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

Sec. 7.7 Binsort and Radix Sort 249The first inner for loop initializes array cnt. The second loop counts thenumber of records to be assigned to each bin. The third loop sets the values in cntto their proper indices within array B. Note that the index stored in cnt[j] is thelast index for bin j; bins are filled from high index to low index. The fourth loopassigns the records to the bins (within array B). The final loop simply copies therecords back to array A to be ready for the next pass. Variable rtoi stores r i foruse in bin computation on the i’th iteration. Figure 7.19 shows how this algorithmprocesses the input shown in Figure 7.17.This algorithm requires k passes over the list of n numbers in base r, withΘ(n + r) work done at each pass. Thus the total work is Θ(nk + rk). What isthis in terms of n? Because r is the size of the base, it might be rather small.One could use base 2 or 10. Base 26 would be appropriate for sorting characterstrings. For now, we will treat r as a constant value and ignore it for the purpose ofdetermining asymptotic complexity. Variable k is related to the key range: It is themaximum number of digits that a key may have in base r. In some applications wecan determine k to be of limited size and so might wish to consider it a constant.In this case, Radix Sort is Θ(n) in the best, average, and worst cases, making it thesort with best asymptotic complexity that we have studied.Is it a reasonable assumption to treat k as a constant? Or is there some relationshipbetween k and n? If the key range is limited and duplicate key values arecommon, there might be no relationship between k and n. To make this distinctionclear, use N to denote the number of distinct key values used by the n records.Thus, N ≤ n. Because it takes a minimum of log r N base r digits to represent Ndistinct key values, we know that k ≥ log r N.Now, consider the situation in which no keys are duplicated. If there are nunique keys (n = N), then it requires n distinct code values to represent them.Thus, k ≥ log r n. Because it requires at least Ω(log n) digits (within a constantfactor) to distinguish between the n distinct keys, k is in Ω(log n). This yieldsan asymptotic complexity of Ω(n log n) for Radix Sort to process n distinct keyvalues.It is possible that the key range is much larger; log r n bits is merely the bestcase possible for n distinct values. Thus, the log r n estimate for k could be overlyoptimistic. The moral of this analysis is that, for the general case of n distinct keyvalues, Radix Sort is at best a Ω(n log n) sorting algorithm.Radix Sort can be much improved by making base r be as large as possible.Consider the case of an integer key value. Set r = 2 i for some i. In other words,the value of r is related to the number of bits of the key processed on each pass.Each time the number of bits is doubled, the number of passes is cut in half. Whenprocessing an integer key value, setting r = 256 allows the key to be processed onebyte at a time. Processing a 32-bit key requires only four passes. It is not unreasonableon most computers to use r = 2 16 = 64K, resulting in only two passes for