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, sn 1} These are two of the three skip-3-subsequences Algorithm 144 The Shell Sort (Precondition: s = {s0 sn 1} is a sequence of n ordinal values) (Postcondition: The entire sequence s is sorted) 1 Set d = 1 2 Repeat step 3 until 9d > n 3 Set d = 3d + 1 4 Do steps 5 6 until d = 0 5 Apply the insertion sort to each of the d skip-d-subsequences of s 6 Set d = d/3 Suppose that s has n = 200 elements Then the loop at step 2 would iterate three times, increasing d from 1 to d = 4, 13, and 40 The first iteration of the loop at step 4 would apply the insertion sort to each of the 40 skip-40subsequences {s0, s40, s80, s120, s160}, {s1, s41, s81, s121, s161}, {s2, s42, s82, s122, s162}, .

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which are inside the unit circle at a radius of r = 0.5. Therefore, the poles of G ( z ) are inside the unit circle, at a radius of r' = (0.5)-"d, and G ( z )is realizable.

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, {s39, s79, s119, s159, s199} Then step 6 would reduce d to 13, and then the second iteration of the loop at step 4 would apply the insertion sort to each of the thirteen skip-13-subsequences {s0, s13, s26, s39, s52, s65, , s194}, {s1, s14, s27, s40, s53, s66, , s195}, , {s12, s25, s38, s51, s64, s77, , s193} Then step 6 would reduce d to 4, and the third iteration of the loop at step 4 would apply the insertion sort to each of the four skip-4-subsequences {s0, s4, s8, s12, , s196}, {s1, s5, s9, s13, , s197}, {s2, s6, s10, s14, , s198}, and {s3, s7, s11, s15, , s199} Then step 6 would reduce d to 1, and, and the fourth.

A causal system with areal-valued unit sample response has a frequency response with areal part given by

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For n = 2, F2 = Fn = Fn 1 + Fn 2 = F(2) 1 + F(2) 2 = F1 + F0 = 1 + 0 = 1. For n = 3, F3 = Fn = Fn 1 + Fn 2 = F(3) 1 + F(3) 2 = F2 + F1 = 1 + 1 = 2. For n = 4, F4 = Fn = Fn 1 + Fn 2 = F(4) 1 + F(4) 2 = F3 + F2 = 2 + 1 = 3. For n = 5, F5 = Fn = Fn 1 + Fn 2 = F(5) 1 + F(5) 2 = F4 + F3 = 3 + 2 = 5. For n = 6, F6 = Fn = Fn 1 + Fn 2 = F(6) 1 + F(6) 2 = F5 + F4 = 5 + 3 = 8. For n = 7, F7 = Fn = Fn 1 + Fn 2 = F(7) 1 + F(7) 2 = F6 + F5 = 8 + 5 = 13.

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iteration of the loop at step 4 would apply the insertion sort to the entire sequence. This entire process would apply the insertion sort 58 times: 40 times to subsequences of size n1 = 5, 13 times to subsequences of size n2 = 15, 4 times to subsequences of size n3 = 50, and once to the entire sequence of size n4 = n = 200. At first glance, the repeated use of the insertion sort within the shell sort would seem to take longer than simply applying the insertion sort directly just once to the entire sequence. Indeed, a direct calculation of the total number of comparisons, using the complexity function n2, yields 40(n12) + 13(n22) + 4(n32) + 1(n42) = 40(52) + 13(152) + 4(502) + 1(2002) = 53,925 which is quite a bit worse than the single n2 = 2002 = 40,000 But after the first iteration of step 4, the subsequent subsequences are nearly sorted. So the actual number of comparisons needed there is closer to n. Thus, the actual number of comparisons is more like 40(n12) + 13(n2) + 4(n3) + 1(n4) = 40(52) + 13(15) + 4(50) + 1(200) = 1,595 which is quite a bit better than 40,000. Theorem 14.8 The shell sort runs in O(n1.5) time. Note that, for n = 200, n1.5 = 2001.5 = 2,829, which is a lot better than n2 = 2002 = 40,000. EXAMPLE 14.7 The Shell Sort

Find h ( n ) and H ( ~ J " ) .

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