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Understanding Asymptotic Analysis in Data Structures

Learn about homework, teamwork, proofs, and asymptotic analysis with Steve Wolfman in this lecture from Winter Quarter 2000. Discover techniques such as proof by counterexample, contradiction, and induction for analyzing algorithms. Understand time and space complexity, rates of growth, and order notation with practical examples and next steps. Join in and enhance your understanding of data structures!

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Understanding Asymptotic Analysis in Data Structures

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  1. CSE 326: Data StructuresLecture #2Asymptotic Analysis Steve Wolfman Winter Quarter 2000

  2. Today’s Outline • How Homework Works • Forming Teams • Things Steve Didn’t Finish on Wednesday (Cross Lists) • Proofs • Asymptotic Analysis • Quiz (don’t be scared!)

  3. Homework • Quiz on Friday • lasts about 10 minutes in class • what you get right counts toward homework • what you get wrong becomes short answer for homework • quiz returned on Monday • Homework due Wednesday • turn in at the start of class

  4. Teams

  5. Proof by... • Counterexample • show an example which does not fit with the theorem • QED (the theorem is disproven) • Contradiction • assume the opposite of the theorem • derive a contradiction • QED (the theorem is proven) • Induction • prove for a base case (e.g., n = 1) • assume for an anonymous value (n) • prove for the next value (n + 1) • QED

  6. Example Proof by Induction • A number is divisible by 3 iff the sum of its digits is divisible by three • Base Case:

  7. Example Continued • Induction Hypothesis:

  8. Example Still Continued • Induction Step:

  9. Asymptotic Analysis • Find a student’s name in a class given her UWID

  10. Analysis of Algorithms • Analysis of an algorithm gives insight into how long the program runs and how much memory it uses • time complexity • space complexity • Analysis can provide insight into alternative algorithms • Input size is indicated by a number n (sometimes there are multiple inputs) • Running time is a function of n such as T(n) = 4n + 5 T(n) = 0.5 n log n - 2n + 7 T(n) = 2n + n3 + 3n • But...

  11. Asymptotic Analysis • Eliminate low order terms • 4n + 5  4n • 0.5 n log n - 2n + 7  0.5 n log n • 2n + n3 + 3n  2n • Eliminate coefficients • 4n  n • 0.5 n log n  n log n • n log n2 = 2 n log n  n log n

  12. Rates of Growth • Suppose a computer executes 1012 ops per second: 104s = 2.8 hrs 1018s = 30 billion years

  13. Order Notation • T(n)  O(f(n)) if there are constants c and n0 such that T(n)  c f(n) for all n  n0 • T(n)  (f(n)) if there are constants c and n0 such that T(n) c f(n) for all n  n0 • T(n)  (f(n)) if T(n)  O(f(n)) and T(n)   (f(n)) • T(n) o(f(n)) if T(n)  O(f(n)) and T(n)  (f(n)) • T(n) (f(n)) if T(n)  O(f(n)) and T(n)  (f(n))

  14. Examples 10,000 n2 + 25 n (n2) 10-10 n2(n2) n log n  O(n2) n log n (n) n3 + 4  o(n4) n3 + 4 (n2)

  15. To Do • Start Project I • Answer the Quiz questions you think you missed • Read chapters 1-3 in the book • Think about whether you like this homework format

  16. Coming Up • Asymptotic Analysis of Recursion • Priority Qs • First homework assignment due (January 12th) • First project due (January 14th) • A day off (January 17th)!

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