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[ 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 ].
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[ 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) = logb n
[ Section 4.1 ] Analysis of Algorithms • Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = logb 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) = logb 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) = logb n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n2
[ Section 4.1 ] Analysis of Algorithms • Examples of functions important in CS: • the constant function: f(n) = c • the logarithm function: f(n) = logb n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n2 • the cubic function and other polynomials • f(n) = n3 • 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) = logb n • the linear function: f(n) = n • the n-log-n function: f(n) = n log n • the quadratic function: f(n) = n2 • the cubic function and other polynomials • f(n) = n3 • the exponential function: • f(n) =
[ Section 4.1 ] Analysis of Algorithms • Comparing growth rates • f(n) = c • f(n) = logb n • f(n) = n • f(n) = n log n • f(n) = n2 • f(n) = n3 • f(n) = bn
[ 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 n0>0 such that f(n) · c g(n) for every n ¸ n0 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) = 7n2+(n3)/5 g(n) = n4
[ 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) = (g(n)) if there are constants c>0 and n0>0 such that f(n) ¸ c g(n) for every n ¸ n0
[ 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) = £(g(n)) if f(n) = O(g(n)) and f(n) = (g(n)).
[ Section 4.2.5 ] Analysis of Algorithms • Words of caution: • what is 10100n ? • exponential algorithms are a big NO (unless the input is really small)
