Time Complexity of an algorithm/code - Part 2.pptx

Time Complexity of an algorithm/code - Part 2.pptx

• 1.
  Time Complexity Part 2 Instructors FarhatJalil
• 2.
   Slide courtesy
• 3.
  Asymptotic Complexity  The5N+3 time bound is said to "grow asymptotically" like N  This gives us an approximation of the complexity of the algorithm  Ignores lots of (machine dependent) details, concentrate on the bigger picture
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  Comparing Functions: AsymptoticNotation  Big Oh Notation: Upper bound  Omega Notation: Lower bound  Theta Notation: Tighter bound
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  Big Oh Notation To denote asymptotic upper bound, we use O- notation.  For a given function g(n), we denote by O(g(n)) (pronounced “big-oh of g of n”) the set of functions:  O(g(n))= { f(n) : there exist positive constants c and n0 such that 0 f(n) c g(n) ≤ ≤ ∗ for all n n0 ≥ }
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  Omega Notation  Todenote asymptotic lower bound, we use Ω- notation.  For a given function g(n), we denote by Ω(g(n)) (pronounced “big-omega of g of n”) the set of functions:  Ω(g(n))= { f(n) : there exist positive constants c and n0 such that 0 c g(n) f(n) ≤ ∗ ≤ for all n n0 ≥ }
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  Theta Notation  Todenote asymptotic tight bound, we use Θ-notation.  For a given function g(n), we denote by Θ(g(n)) (pronounced “big-theta of g of n”) the set of functions:  Θ(g(n))= { f(n) : there exist positive constants c1,c2 and n0 such that 0 c1 g(n) f(n) c2 g(n) ≤ ∗ ≤ ≤ ∗ for all n>n0 }
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  Example : ComparingFunctions  Which function is better? 10 n2 Vs n3 0 500 1000 1500 2000 2500 3000 3500 4000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10 n^2 n^3
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  Comparing Functions  Asinputs get larger, any algorithm of a smaller order will be more efficient than an algorithm of a larger order
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  Big-Oh Notation  Eventhough it is correct to say “7n - 3 is O(n3 )”, a better statement is “7n - 3 is O(n)”, that is, one should make the approximation as tight as possible  Simple Rule: Drop lower order terms and constant factors 7n-3 is O(n) 8n2 log n + 5n2 + n is O(n2 log n)