Computational Complexity

Computational Complexity

• 1.
  Computational Complexity Kasun RangaWijeweera (Email: krw19870829@gmail.com)
• 2.
  Background • The runningtime of a program is proportional to some constant multiplied by one of these terms plus some smaller terms Leading Term + Smaller Terms (Negligible for larger N)
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  Examples • N3 +5 * N2 – 3 * N + 7 N3 is the leading term • N = 108 3 * N2 and 2 * N3 Constant coefficients can be ignored
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  Computational Complexity • Herethe worst case performance of algorithms is studied • Constant factors are ignored • Determine the functional dependence of the running time
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  Definition of Big-OhNotation • A function g(N) is said to O(f(N)) if there exists constants c0 and N0 such that g(N) is less than c0 * f (N) for all N > N0
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  Examples • 7 *N - 2 7 *N – 2 is O (N) Take c0 = 7, N0 = 1 • 3 * N3 + 20 * N2 + 5 3 * N3 + 20 * N2 + 5 is O (N3) Take c0 = 4, N0 = 21 • 3 * log N + 5 3 * log N +5 is O (log N) Take c0 = 11, N0 = 2
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  Problem • 7 *N – 2 < N2 Take c0 = 7, N0 = 1 ?
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  Growth Rate • Functionsin order of increasing growth rate is as follows  1  log N  N  N log N  N2  N3  2N
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  Examples of AlgorithmRunning Times • Min element of an array: O (N) • Closest points in the plane, i.e. smallest distance pairs: N (N - 1)/2  O (N2)
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  Comparing Algorithms Experimentally •Implement each algorithm – Lots of work – Error prone • Run it with sample data • Count the time – Same hardware and software are used
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  A Sample CProgram void main () { clrscr (); clock_t start, end; start = clock (); delay (1000); end = clock(); cout << “Time = ” << (end - start); getch(); }
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  Required Header Files #include <conio.h> # include <fstream.h> # include <dos.h> # include <time.h>
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  Reference
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  Any Questions?
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  Thank You!