CS 310 Exam I Review - CS 310 Exam I Review: Units 1-8...

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CS 310 Exam I Review: Units 1-8 Furman Haddix Ph.D. Assistant Professor Minnesota State University,  Mankato Spring 2008
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Unit 1 Simple Sort Examples Bubble Sort Insertion Sort Merge Sort Recurrence Tree Notation Running Time:  Θ (n) Floor and Ceiling Functions:   
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Unit 2 Asymptotic Notation O(n) – “Big Oh” (or “Oh”) of n (n) – Omega (or “Big Omega”) of n Θ (n) – Theta of n o(n) – Little oh of n ϖ (n) – Little omega of n Solving Recurrence Relations Substitution method Recursion-tree method Master method
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Substitution Method The substitution method can be used for  establishing upper or lower bounds for a  recurrence relation The substitution method uses the following  steps: Guess the solution Prove the inductive step for the solution Prove the basis step for the solution
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Recurrence-Tree  Method Example T(n) = 3T(n/4) +  Θ (n 2 ) cn 2 (3/16)cn 2 (3/16) 2 cn 2   … … …  … … … … … … … … … … … … … … … …  … … … … … … … … Θ (3 log 4 n )   T(1) T(1) T(1) T(1) T(1) T(1)    …   T(1) T(1) T(1) T(1)  T(1) T(1)  Θ (n log 4 3 T(n) = cn 2  + (3/16)cn 2  + (3/16) 2 cn 2  + (3/16) 3 cn 2  + … +  Θ (n log 4 3 log a T(n) cn 2 T(n/4) T(n/4) T(n/4) c(n/4) 2 T(n/16) T(n/16) T(n/16) c(n/4) 2 T(n/16) T(n/16) T(n/16) c(n/4) 2 T(n/16) T(n/16) T(n/16) c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2 c(n/16) 2
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Master Method – Another Recurrence  Tool Works for recurrences of the form: T(n) = a T(n/b) + f(n), where a   1, b > 1,  and f(n) is asymptotically positive    positive for n   n 0   Master Method: 1. If f(n) = O(n log b a- ε ) for some constant  ε  >0, then  T(n) =  Θ (n log b a ) 2. If f(n) =  Θ (n log b a ), then T(n) =  Θ (n log b a  lg n) 3. If f(n) =  (n log b a+ ε ) for some constant  ε  > 0, and if          a f(n/b)   cf(n) for some constant c > 1 and  sufficiently large n, then T(n) =  Θ (f(n)) Note that some recurrences appear to 
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Unit 3 The divide-and-conquer design paradigm Merge sort Binary search Powering   number Fibonacci numbers VLSI tree layout
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Homework 1 Problem 2-4 in textbook
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Homework 2 Problem 4-1 is due on Tuesday, February  5, 2008.
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This note was uploaded on 06/09/2008 for the course CS 310 taught by Professor Furmanhaddix during the Spring '08 term at Minnesota State University, Mankato.

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CS 310 Exam I Review - CS 310 Exam I Review: Units 1-8...

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