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Foundations II: Data Structures and Algorithms

Foundations II: Data Structures and Algorithms. Instructor : Yusu Wang Lecture 1 : Introduction and Asymptotic notation. Course Information. Course webpage http://www.cse.ohio-state.edu/~yusu/courses/2331 Office hours TuTh 11:00 – 11:30 am Grading policy:

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Foundations II: Data Structures and Algorithms

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  1. Foundations II: Data Structures and Algorithms Instructor : Yusu Wang Lecture 1 : Introduction and Asymptotic notation CSE 2331 / 5331

  2. Course Information • Course webpage http://www.cse.ohio-state.edu/~yusu/courses/2331 • Office hours • TuTh 11:00 – 11:30 am • Grading policy: • homework: 20%, • two midterms: 40%, • final: 40% CSE 2331 / 5331

  3. Exams • Midterm 1 • Sept 30th in class. • Midterm 2 • Nov. 4th in class. • Final exam: • Dec. 12th, 10:0am--11:45am, Room DL0264 Mark you calendar. Contact me within first two full weeks if any schedule conflict for midterms CSE 2331 / 5331

  4. Notes • All homework should be submitted before or in class on the due date. Late homework during the same day receives a 10% penalty. Late homework submitted the next day receives a 25% penalty. • You may discuss homework with your classmates. It is very important that you write up your solutions individually, and acknowledge any collaboration / discussion with others. CSE 2331 / 5331

  5. More Information • Textbook • Introduction to Algorithms byCormen, Leiserson, Rivest and Stein (third edition) • Others • CSE 2331 course notes • By Dr. Rephael Wenger (SBX) • The Art of Computer Programming By Donald Knuth CSE 2331 / 5331

  6. Introduction • What is an algorithm? • Step by step strategy to solve problems • Should terminate • Should return correct answer for any input instance CSE 2331 / 5331

  7. This Course • Various issues involved in designing good algorithms • What do we mean by “good”? • Algorithm analysis, language used to measure performance • How to design good algorithms • Data structures, algorithm paradigms • Fundamental problems • Graph traversal, shortest path etc. CSE 2331 / 5331

  8. Asymptotic Notation CSE 2331 / 5331

  9. A = < , , …. > a a a n 2 1 Sorting problem • Input: • Output: permutation of A that is sorted • Example: • Input: < 3, 11, 6, 4, 2, 7, 9 > • Output: < 2, 3, 4, 6, 7, 9, 11 > CSE 2331 / 5331

  10. 3, 6, 11, 4, 2, 7, 9 3, 4, 6, 11, 2, 7, 9 2, 3, 4, 6, 11, 7, 9 Insertion Sort 1 i i+1 n 3, 11, 6, 4, 2, 7, 9 CSE 2331 / 5331

  11. Pseudo-code InsertionSort(A, n) for i = 1 to n-1 do key = A[i+1] j = i while j > 0 and A[j] > key do A[j+1] = A[j] j = j - 1 A[j+1] = key CSE 2331 / 5331

  12. Analysis • Termination • Correctness • beginning of for-loop: if A[1.. i] sorted, then • end of for-loop: A[1.. i+1] sorted. • when i = n-1, A[1.. i+1] is sorted. • Efficiency: time/space • Depends on input size n • Space: roughly n CSE 2331 / 5331

  13. Running Time • Depends on input size • Depends on particular input • Best case • Worst case First Idea: Provide upper bound (as a guanrantee) Make it a function of n : T(n) CSE 2331 / 5331

  14. Running Time • Worst case • T(n) = max time of algorithm on any input of size n • Average case • T(n) = expected time of algorithm over all possible inputs of size n • Need a statistical distribution model for input • E.g, uniform distribution CSE 2331 / 5331

  15. c c c c c c c 1 2 4 2 1 3 5 n T(n) =  ( + i ) = + n + i=2 2 n Insertion Sort InsertionSort(A, n) for i = 1 ton-1do key = A[i] j = i while j > 0 and A[j] > key do A[j+1] = A[j] j = j - 1 A[j+1] = key CSE 2331 / 5331

  16. Asymptotic Complexity • Why shall we care about constants and lower order terms? • Should focus on the relative growth w.r.t. n • Especially when n becomes large ! • E.g: 0.2n1.5 v.s 500n Second Idea: Ignore all constants and lower order terms Only order of growth --- Asymptotic time complexity CSE 2331 / 5331

  17. f(n) lim ≤ c g(n) n Big O Notation • f(n)  O(g(n))if and only if c > 0, and n0 , so that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 • f(n) = O(g(n)) means f(n)  O(g(n))(i.e, at most) • We say that g(n) is an asymptotic upper bound for f(n). • It also means: • E.g, f(n) = O(n2) if there exists c, n0 > 0 such that f(n) ≤ cn2, for all n ≥ n0. CSE 2331 / 5331

  18. Illustration CSE 2331 / 5331

  19. More Examples • ? • ? • ? • ? • ? • ? • ? CSE 2331 / 5331

  20. f(n) lim ≥ c g(n) n Omega  Notation • f(n)  (g(n)) if and only if  c > 0, and n0 , so that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 • f(n) =  (g(n)) means f(n)  (g(n)) (i.e, at least) • We say g(n) is an asymptotic lower bound of f(n). • It also means: CSE 2331 / 5331

  21. Illustration CSE 2331 / 5331

  22. More Examples • ? • ? • ? • ? • ? • ? • ? CSE 2331 / 5331

  23. Theta  Notation • Combine lower and upper bound • Means tight: of the same order • if and only if , and, such that for any • means • We say g(n) is an asymptotically tight bound for f(n). • It also means: CSE 2331 / 5331

  24. CSE 2331 / 5331

  25. Remarks -- 1 • if and only if , and . • Insertion sort algorithm as described earlier has time complexity . • Transpose symmetry: • If , if and only if • Transitivity: • If and , then . • Same for and CSE 2331 / 5331

  26. Remarks -- 2 • It is convenient to think that • Big-O is smaller than or equal to • Big- is larger than or equal to • Big- is equal • However, • These relations are modulo constant factor scaling • Not every pair of functions have such relations If , this does not imply that . CSE 2331 / 5331

  27. n n 2 2 Remarks -- 3 • Provide a unified language to measure the performance of algorithms • Give us intuitive idea how fast we shall expect the alg. • Can now compare various algorithms for the same problem • Constants hidden! • O( n lg lg n ), 2n lg n • T(n) =+ O (n) means T(n) = + h(n), and h(n) = O(n) CSE 2331 / 5331

  28. Pitfalls OK • O(n) + O(n) = O(n) • T(n) = n +  O(n) = n + O(n) k ? i = 1 k should not depend on n ! It can be an arbitrarily large constant … CSE 2331 / 5331

  29. f(n) lim = 0 g(n) n Yet Another Notation • Small o notation: • f(n) = o(g(n))means for any s.t. for all In other words, • Examples: • Is n - lg n = o(n) ? • Is n = o (n lg n) ? CSE 2331 / 5331

  30. f(n) g(n) lim = lim = 0 g(n) f(n) n n Yet Another Notation • Small ω notation: • f(n) = ω(g(n)) means for any s.t. for all In other words, or • Examples: • Is n - lg n = ω(n) ? • Is n = ω(n / lgn) ? CSE 2331 / 5331

  31. Review some basics • Chapter 3.2 of CLRS • See the board CSE 2331 / 5331

  32. Summary • Algorithms are useful • Example: sorting problem • Insertion sort • Asymptotic complexity • Focus on the growth of complexity w.r.t input size CSE 2331 / 5331

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