Data Structures and Algorithm Analysis - Computer Science at ...

72 Chap. 3 Algorithm AnalysisPosition 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Key 11 13 21 26 29 36 40 41 45 51 54 56 65 72 77 83Figure 3.4 An illustration of binary search on a sorted array of 16 positions.Consider a search for the position with value K = 45. Binary search first checksthe value at position 7. Because 41 < K, the desired value cannot appear in anyposition below 7 in the array. Next, binary search checks the value at position 11.Because 56 > K, the desired value (if it exists) must be between positions 7and 11. Position 9 is checked next. Again, its value is too great. The final searchis at position 8, which contains the desired value. Thus, function binary returnsposition 8. Alternatively, if K were 44, then the same series of record accesseswould be made. After checking position 8, binary would return a value of n,indicating that the search is unsuccessful.The final example of algorithm analysis for this section will compare two algorithmsfor performing search in an array. Earlier, we determined that the runningtime for sequential search on an array where the search value K is equally likelyto appear in any location is Θ(n) in both the average and worst cases. We wouldlike to compare this running time to that required to perform a binary search onan array whose values are stored in order from lowest to highest.Binary search begins by examining the value in the middle position of the array;call this position mid and the corresponding value k mid . If k mid = K, thenprocessing can stop immediately. This is unlikely to be the case, however. Fortunately,knowing the middle value provides useful information that can help guidethe search process. In particular, if k mid > K, then you know that the value Kcannot appear in the array at any position greater than mid. Thus, you can eliminatefuture search in the upper half of the array. Conversely, if k mid < K, thenyou know that you can ignore all positions in the array less than mid. Either way,half of the positions are eliminated from further consideration. Binary search nextlooks at the middle position in that part of the array where value K may exist. Thevalue at this position again allows us to eliminate half of the remaining positionsfrom consideration. This process repeats until either the desired value is found, orthere are no positions remaining in the array that might contain the value K. Figure3.4 illustrates the binary search method. Figure 3.5 shows an implementationfor binary search.To find the cost of this algorithm in the worst case, we can model the runningtime as a recurrence and then find the closed-form solution. Each recursive callto binary cuts the size of the array approximately in half, so we can model theworst-case cost as follows, assuming for simplicity that n is a power of two.T(n) = T(n/2) + 1 for n > 1; T(1) = 1.