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# prove that counting sort is stable

Conclude that d(k) = (k lg k). Go to Chapter 10 Back to Table of Contents. 9] of sorted lists (buckets) after line 5 of the algorithm. An operator can then gather the cards bin by bin, so that cards with the first place punched are on top of cards with the second place punched, and so on. Thus, the overall time is O(k + n). Suppose that the for loop header in line 9 of the COUNTING-SORT procedure is rewritten as 9 for j ← 1 to length [ A ] Show that the algorithm still works properly. 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. Prove that COUNTING-SORT is stable. Since insertion sort runs in quadratic time (see Section 1.2), the expected time to sort the elements in bucket . Each leaf is annotated by a permutation (1),(2), . Suppose that we have an array of n data records to sort and that the key of each record has the value 0 or 1. Where is a "free" place to keep the pointers for the linked list?) . The total time to examine all buckets in line 5 is O(n), and so the only interesting part of the analysis is the time taken by the insertion sorts in line 5. 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. (Hint: Design the bucket sizes in BUCKET-SORT to reflect the uniform distribution of the points in the unit circle.) b. You may use O(k) storage outside the input array. 9.2-3 When each digit is in the range 1 to k, and k is not too large, counting sort is the obvious choice. The sort performed by a card sorter is stable, but the operator has to be wary about not changing the order of the cards as they come out of a bin, even though all the cards in a bin have the same digit in the chosen column. 9.4-2 Let i be the number of leaves in RT and k - i the number of leaves in LT.) Assuming the contrary, we have 720 329 457 720 For example, we might wish to sort dates by three keys: year, month, and day. An important property of counting sort is that it is stable: numbers with the same value appear in the output array in the same order as they do in the input array. 355 839 657 839 The sort performed by a card sorter is stable, but the operator has to be wary about not changing the order of the cards as they come out of a bin, even though all the cards in a bin have the same digit in the chosen column. e. Prove that D(TA) = (n! without using Stirling's approximation. a. Suppose that each leaf of TA is labeled with the probability that it is reached given a random input. Can your sort from part (a) be used to radix sort n records with b-bit keys in O(bn) time? To see that this algorithm works, consider two elements A[i] and A[j]. Counting sort is a stable sort, and runs in O(n + k), or linear, time where n is the size of the input list and k is the value of the max element in the input array. Unfortunately, since the cards in 9 of the 10 bins must be put aside to sort each of the bins, this procedure generates many intermediate piles of cards that must be kept track of. 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. 5 do sort list B[i] with insertion sort Which of the following sorting algorithms are stable: insertion sort, merge sort, heapsort, and quicksort? 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. To see that this algorithm works, consider two elements A[i] and A[j]. Thus, the entire bucket sort algorithm runs in linear expected time. (See Section 6.2 for a definition of uniform distribution.) Show that d(k) = min1ik {d (i)+d(k - i)+k}. leaves. 4. Give a simple, linear-time algorithm for sorting the n data records in place. b. How much additional time and space does your scheme entail? (a) The input array A[1 . Sorting on digit n orders the elements by their n-th digit; since the sort is stable, the order of those elements whose n-th digits are equal do not change. lg(n!)) lg(n!)) Show how to sort n integers in the range 1 to n2 in O(n) time. leaves are labeled 1/n! Unfortunately, the version of radix sort that uses counting sort as the intermediate stable sort does not sort in place, which many of the (n lg n) comparison sorts do. We shall see why stability is important in the next section. 9-1 Average-case lower bounds on comparison sorting e. Prove that D(TA) = (n! An operator can then gather the cards bin by bin, so that cards with the first place punched are on top of cards with the second place punched, and so on. To analyze the running time, observe that all lines except line 5 take O(n) time in the worst case. 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. Give a simple, linear-time algorithm for sorting the n data records in place. The remaining columns show the list after successive sorts on increasingly significant digit positions. The probability that a given element falls into bucket B[i] is 1/n, since each bucket is responsible for 1/n of the interval [0,1).

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