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CSE 326: Data Structures Lecture #3 Asymptotic Analysis

This lecture covers asymptotic analysis and the concept of Silicon Downs. It discusses the different types of analysis and their applications in analyzing algorithms.

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CSE 326: Data Structures Lecture #3 Asymptotic Analysis

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  1. CSE 326: Data StructuresLecture #3Asymptotic Analysis Steve Wolfman Winter Quarter 2000

  2. Today’s Outline • Silicon Downs • Asymptotic Analysis • Return Quiz

  3. Silicon Downs Post #1 n3 + 2n2 n0.1 n + 100n0.1 5n5 n-152n/100 82log n mn3 Post #2 100n2 + 1000 log n 2n + 10 log n n! 1000n15 3n7 + 7n 2mn Winner O(n2) O(log n) TIE O(n) O(n5) O(n15) O(n6) IT DEPENDS

  4. Race I n3 + 2n2 vs. 100n2 + 1000

  5. Race II n0.1 vs. log n

  6. Race III n + 100n0.1 vs. 2n + 10 log n

  7. Race IV 5n5 vs. n!

  8. Race V n-152n/100 vs. 1000n15

  9. Race VI 82log(n) vs. 3n7 + 7n

  10. FBI Finds Silicon Downs Fixed • The fix sheet (typical growth rates in order) • constant: O(1) • logarithmic: O(log n) (logkn, log n2 O(log n)) • poly-log: O(logk n) • linear: O(n) • log-linear: O(n log n) • superlinear: O(n1+c) (c is a constant > 0) • quadratic: O(n2) • cubic: O(n3) • polynomial: O(nk) (k is a constant) • exponential: O(cn) (c is a constant > 1)

  11. Terminology Given an algorithm whose running time is T(n) • T(n) O(f(n)) if there are constants c and n0 such that T(n)  c f(n) for all n  n0 • 1, log n, n, 100n  O(n) • T(n) (f(n)) if there are constants c and n0 such that T(n) c f(n) for all n  n0 • n, n2, 100 . 2n, n3 log n  (n) • T(n)  (f(n)) if T(n)  O(f(n)) and T(n)  (f(n)) • n, 2n, 100n, 0.01 n + log n  (n) • T(n)  o(f(n)) if T(n)  O(f(n)) and T(n)  (f(n)) • 1, log n, n0.99  o(n) • T(n)  (f(n)) if T(n)  O(f(n)) and T(n)  (f(n)) • n1.01, n2, 100 . 2n, n3 log n  (n)

  12. Types of analysis Orthogonal axes • bound flavor • upper bound (O, o) • lower bound (, ) • asymptotically tight () • analysis case • worst case (adversary) • average case • best case • “common” case • analysis quality • loose bound (any true analysis) • tight bound (no better bound which is asymptotically different)

  13. Analyzing Code • C++ operations - constant time • consecutive stmts - sum of times • conditionals - sum of branches, condition • loops - sum of iterations • function calls - cost of function body Above all, use your head!

  14. More Examples Than You Can Shake a Stick At (#1) for i = 1 to n do for j = 1 to n do sum= sum+ 1

  15. METYCSSA (#2) for i = 1 to n do for j = i to n do sum= sum+ 1

  16. METYCSSA (#3) • Conditional if C then S1 else S2 • Loops while C do S

  17. METYCSSA (#4) • Recursion • Almost always yields a recurrence • Recursive max T(0) <= b T(n) <= c + T(n - 1) ifn > 0 • Analysis T(n) <= c + c + T(n - 2) (by substitution) T(n) <= c + c + c + T(n - 3) (by substitution, again) T(n) <= kc + T(n - k) (extrapolating 0 < k n) T(n) <= nc + T(0) = nc + b (for k = n) • T(n)

  18. METYCSSA (#5): Mergesort • Mergesort algorithm • split list in half, sort first half, sort second half, merge together • T(1) <= b T(n) <= 2T(n/2) + cnifn > 1 • Analysis T(n) <= 2T(n/2) + cn <= 2(2T(n/4) + c(n/2)) + cn = 4T(n/4) + cn + cn <= 4(2T(n/8) + c(n/4)) + cn + cn = 8T(n/8) + cn + cn + cn <= 2kT(n/2k) + kcn(extrapolating 1 < k n) <= nT(1) + cn log n(for 2k = n or k = log n) • T(n)

  19. METYCSSA (#6): Fibonacci • Recursive Fibonacci: int Fib(n) if (n == 0 or n == 1) return 1 else return Fib(n - 1) + Fib(n - 2) • Lower bound analysis • T(0), T(1) >= b T(n) <= T(n - 1) + T(n - 2) + c ifn > 1 • Analysis let  be (1 + 5)/2 which satisfies 2 =  + 1 show by induction on n that T(n) >= bn - 1

  20. Example #6 continued • Basis: T(0)  b > b-1 and T(1)  b = b0 • Inductive step: Assume T(m)  bm - 1 for all m < n T(n)  T(n - 1) + T(n - 2) + c  bn-2 + bn-3 + c  bn-3( + 1) + c = bn-32 + c  bn-1 • T(n) • Why? Same recursive call is made numerous times.

  21. Example #6: Learning from Analysis • To avoid recursive calls • store all basis values in a table • each time you calculate an answer, store it in the table • before performing any calculation for a value n • check if a valid answer for n is in the table • if so, return it • This strategy is called “memoization” and is closely related to dynamic programming • How much time does this version take? Ask Zasha about me!

  22. Final Example (#7):Longest Common Subsequence • Problem: given two strings (m and n), find the longest sequence of characters which appears in order in both strings • lots of applications, DNA sequencing, blah, blah, blah • Example: • “search me” and “insane method” = “same”

  23. Asymptotic Analysis Summary • Determine what characterizes a problem’s size • Express how much resources (time, memory, etc.) an algorithm requires as a function of input size using O(•), (•), (•) • worst case • best case • average case • common case • overall

  24. Remind Darth Wolfman to Hand Back the Quizzes • Well, what are you waiting for?

  25. To Do • Write up homework (for Wed. beginning of class) • Finalize Project I teams • Finish Project I (for Friday 10PM) • Read chapter 6 in the book • Keep thinking about whether you like this homework format

  26. Coming Up • Priority Qs • Self-balancing trees • First homework assignment due (January 12th) • First project due (January 14th) • A day off (January 17th)!

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