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If q p > 1, do steps 2 5 2 Apply Algorithm 147 to s, obtaining the pivot index m 3 (Invariant: the pivot element sm is in its correct sorted position) 4 Apply the quick sort to {s0, s1, , sm 1} 5 Apply the quick sort to {sm+1, si+2, , sn 1} Algorithm 147 Partition (Precondition: s = {sp sq 1} is a sequence of q p ordinal values) (Postcondition: Return m, where p m < q and si sm sj for p i 1 Set x = s p (the pivot element) 2 Set i = p and j = q..

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a(n)

Theorem 1411 The quick sort runs in O(n lgn) time in the best case The best case is when the sequence values are uniformly randomly distributed so that each call to the quick partition algorithm will result in balanced split of the sequence In that case, each recursive call to the quick sort algorithm divides the sequence into two subsequences of nearly equal length As with the binary search and the merge sort (Algorithm 145 on page 261), this repeated subdivision takes lgn steps to get down to size 1 subsequences, as illustrated in the diagram in Figure 142 on page 262 So there are O(lgn) calls made to the quick partition algorithm which runs in O(n) time, so the total running time for the quick sort algorithm is O(n lgn) Theorem 1412 The quick sort runs in O(n2) time in the worst case.

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The symmetry that exists between the numerator and denominator polynomials allows for special structures that are more efficient than direct form realizations in terms of the number of multiplies required to compute each output value y(n ).

j < q.)

(a) Consider the first-order allpass filter,

The worst case is when the sequence is already sorted (or sorted in reverse order) In that case, the quick partition algorithm will always select the last element (or the first element, if the sequence is sorted in reverse order), resulting in the most unbalanced split possible: One piece has n 2 elements, and the other piece has 1 element Repeated division of this type will occur O(n) times before both pieces get down to size 1 So there are O(n) calls made to the quick partition algorithm which runs in O(n) time, so the total running time for the quick sort algorithm is O(n2) Note that in the worst case, the quick sort reverts to the selection sort (Algorithm 142 on page 257) because each call to quick partition amounts to selecting the largest element from the subsequence passed to it So actually, Theorem 14.

H(z) =

3. 4. 5. 6. 7. 8.

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where a! is real. Find an implementation for this system that requires two delays but only one multiplication.

12 is a corollary to Theorem 144 on page 258 Theorem 1413 The quick sort runs in O(n lgn) time in the average case The proof of this fact is beyond the scope of this outline Theorem 1414 The quick sort is correct The invariant inside the while loop proof claims that all the elements to the left of a[i] are less than or equal to the pivot element and that all the elements to the right of a[j] are greater than or equal to the pivot.

where a and B are real. Derive a structure that implements this system using four delays but only two multiplies.

Repeat steps 4 7 while i < j. Decrement j until either sj < x or j = i. If j > i, copy sj into si . Increment i until either si > x or i = j. If j > i, copy sj into si . Copy x into sj.

(a) The direct form realization of a first-order allpass filter requires two multiplies and one delay as shown in the figure below.

This is true because every element to the left of a[i] that is greater than the pivot was swapped with some element to the right of a[j] that is less than the pivot, and conversely (every element to the right of a[j] that is less than the pivot was swapped with some element to the left of a[i] that is greater than the pivot When that loop terminates, j i, so at that point all the elements that are greater than the pivot have been moved to the right of a[i], and all the elements that are less than the pivot have been moved to the left of a[i] This is the invariant in step 7 of the quick partition algorithm.

IMPLEMENTATION OF DISCRETE-TIME SYSTEMS To see how the two multiplies may be combined, consider the difference equation for this system:

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