Finally, in lines 9-11, we place each element A[j] in its correct sorted position in the output array B. Give a simple, linear-time algorithm for sorting the n data records in place. It is essential that the digit sorts in this algorithm be stable. You may use O(k) storage outside the input array. lg(n!)) The idea of bucket sort is to divide the interval [0, 1) into n equal-sized subintervals, or buckets, and then distribute the n input numbers into the buckets. We begin by examining a deterministic comparison sort A with decision tree TA. Show that d(k) = min1ik {d (i)+d(k - i)+k}. To see that this algorithm works, consider two elements A[i] and A[j]. In a decision tree, each internal node is annotated by ai : aj for some i and j in the range 1 i, j n, where n is the number of elements in the input sequence. large but relatively straightforward, Counting sort and Radix sort are two relatively small but ingenious and nonstandard algorithms with inherently complex correctness proofs. Radix sorting by the least-significant digit first appears to be a folk algorithm widely used by operators of mechanical card-sorting machines. 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. a. c. Suppose that the n records have keys in the range from 1 to k. Show how to modify counting sort so that the records can be sorted in place in O(n + k) time. Prove that exactly n! 9 for j 1 to length[A] 9-1 Average-case lower bounds on comparison sorting (b) The array C after line 7. Let these buckets be B[i'] and B[j'], respectively, and assume without loss of generality that i' < j'. The correctness of radix sort follows by induction on the column being sorted (see Exercise 9.3-3). In practice, we usually use counting sort when we have k = O(n), in which case the running time is O(n). You may use O(k) storage outside the input array. Exercises Using Figure 9.2 as a model, illustrate the operation of COUNTING-SORT on the array A = 7, 1, 3, 1, 2, 4, 5, 7, 2, 4, 3. (The other two places are used for encoding nonnumeric characters.) The remaining columns show the list after successive sorts on increasingly significant digit positions. Use no storage of more than constant size in addition to the storage provided by the array. The vertical arrows indicate the digit position sorted on to produce each list from the previous one. Suppose a list of n numbers has a continuous probability distribution function P that is computable in O(1) time. 457 355 329 355 (Section 11.2 describes how to implement basic operations on linked lists.) The first column is the input. Knuth credits H. H. Seward with inventing counting sort in 1954, and also with the idea of combining counting sort with radix sort. It is essential that the digit sorts in this algorithm be stable. 9-2 Sorting in place in linear time 9-1 Average-case lower bounds on comparison sorting Hence, we must show that A[i] A[j]. . The basic idea of counting sort is to determine, for each input element x, the number of elements less than x. Which of the following sorting algorithms are stable: insertion sort, merge sort, heapsort, and quicksort? a. Suppose a list of n numbers has a continuous probability distribution function P that is computable in O(1) time. c. Suppose that the n records have keys in the range from 1 to k. Show how to modify counting sort so that the records can be sorted in place in O(n + k) time. (Section 11.2 describes how to implement basic operations on linked lists.) nA[j] Lower bounds for sorting using generalizations of the decision-tree model were studied comprehensively by Ben-Or [23]. 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