Analysis of Algorithms Examples of functions important in CS: the - PowerPoint PPT Presentation

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) =

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n • the linear function: f(n) =

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n 2

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n 2 • the cubic function and other polynomials f(n) = n 3 f(n) =

[ Section 4.1 ] Analysis of Algorithms Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = log b n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n 2 • the cubic function and other polynomials f(n) = n 3 • the exponential function: f(n) =

[ Section 4.1 ] Analysis of Algorithms Comparing growth rates • f(n) = c • f(n) = log b n • f(n) = n • f(n) = n log n • f(n) = n 2 • f(n) = n 3 • f(n) = b n

[ Section 4.2 ] Analysis of Algorithms How to analyze algorithms - Experimental studies

[ Section 4.2 ] Analysis of Algorithms How to analyze algorithms - Counting the number of primitive operations Note: possible difference between the worst-case running time and the average-case running time

[ Section 4.2.3 ] Analysis of Algorithms Asymptotic notation Big-Oh notation : Let f(n) and g(n) be functions from integers to reals. We say that f(n) = O(g(n)) if there are constants c>0 and n 0 >0 such that f(n) ≤ c g(n) for every n ≥ n 0 We also say that f(n) is order of g(n).

[ Section 4.2.3 ] Analysis of Algorithms Examples: f(n) = 5n-3 g(n) = n

[ Section 4.2.3 ] Analysis of Algorithms Examples: f(n) = 7n 2 +(n 3 )/5 g(n) = n 4

[ Section 4.2.3 ] Analysis of Algorithms Examples: Insert-sort algorithm // input: array A, output: array A is sorted int i,j; int n = A.length; for (i=0; i<n-1; i++) { j = i; while ((j>=0) && (A[j+1]<A[j])) { int tmp = A[j+1]; A[j+1] = A[j]; A[j] = tmp; j--; } }

[ Section 4.2.3 ] Analysis of Algorithms Asymptotic notation continued Big-Omega notation : Let f(n) and g(n) be functions from integers to reals. We say that f(n) = Ω (g(n)) if there are constants c>0 and n 0 >0 such that f(n) ≥ c g(n) for every n ≥ n 0

[ Section 4.2.3 ] Analysis of Algorithms Asymptotic notation continued Big-Theta notation : Let f(n) and g(n) be functions from integers to reals. We say that f(n) = Θ (g(n)) if f(n) = O(g(n)) and f(n) = Ω (g(n)).

[ Section 4.2.5 ] Analysis of Algorithms Words of caution: • what is 10 100 n ? • exponential algorithms are a big NO (unless the input is really small)