Asymptotic notation
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This document discusses asymptotic notations that are used to characterize an algorithm's efficiency. It introduces Big-Oh, Big-Omega, and Theta notations. Big-Oh gives an upper bound on running time. Big-Omega gives a lower bound. Theta gives both upper and lower bounds, representing an algorithm's exact running time. Examples are provided for each notation. Time complexity is also introduced, which describes how the time to solve a problem changes with problem size. Worst-case time analysis provides an upper bound on runtime.
Asymptotic notation
- 1. ASYMPTOTIC NOTATIONS Shashikant V. Athawale Assistant Professor ,Computer Engineering Department AISSMS College of Engineering, Kennedy Road, Pune , MS, India - 411001 1
- 2. WHY IMPORTANT ?? Give a simple characterization of an algorithm’s efficiency. Allow comparison of performances of various algorithms 2
- 3. ASYMPTOTIC NOTATIONS 1. Big-oh Notation (O) 2. Big-Omega Notation (Ω) 3. Theta Notation (Θ ) 3
- 4. BIG-OH NOTATION (O) Gives the upper bound of algorithm’s running time. Let f: N-> R be a function. Then O(f) is the set of functions O(f) = { g: N-> R | there exists a constant c and a natural number n0 such that |g(n)| <= c|f(n)| for all n>= n0 } 4
- 5. BIG-OMEGA NOTATION (Ω) Gives the lower bound of algorithm’s running time. Let f, g: N-> R be functions from the set of natural numbers to the set of real numbers. We write g ∈ Ω(f) if and only if there exists some real number n0 and a positive real constant c such that g(n)| >= c|f(n)| for all n in N satisfying n>= n0. 5
- 6. THETA NOTATION (Θ ) If f and g are functions from S to the real numbers, then we write g ∈ Θ(f) if and only if there exists some real number n0 and positive real constants C and C’ such that C|f(n)|<= |g(n)| <=C’|f(n)| for all n in S satisfying n>= n0 . Thus, Θ(f) = O(f) ∩ Ω(f) 6
- 7. 7 notation intuition O (Big-Oh) f(n) ≤ g(n) Ω (Big-Omega) f(n) ≥ g(n) Θ (Theta) f(n) = g(n) INTUITION ABOUT THE NOTATIONS
- 8. LITTLE OH NOTATION (O) little-Oh Defn: f(n) = o(g(n)) If for all positive constants c there exists an n0 such that f(n) < c· g(n) for all n ≥ n0. 8
- 9. LITTLE OMEGA NOTATION (Ω) little-Omega Defn: f(n) = Ω(g(n)) If for all positive constants c there exists an n0 such that f(n) > c· g(n) for all n ≥ n0. 9
- 10. TIME COMPLEXITY Dependency of the time it takes to solve a problem as a function of the problem dimension/size Examples: Sorting a list of length n Searching a list of length n Multiplying a n×n matrix by an n×1 vector Time to solve problem might depend on data Average-case time Best-case time data is well suited for algorithm (can’t be counted on) Worst-case time data is such that algorithm performs poorly (time-wise) Worst-Case gives an upper bound as to how much time will be needed to solve any instance of the problem 10
- 11. Thank You 11