1 / 19

Analysis of Algorithms

This analysis compares the efficiency of two algorithms, algo1 and algo2, to determine which algorithm is better suited for a specific task. The efficiency is measured by analyzing the number of operations performed as a function of input size.

dionr
Download Presentation

Analysis of Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Analysis of Algorithms CPS212 Gordon College

  2. Measuring the efficiency of algorithms • There are 2 algorithms: algo1 and algo2 that produce the same results. • How do you choose which algorithm to keep. • Algorithm Efficiency

  3. Measuring the efficiency of algorithms • How do you determine an algorithm’s inherent efficiency? • Code them up and compare their running times on a lab machine. • How were they coded? We want to compare the algorithms - not their implementations • What computer do we use? What else is running? • What data is a fair test of an algorithm’s efficiency.

  4. Measuring the efficiency of algorithms • Goal: • Measure the efficiency of an algorithm - independent of implementation, hardware, or data. • Solution: • Analyze and represent the number of operations the algorithm will perform as a function of input. • Example: • Copying an array with n elements requires x operations. What is x?

  5. Growth rates • Algo1 requires n2 / 2 operations to solve a problem with input size n • Algo2 requires 5n + 10 operations Which is the better algorithm for the job?

  6. Order of magnitude growth analysis • A function f(x) is O(g(x)) if and only if there exist 2 positive constants, c and n, such that | f(x) |≤ cg(x) for all x > n cg(x) f(x) number of operations size of input set

  7. Order of magnitude growth analysis • Asymptotic growth - as the growth approaches infinity • An asymptote of a real-valued function y = f(x) is a curve which describes the behavior of f as either x or y tends to infinity. cg(x) f(x) number of operations size of input set

  8. Order of magnitude growth analysis • Important points: • Focus only on the growth • Shape of g(x) is essential • As the input set grows large (ignore the shape for small x) cg(x) f(x) number of operations size of input set

  9. Standard function shapes: constant • O(1) Examples? f(x) = cg(x) number of operations size of input set

  10. Standard function shapes: linear • O(x) • Examples? cg(x) f(x)=ax+b number of operations size of input set

  11. Standard function shapes: logarithmic • O(log x) (base 2) bc = a logba = c 23 = 8 log28 = 3 log(x*y) = log x + log y log(xa) = a log x Examples?

  12. Standard function shapes: Polynomial • O(x2) Examples?

  13. Polynomial vs. Linear

  14. Polynomial vs. Linear Example: 50n + 20 & n2 At what point will n2 surpass 50n+20? n2 = 50n+20 Solve for x…quadratic formula n2 - 50n - 20 = 0 n = 101/2 = 50.5 n = -1/2

  15. Standard function shapes: Exponential • O(cx) Examples? Hamiltonian Circuit Traveling Salesman Optimization problems Solution: Limit to small input sets Isolate special cases Find approximate solution (near optimal)

  16. Complexity in action

  17. Real Examples • Searching (sequential) • Unit of work: comparisons • Best case: O(1) [theta] • Worst case: O(n) [theta] • Average Case: O(n) [theta]

  18. Real Examples • Sort (selection) • Unit of work: comparisons and exchanges • Best case: O(n2) [theta] • Worst case: O(n2) [theta] • Average Case: O(n2) [theta]

  19. Real Examples • Search (binary) • Unit of work: comparisons • Best case: O(1) [theta] • Worst case: O(log n) [theta] • Average Case: O(log n) [theta]

More Related