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Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency

The Design and Analysis of Algorithms. Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency. Asymptotic Notations and Basic Efficiency Classes. Section 2.2. Asymptotic Notations and Basic Efficiency Classes. Classifying Functions by Their Asymptotic Growth

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Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency

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  1. The Design and Analysis of Algorithms Chapter 2:Fundamentals of the Analysis of Algorithm Efficiency Asymptotic Notations and Basic Efficiency Classes

  2. Section 2.2. Asymptotic Notations and Basic Efficiency Classes • Classifying Functions by Their Asymptotic Growth • Formal Definitions of Theta, Little oh and Little omega • Big Oh and Big Omega • Rules to Manipulate Big-Oh Expressions • Typical Growth Rates

  3. Classifying functions by their asymptotic growth • Given a particular function g(n) , the set of all functions can be partitioned into three sets: • Little oh : o(g(n)): the set of functions f(n) that grow slower than g(n) • Theta:Θ(g(n)): the set of functions f(n) that grow at same rate as g(n) • Little omega: ω(g(n)): the set of functions f(n) that grow faster than g(n)

  4. Formal Definition of Theta f(n) and g(n) have the same rate of growth, if lim( f(n) / g(n) ) = c, 0 < c < ∞, n → ∞ Notation: f(n) = Θ(g(n)) There exist constants c1 and c2 and a nonnegative integer n0 such that c2g(n) ≤ f(n) ≤ c1g(n) for all n ≥ n0

  5. Formal Definition of Little oh f(n) grows slower than g(n) ( or g(n) grows faster than f(n)) if lim(f(n)/g(n)) = 0, n→ ∞ Notation: f(n) = o(g(n)) There exists a constant c and a nonnegative integer n0 such that f(n) < c.g(n) for all n ≥ n0

  6. Formal Definition of Little omega f(n) grows faster than g(n) ( or g(n) grows slower than f(n)) if lim(f(n)/g(n)) = ∞, n→ ∞ Notation: f(n) = (g(n)) There exists a constant c and a nonnegative integer n0 such that c.g(n) < f(n) for all n ≥ n0

  7. Big Oh and Big Omega • O(g(n)) = o(g(n)  Θ(g(n)) •  (g(n)) = ω (g(n))  Θ(g(n))

  8. f(n) = ω (g(n)) f(n) grows faster than g(n) f(n) =Ω(g(n) f(n) = Θ (g(n)) f(n ) grows with same rate f(n) = O(g(n)) f(n) =o(g(n)) f(n) grows slower than g(n)

  9. Rules to Manipulate Big-Oh Expressions • Let T1(N) = O(f(N)) and T2(N) = O(g(N)) • T1(N) + T2(N) = max(O(f(N)), O(g(N))) where max(O(f(N)), O(g(N))) = f(N) if g(N) = O(f(N)) g(N) if f(N) = O(g(N)), • T1(N) * T2(N) = O(f(N) * g(N)) • If T(N) is a polynomial of degree k, T(N) = Θ(Nk) = O(Nk) • logkN = O(N) for any constant k.

  10. Typical Growth Rates C constant, we write O(1) logN logarithmic log2N log-squared N linear NlogN N2 quadratic N3 cubic 2N exponential N! factorial

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