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Algorithm Complexity

Algorithm Complexity. Level II, Term ii CSE – 243 Md. Monjur-ul-hasan Lecturer Dept of cse , cuet Email: mail@monjur-ul-hasan.info. A motivation for complexity. Talking about running time and memory space of algorithms is difficult.

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Algorithm Complexity

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  1. Algorithm Complexity Level II, Term ii CSE – 243 Md. Monjur-ul-hasan Lecturer Dept of cse, cuet Email: mail@monjur-ul-hasan.info

  2. A motivation for complexity • Talking about running time and memory space of algorithms is difficult. • Different computers will run them at different speeds and memory • Different problem sizes run for a different amount of time and memory(even on same computer). • Different inputs of the same size run for a different amount of time and memory. Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  3. Definition: Complexity • Efficiency or complexity of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity). Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  4. exp (n) n log n More Curve Fig 1.3 Fundamental Computer Algorithm By, E. Horowitz S. Sahni S. Rajasekaran f(n) n Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  5. Using Upper and Lower Bounds to talk about running time. Running time (and memory) on some computer Upper Bound Different inputs Lower Bound 5 10 15 20 Md. Monjur-ul-hasan. Lecturer, Dept of CSE CUET Problem Size

  6. Bounds on running time • We want to bound the running time: • “for arrays of size n it takes my computer 45·n2 milliseconds at most to sort the array with bubblesort” • But what about other computers? (faster/slower) • Instead we say: • “On any computer, there is a constant c such that it takes at most c·n2 time units to sort an array of size n with bubblesort” • How is this sentence useful? Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  7. Big O notation. • We say that running time of an alg. is O(f(n)) If there exists c such that for large enough n: RunningTime(n) < c·f(n) Translation: From some point c·f(n) is an upper bound on the running time. • Why only for “large enough n”? Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  8. Big-O notation • Do not assume that because we analyze running time asymptotically we no longer care about the constants. • If you have two algorithms with different asymptotic limits – easy to choose. • (may be misleading if constants are really large). • Usually we are happy when something runs in polynomial time and not exponential time. O(n2) is much much better than O(2n). Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  9. Use of big-O notation • Example: The running time of bubble sort on arrays of size n is O(n2). Translation: for a sufficiently large array, the running time is less than c·n2 for some c (No matter how bad the input is). Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  10. Big Omega • For lower bounds we use Ω(f(n)). • I.e., if we say the running time of an algorithm is Ω(f(n)), we mean thatRunningTime(n)>c·f(n) for some c. Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  11. An Example: • Checking a the number n is prime. • We can try and divide the number n by all integers in the range 2,…,n1/2. • Best case: sometimes we find it isn’t prime really quickly (e.g. it divides by 2). • Worst case:sometimes the number is prime and we try n1/2 different divisions. • Running time is O(n1/2), and is also Ω(1). Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  12. Bounds might not be tight • We saw that finding out if a number n is prime takes us O(n1/2) time. • It also takes us O(n5) time. • The second bound guarantees less. It is not tight. Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  13. Big Theta Notation • The function f(n) = Θ(g(n)) iff there exists positive constants c1 and c2 such that c1Xg(n) < f(n) < c2Xg(n). • Average time Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  14. Measurement: Tabular Method Complexity: O(n) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

  15. Bounds might not be tight Complexity: O(mn) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

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