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Unformatted text preview: counting costs
want a coarse comparison of algorithms \speed" that ignores hardware, programmer virtuosity which speed do we care about: best, worst, average? why? dene idealized \step" that doesn't depend on particular hardware and idealized \time" that counts the number of steps for a given input. slide 9 linear search
def LS(A,x) : """ Return index i such that x == L[i]. 1. i = 0 2. while i < len(A) : 3. if A[i] == x : 4. return i 5. i = i + 1 6. return 1 Otherwise, return 1 """ Trace LS([2,4,6,8],4), and count the time complexity What is What is
tLS A; x tLS ([2 4 6 8] 4)
; ; ; ; ( ), if the rst index where is found is ?
x j tLS A; x ( ) is isn't in at all?
x A slide 10 worst case
denote the worstcase complexity for program with input where the input size of is as size( ) = P ( ) = max P ( )
P x n W n ft x j x P I x ng x P I , The upper bound That is: WP P y U ( ) means N N
; Vn P Wc P Wc P R+ R+ ; WB P ; WB P ; Vx P I ; N size( )
x ;n ! B ! B A A max P ( ) size( ) = (size( )) P( )
ft x j x P I x x t x cU ng cU n ( ) The lower bound That is: WP P ( ) means
L Wc P Wc P R+ R+ ; WB P ; WB P N N ; Vn P ; Vn P N N ;n ! B ;n ! B A A max P ( ) size( ) =
ft x Wx P I ; x j x P I n t size( ) = ( ) ( ) P( )
x ng ! cU n x ! cL n slide 11 bounding a sort
def IS(A) : """ IS(A) sorts the elements of A in nondecreasing order """ 1. i = 1 2. while i < len(A) : 3. t = A[i] 4. j = i 5. while j > 0 and A[j1] > t : 6. A[j] = A[j1] # shift up 7. j = j1 8. A[j] = t 9. i = i+1 I want to prove that WIS P y n ( 2). slide 12 scratch slide 13 ...
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 Fall '10
 DannyHeap
 Computer Science, Algorithms, Analysis of algorithms, Computational complexity theory, size, max p, ne idealized \step

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