Algorithm Analysis and Notations

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Algorithm Analysis

  • Task of measuring how much storage and determining how much computing time required from algorithm.

Time complexity

  • The time complexity of algorithm is defined as amount of time taken by algorithm to run its complexity

  • Number of instruction which one algorithm to execute during its running time is called time complexity.

  • It is machine independent

  • Time Complexity represent as a function of input t(n) time required to execute number of steps.

  • Sometimes it may happen some algorithm have different time complexity just because of its input size and number of steps executed.

Space Complexity

  • When we design an algorithm to solve problem it needs some computer memory to complete its execution.

  • Total amount of computer memory required by an Algorithm to complete its execution is called space complexity.

  • It is machine dependent.

  • The memory is primary memory not a hard drive or removable media.


  1. Asymptotic Notation
  2. Big-O-Notation
  3. Omega Notation
  4. Theta Notation


Asymptotic Notation

  • A Asymptotic notation is used to describe the running time of an algorithm its a order of growth of function


  • They can be used to represent the complexities of algorithms for asymptotic analysis.

  • They allow the comparison of the performance of an algorithm.

  • To choose the best algorithm we need to check the efficiency of each algorithm.

  • Asymptotic notation a short hand way to represent time complexity.

Big-O-Notation (upper-bound; worst case)

  • The big-O notation is a method representing the upper bond of algorithm running time.

  • Using big-O-Notation we can give longest amount of time taken by the algorithm to complete.

  • Let f(n) and g(n) being two non-negative functions

  • Similarly C is some constant such that C > 0 we can write f(n) <= C * g(n)

  • It is also denoted as f(n (- Og(n))

  • In other words f(n) < g(n) if g(n) is multiple of some constant.

Omega Notation (lower bond; best case)

  • Omega notation is denoted by Ω (omega).

  • This notation is used to represent lower bound of algorithm running time.

  • Using omega notation we can denote shortest amount of time taken by algorithm.

  • A function f(n) is said to be in Ω(g(n)) if f(n) is bounded below some positive constant multiple of y(n) such that f(n) >= C * g(n)

  • For all n >= no is denoted as f(n) (- g(n)

Theta Notation

  • Theta notation is denoted by Θ

  • In this method the running time is between upper bound and lower bound.

  • Let f(n) and g(n) be two non-negative functions.

  • There are two positive constant named C1 and C2

C1 * g(n) <= f(n) <= C2 * g(n)

  • Then we can say that f(n) (- Θ g(n)

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