK < L[⌈pn⌉] then QBS will sequentially probe by steps of size √ n, that is, we stepthroughL[⌈pn − i √ n⌉], i = 1, 2, 3, ...until we reach a value less than or equal to K. Similarly for K > L[⌈pn⌉] we willstep forward by √ n until we reach a value in L that is greater than K. We are nowwithin √ n positions of K. Assume (for now) that it takes a constant number ofcomparisons to bracket K within a sublist of size √ n. We then take this sublist andrepeat the process recursively.What is the cost for QBS? Note that √ c n = c n/2 , and we will be repeatedlytaking square roots of the current sublist size until we find the item we are lookingfor. Because n = 2 log n and we can cut log n in half only log log n times, the costis Θ(log log n) if the number of probes on jump search is constant.The number of comparisons needed is√ n∑iP(need exactly i probes)i=1