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ASYMPTOTIC COMPLEXITY CLASSES In computer science, algorithms are classified by their complexity functions. These are functions that describe an algorithm s running time relative to the size of the problem. For example, the Bubble Sort belongs to the complexity class (n2). This means that if the Bubble Sort takes T milliseconds to sort an array of n elements, then it will take about 4T milliseconds to sort an array of 2n elements because (2n)2 = 4n2. The symbol () is one of five symbols used to describe complexity functions. They all can be defined in terms of the ratios of f(n) and g(n), where f(n) is the algorithm s timing function and g(n) is a characterizing function such as lg n or n2. For a given function g(n), the five asymptotic complexity classes are O(g(n)) = { f(n) S | f(n)/g(n) is bounded } (g(n)) = { f(n) S | g(n)/f(n) is bounded } (g(n)) = { f(n) S | f(n)/g(n) is bounded and g(n)/f(n) is bounded } } o(g(n)) = { f(n) S | f(n)/g(n) 0 as n (g(n)) = { f(n) S | g(n)/f(n) 0 as n } These definitions assume that f(n) and g(n) that are positive ascending functions. As sets of functions, o(g) O(g) (g) (g) (g) = O(g) (g) EXAMPLE A.2 Asymptotic Growth Classes

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or antisymmetric (see Sec. 5.3). h ( n ) = -h(N - n) This symmetry may be exploited to simplify the network structure. For example, if N is even and h(n) is symmetric (type 1 filter),

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a. The adjacency matrix for a wheel graph looks like matrix A shown in Figure 15.58 on page 316. b. The incidence matrix for a wheel graph looks like matrix B shown in Figure 15.58 (for the case n = 4). In general, it will have n 1s followed by n 0s on the first row. Below that will lie the identity matrix (all 1s on the diagonal and 0s elsewhere) followed by the square matrix with 1s on the diagonal and the subdiagonal. Compare this with the recursive solution to Problem 15.9 on page 308. c. The adjacency list for a wheel graph looks like the list shown in Figure 15.58 on page 316. The edge list for the first vertex (the central vertex) has n edge nodes, one for every other vertex. Every other edge list has three edge nodes: one pointing to the central vertex (labeled a in Figure 15.58) and one to each of its neighbors. a. The two graphs are isomorphic. The bijection is defined by the vertex labels shown in Figure 15.59 on page 316. b. An euler cycle for G2 is ABCDEBFCADFEA. c. A hamiltonian cycle for G2 is ABCDFEA. The trace of Dijkstra s algorithm is shown in Figure 15.60 on page 316. The trace of Dijkstra s algorithm is shown in Figure 15.61 on page 316. a. If the depth-first search is applied to a tree, it does a preorder traversal. b. If the breadth-first search is applied to a tree, it does a level-order traversal.

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Therefore, forming the sums [x(n - k) x(n - N k)] prior to multiplying by h(k) reduces the number of multiplications. The resulting structure is shown in Fig. 8-4(a). If N is odd and h(n) is symmetric (type I1 filter), the structure is as shown in Fig. 8-4(h). There are similar structures for the antisymmetric (types 111 and IV) linear phase filters.

For every k > 0, nk = o(2n ), because nk/2n 0. For every k > 0, (lgn)k = o(n), because (lgn)k/n 0. For every base b > 1, log b n = (lgn), because log b n/lg n = log b 2. The factorial numbers n! = (2n ), because 2n/n! 0.

29 1

15.15 15.16 15.17

The frequency sampling structure is an implementation that parameterizes a filter in terms of its DFT coefficients. Specifically, let H ( k ) be the N-point DFT of an FIR filter with h ( n ) .= 0 for n < 0 and n 2 N.' Because the unit sample response of the filter is 1 N-I h ( n )= ~ ( k ) J ~ ~ ~ ~ / ~ u

APPENDIX]

H ( z )=

CHAP. 15]

h(n)z-" =

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