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Asymptotic Analysis • • • • Motivation Definitions Common complexity functions Example problems Motivation • Lets agree that we are interested in performing a worst case analysis of algorithms • Do we need to do an exact analysis? Exact Analysis is Hard! Even Harder Exact Analysis Simplifications • Ignore constants • Asymptotic Efficiency Why ignore constants? • Implementation issues (hardware, code optimizations) can speed up an algorithm by constant factors – We want to understand how effective an algorithm is independent of these factors • Simplification of analysis – Much easier to analyze if we focus only on n2 rather than worrying about 3.7 n2 or 3.9 n2 Asymptotic Analysis • We focus on the infinite set of large n ignoring small values of n • Usually, an algorithm that is asymptotically more efficient will be the best choice for all but very small inputs. 0 infinity “Big Oh” Notation • O(f(n)) = {g(n) : there exists positive constants c and n0 such that 0 <= g(n) <= c f(n) } – What are the roles of the two constants? • n0: • c: Set Notation Comment • O(f(n)) is a set of functions. • However, we will use one-way equalities like n = O(n2) • This really means that function n belongs to the set of functions O(n2) • Incorrect notation: O(n2) = n • Analogy – “A dog is an animal” but not “an animal is a dog” Three Common Sets g(n) = O(f(n)) means c f(n) is an Upper Bound on g(n) g(n) = (f(n)) means c f(n) is a Lower Bound on g(n) g(n) = (f(n)) means c1 f(n) is an Upper Bound on g(n) and c2 f(n) is a Lower Bound on g(n) These bounds hold for all inputs beyond some threshold n0. O(f(n)) (f(n)) (f(n)) (f(n)) O(f(n)) and (f(n)) O( f (n)) 1 100 ( f (n)) 1 25 n2 n Example Function f(n) = 2 3n - 100n + 6 Quick Questions c 3n2 - 100n + 6 = O(n2) 3n2 - 100n + 6 = O(n3) 3n2 - 100n + 6 O(n) 3n2 - 100n + 6 = (n2) 3n2 - 100n + 6 (n3) 3n2 - 100n + 6 = (n) 3n2 - 100n + 6 = (n2)? 3n2 - 100n + 6 = (n3)? 3n2 - 100n + 6 = (n)? n0 “Little Oh” Notation • o(g(n)) = {f(n) : for any positive constant c >0, there exists a constant n0 > 0 such that 0 <= f(n) < cg(n) for all n >= n0} – Intuitively, limn f(n)/g(n) = 0 – f(n) < c g(n) Two Other Sets g(n) = o(f(n)) means c f(n) is a strict upper bound on g(n) g(n) = w(f(n)) means c f(n) is a strict lower bound on g(n) These bounds hold for all inputs beyond some threshold n0 where n0 is now dependent on c. Common Complexity Functions Complexity 10 20 30 40 50 60 n 110-5 sec 210-5 sec 310-5 sec 410-5 sec 510-5 sec 610-5 sec n2 0.0001 sec 0.0004 sec 0.0009 sec 0.016 sec 0.025 sec 0.036 sec n3 0.001 sec 0.008 sec 0.027 sec 0.064 sec 0.125 sec 0.216 sec n5 0.1 sec 3.2 sec 24.3 sec 1.7 min 5.2 min 13.0 min 2n 0.001sec 1.0 sec 17.9 min 12.7 days 35.7 years 366 cent 3n 0.59sec 58 min 6.5 years 3855 cent 2108cent 1.31013cent log2 n 310-6 sec 410-6 sec 510-6 sec 510-6 sec 610-6 sec 610-6 sec n log2 n 310-5 sec 910-5 sec 0.0001 sec 0.0002 sec 0.0003 sec 0.0004 sec Complexity Graphs n log(n) Complexity Graphs n log(n) n n log(n) Complexity Graphs n10 n3 n2 n log(n) Complexity Graphs (log scale) nn 3n n20 2n n10 1.1n Logarithms Properties: bx = y x = logby blogbx = x logab = b loga logax = c logbx (where c = 1/logba) Questions: * How do logan and logbn compare? * How can we compare n logn with n2? Example Problems 1. What does it mean if: f(n) O(g(n)) and g(n) O(f(n)) 2. Is 2n+1 = O(2n) ? Is 22n = O(2n) ? 3. Does f(n) = O(f(n)) ? 4. If f(n) = O(g(n)) and g(n) = O(h(n)), can we say f(n) = O(h(n)) ? ???