elimence / Algorithms

Implementation of common algorithms in various languages

Analysis Of Algorithms

• The efficiency of an algorithm can be measured in terms of:

  • Execution time time complexity
  • The amount of memory required space complexity

• The measure which is most important depends on the limitations of the technology available at time of analysis

Time Complexity

• For most algorithms analyzed here, time complexity comparisons are more interesting than space complexity comparisons.

• Time complexity: A measure of the amount of time required to execute an algorithm

• Factors that should not affect time complexity analysis:

  • The programming language chosen to implement the algorithm
  • The quality of the compiler
  • The speed of the computer on which the algorithm is to be executed

• Objectives of time complexity analysis:

  • To determine the feasibility of an algorithm by estimating an upper bound on the amount of work performed
  • To compare different algorithms before deciding on which one to implement

• Time complexity expresses the relationship between the size of the input and the run time for the algorithm and is usually expressed as a proportionality rather than an exact function

• To simplify analysis, we sometimes ignore work that takes a constant amount of time, independent of the input size

• Simplifies analysis can be based on number of:

  • Arithmetic operations performed
  • Comparisons made
  • Times through a critical loop etc

Big-O Notation

• Formally, the time complexity T(n) of an algorithm is O(f(n)) of the order f(n) if, for some positive constants C1 and C2 for all but finitely many values of n

• This gives upper and lower bounds on the amount of work done for all sufficiently large n

• We want an easily recognized elementary function to describe the performance of the algorithm, so we use the dominant term of T(n): it determines the basic shape of the function

Worst Case vs. Average Case

• Worst case analysis is used to find an upper bound on an algorithm's performance for large problems (large n)
• Average case analysis determines the average (or expected) performance
• Worst case time complexity is usually simpler to work out

Big-O Analysis in General

• With independent nested loops: The number of iterations of the inner loop is independent of the number of iterations of the outer loop

  • Example

   int x = 0;
  
   for (int j = 1; j <= n/2; j++)
   	for (int k = 1; k <= n*n; k++)
   		x = x + j + k;