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14.1 The O(n2) algorithm should take: a. 12.4 milliseconds (4 times as long) to run on the 400-element array. b. 124 seconds (40,000 times as long) to run on the 40,000-element array. That s about 2 minutes. This answer can be computed algebraically as follows. The running time t is proportional to n2, so there is some constant c for which t = c n2. If it takes t = 3.1 milliseconds to sort n = 200 elements, then (3.1 milliseconds) = c (200 elements)2, so c = (3.1 milliseconds)/(200 elements)2 = 0.0000775 milliseconds/element2. Then, for n = 40,000, t = c n2 = (0.0000775 milliseconds/ element2) (40,000 elements)2 = 124,000 milliseconds = 124 seconds. The O(n lg n) algorithm should take 1.24 seconds (400 times as long) to run on the 40,000-element array. This answer can be computed algebraically. The running time t is proportional to nlgn, so there

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Given a linear shift-invariant system with a rational system function H ( z ) , the input and output are related by a linear constant coefficient difference equation. For example, with a system function

14.2 14.3

1)-bit numbers is rounded to B I bits before any additions are performed, find the variance of the round-off noise at the output of the filter.

CHAP. 14]

IMPLEMENTATION OF DISCRETE-TIME SYSTEMS (a) The system function for a pth-order allpass filter has the form

14.4 14.5 14.6 14.7 14.8 14.9

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is some constant c for which t = c nlgn If it takes t = 31 milliseconds to sort n = 200 elements, then (31) = c (200) lg(200), so c = (31 milliseconds)/(200 lg(200)) = 00155/lg(200) Then, for n = 40,000, t = c n lg n = (00155/lg(200))(40,000 lg(40,000)) = 620 (lg(40,000)/lg(200)) Now 40,000 = 2002, so lg(40,000) = lg(2002) = 2 lg 200 Thus, lg(40,000)/lg(200) = 2, so t = 620 2 milliseconds = 1240 milliseconds = 124 s 143 144 The insertion sort has its worst performance on an array that is sorted in reverse order, because each new element inserted requires all of the elements on its left to be shifted one position to the right The bubble sort, as implemented in Algorithm 141 on page 257, is insensitive to input.

H (z) =

a(p)

14.10 14.11

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That means that it will execute the same number n(n 1)/2 of comparisons regardless of the original order of the elements in the array So it doesn t matter whether the array is already sorted or whether it is sorted in reverse order; it is still very slow The selection sort is also insensitive to input: It takes about the same amount of time to sort arrays of the same size, regardless of their initial order The merge sort is also insensitive to input: It takes about the same amount of time to sort arrays of the same size, regardless of their initial order The quick sort is quite sensitive to input As implemented in Algorithm 146 on page 263, the quick sort will degrade into an O(n2) algorithm in the special cases where the array is initially sorted in either order.

With a direct form I1 implementation of this system, rounding each product to 5 any additions, we have the round-off noise model shown in the following figure:

2-28

14.12 14.13

That is because the pivot element will always be an extreme value within its subarray, so the partitioning splits the subarray very unevenly, thereby requiring n steps instead of lg n The heap sort is a little sensitive to input, but not much The heapify() function may require fewer than lg n iterations The bubble sort is slower than the selection sort, and the insertion sort (in most cases) is a little faster The merge sort is slower than the heap sort, and the quick sort (in most cases) is faster a Trace of the quick sort:.

u2 =

Note, however, that this noise model may be simplified as illustrated in the following figure:

14.14 14.15 14.16 14.17 14.18 14.19

a[0] 44 22 a[1] 88 33 a[2] 55 44 a[3] 99 77 55 a[4] 66 a[5] 33 55 77 a[6] 22 88 a[7] 88 a[8] 77 99

IMPLEMENTATION OF DISCRETE-TIME SYSTEMS Assuming that each noise source is uncorrelated with the others, the variances of e,(n) and eh(n)are u,Z=,,u; Because the output noise is

~ ; = ~ u ;

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