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Extracting Performance Functions Basic Operations The number of Basic Operations performed must be proportional to the run time Counting techniques depend on control structures The Worst Case assumption is most common Average Case can be done for some algorithms Basic Operations Examples Sorting Key to Key Comparisons Searching Key to Unknown Compares Matrix Multiply Adds or Multiplies Graph Operations Processing a Vertex Polynomial Evaluation Arithmetic Ops Counting Procedures Straight Line Code Simply Count the operations you see Assume basic operation is addition a b c d c x 5 d d 3 e e 1 Total Operations 3 Counting IF Statements Basic Operation is Addition Assume Worst Case Count Only One Side If a b c Then c d e f g c 1 Else c e f Endif Total Operations 4 Counting Loops Assume Worst Case Number of Iterations Count Body Multiply by Iteration Count Assume Basic Operation is Addition For i 1 to 12 do a b c d d 7 End For Total Operations 24 Input Dependent Loops If the number of iterations depends on the size of the input n then the count is a function of n For i 1 to n do a b c d d 7 End For Total Operations 2n Counting Nested Loops The following rules of thumb usually apply A single loop yields a linear function of n A doubly nested loop yields a function of n2 A triply nested loop yields a function of n3 Be Careful when applying these rules For i 1 to n do For j 1 to n do a a 1 End For End For Total Operations n2 For i 1 to n do For j 1 to 3 do a a 1 End For End For Total Operations 3n Algorithm Peculiarities It is necessary to take the peculiarities of an algorithm into account when counting operations i 1 Cond ExternalFunction While i n And Cond do a a 1 Cond ExternalFunction i i 1 End While For j i to n do a a 1 End For Total Operations n Recursive Functions Define W n as the number of operations done for input of size n When encountering a recursive call add W x where x is the size of the input for the recursive call More work must be done to obtain a usable