A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 3.5 Calculating the Running Time for a Program 77then multiplied by the input value, which takes constant time. Thus, the cost ofthe factorial function, if we wish to measure cost in terms of the number of multiplicationoperations, is one more than the number of multiplications made by therecursive call on the smaller input. Because the base case does no multiplications,its cost is zero. Thus, the running time for this function can be expressed asT(n) = T(n − 1) + 1 for n > 1; T (1) = 0.We know from Examples 2.8 and 2.13 that the closed-form solution for this recurrencerelation is Θ(n).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, then youknow that you can ignore all positions in the array less than mid. Either way, halfof the positions are eliminated from further consideration. Binary search next looksat the middle position in that part of the array where value K may exist. The valueat this position again allows us to eliminate half of the remaining positions fromconsideration. This process repeats until either the desired value is found, or thereare no positions remaining in the array that might contain the value K. Figure 3.4illustrates the binary search method. Here is an implementation for binary search:// Return the position of an element in sorted array "A"// with value "K". If "K" is not in "A", return A.length.static int binary(int[] A, int K) {int l = -1;int r = A.length; // l and r are beyond array boundswhile (l+1 != r) { // Stop when l and r meetint i = (l+r)/2; // Check middle of remaining subarray}if (K < A[i]) r = i; // In left halfif (K == A[i]) return i; // Found itif (K > A[i]) l = i; // In right half}return A.length; // Search value not in A