e. Prove that D(TA) = (n! 9.2 Counting sort Show how to sort the numbers in linear expected time. What simple change to the algorithm preserves its linear expected running time and makes its worst-case running time O(n lg n)? i' = nA[i] Heapsort and merge sort are asymptotically optimal comparison sorts. Instead, evaluate the summation using techniques from Section 3.2. 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. For any random variable X, equation (6.30) gives Figure 9.4 shows the operation of bucket sort on an input array of 10 numbers. 9.3-3 It is a non-comparison based sorting algorithm that sorts a collection of integers. 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 are given n points in the unit circle, pi = (xi, yi), such that . Counting sort beats the lower bound of (n 1g n) proved in Section 9.1 because it is not a comparison sort. b. c. Let d(m) be the minimum value of D(T) over all trees T with m leaves. e. Prove that D(TA) = (n! Your algorithm should use O(n + k) preprocessing time. For concreteness, let's say that d lg n is the number of bits, where d is a positive constant. What is a hashing algorithm? 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. Use induction to prove that radix sort works. How much time does counting sort require? Prove that exactly n! 9 for j 1 to length[A] . k] provides temporary working storage. . Of course, the property of stability is important only when satellite data are carried around with the element being sorted. Exercises i' = nA[i] = j', 3. e. Prove that D(TA) = (n! Radix sorting by the least-significant digit first appears to be a folk algorithm widely used by operators of mechanical card-sorting machines. For any random variable X, equation (6.30) gives The elements in a given subsequence are all smaller than the elements in the succeeding subsequence and larger than the elements in the preceding subsequence. 9.2-5 = j', (Hint: Consider a tree T with k leaves that achieves the minimum. Like counting sort, bucket sort is fast because it assumes something about the input. Like counting sort, bucket sort is fast because it assumes something about the input. Describe an algorithm that, given n integers in the range 1 to k, preprocesses its input and then answers any query about how many of the n integers fall into a range [a . Counting sort assumes that each of the n input elements is an integer in the range 1 to k, for some integer k. When k = O(n), the sort runs in O(n) time. Exercises Give a simple scheme that makes any sorting algorithm stable. Show that d(k) = min1ik {d (i)+d(k - i)+k}. 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. . 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. g, (3, 4, 5) tet AN 1. nj... 1. 457 355 329 355 We assume that every permutation of A's inputs is equally likely. There are d passes, so the total time for radix sort is (dn + kd). 9] of sorted lists (buckets) after line 5 of the algorithm. What is the worst-case running time for the bucket-sort algorithm? The Counting sort algorithm is not based on comparisons like most other sorting methods are, and its time complexity is thus not bounded by Ω(nlogn) as all comparison sorts are. What simple change to the algorithm preserves its linear expected running time and makes its worst-case running time O(n lg n)? What is the worst-case running time for the bucket-sort algorithm? The code for radix sort is straightforward. . Bucket sort runs in linear time on the average. The execution of the sorting algorithm corresponds to tracing a path from the root of the decision tree to a leaf. b. b. The execution of the sorting algorithm corresponds to tracing a path from the root of the decision tree to a leaf. , B[n - 1] together in order Show that the algorithm still works properly. How are they carried out in the context of IMS networks? 9.3-5 Is the modified algorithm stable? Assuming the contrary, we have Figure 9.3 The operation of radix sort on a list of seven 3-digit numbers. In this problem, we prove an (n lg n) lower bound on the expected running time of any deterministic or randomized comparison sort on n inputs. 2 do use a stable sort to sort array A on digit i Since insertion sort runs in quadratic time (see Section 1.2), the expected time to sort the elements in bucket . 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. Let D(T) denote the external path length of a tree T; that is, D(T) is the sum of the depths of all the leaves of T. Let T be a tree with k > 1 leaves, and let RT and LT be the right and left subtrées of T. Show that D(T) = D(RT) + D(LT) + k. 457 355 329 355 The vertical arrows indicate the digit position sorted on to produce each list from the previous one. Go to Chapter 10     Back to Table of Contents, We are given n points in the unit circle, pi = (xi, yi), such that . (Hint: Consider a tree T with k leaves that achieves the minimum. for TA, and conclude that the expected time to sort n elements is (n lg n). 9.4-3 9.2-5 For decimal digits, only 10 places are used in each column. If the value of an input element is i, we increment C[i]. h lg(n!) The analysis of the running time depends on the stable sort used as the intermediate sorting algorithm. 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. Show that the algorithm still works properly. 9.3 Radix sort 9] of sorted lists (buckets) after line 5 of the algorithm. Thus, the entire bucket sort algorithm runs in linear expected time. Exercises When d is constant and k = O(n), radix sort runs in linear time. . Chapter notes Prove that exactly n! . k] provides temporary working storage. BUCKET-SORT(A) Radix sorting by the least-significant digit first appears to be a folk algorithm widely used by operators of mechanical card-sorting machines. The elements in a given subsequence are all smaller than the elements in the succeeding subsequence and larger than the elements in the preceding subsequence. 6 concatenate the lists B[0], B[1], . The first column is the input. c. Let d(m) be the minimum value of D(T) over all trees T with m leaves. b. 9] of sorted lists (buckets) after line 5 of the algorithm. Figure 9.3 shows how radix sort operates on a "deck" of seven 3-digit numbers. Lower bounds for sorting using generalizations of the decision-tree model were studied comprehensively by Ben-Or [23]. 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. In order to evaluate this summation, we must determine the distribution of each random variable ni. Give a simple scheme that makes any sorting algorithm stable. Suppose that the points are uniformly distributed; that is, the probability of finding a point in any region of the circle is proportional to the area of that region. k] provides temporary working storage. The cards are then combined into a single deck, with the cards in the 0 bin preceding the cards in the 1 bin preceding the cards in the 2 bin, and so on. Of course, the property of stability is important only when satellite data are carried around with the element being sorted. since the lg function is monotonically increasing. Exercises Go to Chapter 10     Back to Table of Contents. = j', You may use O(k) storage outside the input array. Suppose they fall into different buckets, however. If the value of an input element is i, we increment C[i]. To analyze the running time, observe that all lines except line 5 take O(n) time in the worst case. Show that the algorithm still works properly. Let D(T) denote the external path length of a tree T; that is, D(T) is the sum of the depths of all the leaves of T. Let T be a tree with k > 1 leaves, and let RT and LT be the right and left subtrées of T. Show that D(T) = D(RT) + D(LT) + k. 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. b. 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. Lower bounds for sorting using generalizations of the decision-tree model were studied comprehensively by Ben-Or [23]. To analyze the running time, observe that all lines except line 5 take O(n) time in the worst case. For example, if there are 17 elements less than x, then x belongs in output position 18. The process continues until the cards have been sorted on all d digits. = j', 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. We could run a sorting algorithm with a comparison function that, given two dates, compares years, and if there is a tie, compares months, and if another tie occurs, compares days. Since the inputs are uniformly distributed over [0, 1), we don't expect many numbers to fall into each bucket. Notes: Most students identified correctly that we needed a stable sort, however a … You may use O(k) storage outside the input array. Use induction to prove that radix sort works. 9.3-5 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. Go to Chapter 10     Back to Table of Contents, To see that this algorithm works, consider two elements A[i] and A[j]. It is essential that the digit sorts in this algorithm be stable. . 9-1 Average-case lower bounds on comparison sorting 9.3 Radix sort Give a simple scheme that makes any sorting algorithm stable. 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