436 657 355 657 . 9.3-4 This scheme must be modified slightly to handle the situation in which several elements have the same value, since we don't want to put them all in the same position. What simple change to the algorithm preserves its linear expected running time and makes its worst-case running time O(n lg n)? lg(n!)) 1 n length [A] Figure 9.4 shows the operation of bucket sort on an input array of 10 numbers. . Some computer scientists like to think of the number of bits in a computer word as being (lg n). d. Prove that for a given value of k, the function i lg i + (k - i) lg(k - i) is minimized at i = k/2. . Prove that exactly n! The following theorem establishes such a lower bound. Using this bound in equation (9.1), we conclude that the expected time for insertion sorting is O(n). Prove that COUNTING-SORT is stable. d. Prove that for a given value of k, the function i lg i + (k - i) lg(k - i) is minimized at i = k/2. Bucket i holds values in the interval [i/10,(i + 1)/10). and that the rest are labeled 0. Counting sort is illustrated in Figure 9.2. . Prove that 2n - 1 comparisons are necessary in the worst case to merge two sorted lists containing n elements each. 9-2 Sorting in place in linear time Let i be the number of leaves in RT and k - i the number of leaves in LT.) In the code for counting sort, we assume that the input is an array A[1 . A sorting algorithm is said to be stable if the order of the same values in the output remains the same as in input array. , B[n - 1] together in order lg(n!)) Use no storage of more than constant size in addition to the storage provided by the array. RADIX-SORT(A, d) For concreteness, let's say that d lg n is the number of bits, where d is a positive constant. 3. Consequently, the worst-case number of comparisons for a comparison sort corresponds to the height of its decision tree. . Counting Sort is stable sort as relative order of elements with equal values is maintained. Heapsort and merge sort are asymptotically optimal comparison sorts. After the initialization in lines 1-2, we inspect each input element in lines 3-4. Counting sort is illustrated in Figure 9.2. Since the card sorter can look at only one column at a time, the problem of sorting n cards on a d-digit number requires a sorting algorithm. Thus, the situation is analogous to the ball-tossing example of Section 6.6.2: we have n balls (elements) and n bins (buckets), and each ball is thrown independently with probability p = 1 /n of falling into any particular bucket. Of course, the property of stability is important only when satellite data are carried around with the element being sorted. Since a binary tree of height h has no more than 2h leaves, we have 9.2-1 Show that the algorithm still works properly. Thus, bucket sort works. . For any random variable X, equation (6.30) gives Like counting sort, bucket sort is fast because it assumes something about the input. Each pass over n d-digit numbers then takes time (n + k). f. Show that for any randomized comparison sort B, there exists a deterministic comparison sort A that makes no more comparisons on the average than B does. Now, consider a randomized comparison sort B. 9.4 Bucket sort Hence, we must show that A[i] A[j]. Bucket sorting has been in use since 1956, when the basic idea was proposed by E. J. Isaac and R. C. Singleton. For example, we might wish to sort dates by three keys: year, month, and day. The sorted output consists of a concatenation in order of the lists B[0], B[1], . Prove that exactly n! Knuth credits H. H. Seward with inventing counting sort in 1954, and also with the idea of combining counting sort with radix sort. Figure 9.4 The operation of BUCKET-SORT. a. Exercises The decision-tree model Chapter notes (See Exercise 9.3-5.) . What is the worst-case running time for the bucket-sort algorithm? . , B[9]. Problems To analyze the cost of the insertion sorts, let ni be the random variable denoting the number of elements placed in bucket B[i]. Knuth's comprehensive treatise on sorting [123] covers many variations on the sorting problem, including the information- theoretic lower bound on the complexity of sorting given here. 9.3-3 The sorter can be mechanically "programmed" to examine a given column of each card in a deck and distribute the card into one of 12 bins depending on which place has been punched. Thus, the probability that ni = k follows the binomial distribution b(k; n, p), which has mean E[ni] = np = 1 and variance Var[ni] = np(1 - p) = 1- 1/n. Prove that exactly n! Knuth credits H. H. Seward with inventing counting sort in 1954, and also with the idea of combining counting sort with radix sort. A randomization node models a random choice of the form RANDOM( 1, r) made by algorithm B; the node has r children, each of which is equally likely to be chosen during an execution of the algorithm. Alternatively, we could sort the information three times with a stable sort: first on day, next on month, and finally on year. Figure 9.3 shows how radix sort operates on a "deck" of seven 3-digit numbers. According to Knuth, the first published reference to the method is a 1929 document by L. J. Comrie describing punched-card equipment. The sorted output consists of a concatenation in order of the lists B[0], B[1], . What is the worst-case running time for the bucket-sort algorithm? (Hint: Design the bucket sizes in BUCKET-SORT to reflect the uniform distribution of the points in the unit circle.) 1 for i 1 to d 9.4-4 . , k, how many input elements are less than or equal to i; this is done by keeping a running sum of the array C. (See Exercise 9.3-5.) A probability distribution function P(x) for a random variable X is defined by P(x) = Pr{X x}. 3. c. Let d(m) be the minimum value of D(T) over all trees T with m leaves. Figure 9.4 shows the operation of bucket sort on an input array of 10 numbers. 9 for j 1 to length[A] Chapter notes (Hint: Consider a tree T with k leaves that achieves the minimum. , B[n - 1] together in order In a typical computer, which is a sequential random-access machine, radix sort is sometimes used to sort records of information that are keyed by multiple fields. We are given n points in the unit circle, pi = (xi, yi), such that . 9.4-2 BUCKET-SORT(A) Using Figure 9.4 as a model, illustrate the operation of BUCKET-SORT on the array A = .79, .13, .16, .64, .39, .20, .89, .53, .71, .42. 3 do insert A[ i] into list B [nA[i]] Explain FAMA... 1. 9.3-4 . Modify COUNTING-SORT to produce the output in sorted order without using any substantial additional storage besides that in A and C. (Hint: Link elements of A that have the same key into lists. In order to evaluate this summation, we must determine the distribution of each random variable ni. Of course, the property of stability is important only when satellite data are carried around with the element being sorted. A d-digit number would then occupy a field of d columns. Describe symmetric key and public key encryption. 5 do sort list B[i] with insertion sort (See Section 6.2 for a definition of uniform distribution.) A probability distribution function P(x) for a random variable X is defined by P(x) = Pr{X x}. Let these buckets be B[i'] and B[j'], respectively, and assume without loss of generality that i' < j'. Hence, we must show that A[i] A[j]. for TA, and conclude that the expected time to sort n elements is (n lg n). Lower bounds for sorting using generalizations of the decision-tree model were studied comprehensively by Ben-Or [23]. 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