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Ye Junjie

Asympto t ic Notations & Pseudocode CSC3160 tutorial (1 st week) Office: SHB 913 Office hour: M10:00-12:00 Email: jjye@cse.cuhk.edu.hk. Ye Junjie. About the course . CSCI 3160 Design and Analysis of Algorithms Instructor : Prof. CAI Leizhen 蔡雷震 Tutor: Ye Junjie 叶俊杰 (SHB 913)

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Ye Junjie

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  1. Asymptotic Notations & PseudocodeCSC3160 tutorial (1st week)Office: SHB 913Office hour: M10:00-12:00Email: jjye@cse.cuhk.edu.hk Ye Junjie

  2. About the course CSCI 3160 Design and Analysis of Algorithms Instructor: Prof. CAI Leizhen蔡雷震 Tutor: Ye Junjie叶俊杰 (SHB 913) Office HourM10:00 - 12:00 Course Homepage: http://www.cse.cuhk.edu.hk/csci3160/ Textbook: Algorithms (S.Dasgupta, C.Papadimitriou and U.Vazirani)

  3. Outline Asymptotic Notations Big O natation Other notations: Ω, Θ Pseudocode Examples Sitting Plan Problem

  4. Asymptotic notations: Big O Definition: We say f(n) = O(g(n)) if there exist constants c, n0 such that |f(n)| ≤ c |g(n)| For all n ≥ n0 Examples: n = O(2n), n = O(0.5n), n = O(n2), 10000n = O(0.0000001n2). n2 ≠ O(n), n≠ O(log n).

  5. Asymptotic notations: Big O How to determine? f(n)=O(g(n)) means when , the order of f(n) ≤ g(n) If or constant, then f(n)=O(g(n)) Spectrum of functions… Smaller Larger Exponential functions are faster than polynomial functions Polynomial functions are faster than logarithmic functions log n = O(n) n = O(2n) O(1) … loglog n, … log n, log2 n, … n1/3, n1/2 , n, n2, n3 … 2n, 2n^2 … 22^n …

  6. Asymptotic notations: Ω & Θ Definition: We say f(n) = Ω (g(n)) if there exist constants c, n0 such that |f(n)| ≥ c |g(n)| For all n ≥ n0 We say f(n) = Θ(g(n)) if there exist constants c1, c2, n0 such that c1 |g(n)| ≤ |f(n)| ≤ c2 |g(n)| For all n ≥ n0 Theorem: f(n) = O(g(n)) if and only if g(n) = Ω(f(n)). f(n) =Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)). O means “upper bound”, Ω means “lower bound”.

  7. Examples:O, Ω, Θ f(n)= __(g(n)) (a) f(n) = n-100 ; g(n) = n-200 (b) f(n) = n1/2 ; g(n) = n2/3 (c) f(n) =100n + log n ; g(n) = n + (log n)2 (d) f(n) = n1.01 ; g(n) = n log2n

  8. Use of Pseudocode BEFORE Composing an argument  structures organization BEFORE Writing a program  Pseudocode Like a flow chart: Aims: To make everything clear and readable Attend lecture If tutorial is useful yes no Attend tutorial Skip tutorial Do homework Pseudocode: TAKE_COURSE(n) 1 attend lecture 2 if tutorial is useful attend tutorial else skip tutorial 3do homework

  9. What is Pseudocode Pseudocode is not a programming language. not C/C++, not Java, not Assembly Language, …… Pseudocode is an description of algorithm. Pseudocode is any idea of a computer program intended for human reading rather than machine reading. In this course, we use Pseudocode to present algorithms.

  10. Example 1: Bubble sort • A simple sorting algorithm • Sort a list of numbers from small to large • It works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order.

  11. Example 1: Bubble sort • Pseudocode: • 1 BUBBLE-SORT ( A : list of numbers ) • 2 for i ← length(A) down to 1 • 3 for j ← 1 to i-1 • 4 if A [j] > A [j+1] then • 5 swap A [j] and A[j+1]) • 8 end if • 9 end for • 10 end for

  12. Sitting Plan Problem Sitting Plan: How to remove k students(totally n students) so that the minimum distance between two students is at least d? Seats in a Line: linear time Seats in a loop: hints O(n^2) O(dn) ? General model: vertex cover

  13. Example 2 Seats in a Line: linear time Pseudocode: 1 Seat_Line ( S[1] to S[n] denote the places of n students) 2 preseat← S[1], count ← 0 3 for i← 2 to n-1 4 if S[i] – preseat =< d then 5 remove S[i], count++ 6 if count > k then 7 answer NO! 8 end if 9 else 10 preseat ← S[i] 11 end if 12 end for

  14. Thank you! Q&A

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