Alex. Gerbessiotis  01/24/2025
(c) All rights reserved.  JANUARY 2025. 
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Homework 1 solution key
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Q1. a: 2  b: 1   c: 4  d: 3   
Q2. YES
input is x
t1 =           x**2   1st
t2 = t1 * x =  x**3   2nd
t3 = t1 * t2=  x**5   3rd
t4 = t3 * t1=  x**7   4th 
t5 = t4 * t4=  x**14  5th
t6 = t5 * t5=  x**28  6th
t7 = t6 * t6=  x**56  7th
t8 = t7 * t4=  x**63  8th  
Q3. (a) T(n)=Theta(n) , (b) T(n) = O(lgn),  (c)  T(n) = Theta(n) .
Q4. 63n
   [Total 7n+11n+17n+28n=63n]
   3n,4n,10n, 5n,6n, 
     7n        11n
        17n
            28n
Q5.  little omega , littleomega,  little o, little o, theta, little o
Q6.  a. T(n)=Theta(n)  b. S(n)=Theta(1) (See Problem 20 of Subject1)
Q7.  (a) T(n)=Theta(n*lgn)
 Dup(X,n)  [See Problem 13 of Subject 1(Appendix)]
1. Y=MergeSort(X,n);
2. for(i=0;i<n-1;i++)
3.   if Y[i] == Y[i+1]
4.       return(YES)
5. return(NO)

Q8.  A=1 B=2 C=0 D=0
 In more detail:
    C(n) = 2 C(n/2) + (n/2) =>  C(n) =nlgn/2
    
     In the last iteration we have
     <n/2+1, ... , n >  < 1,  ...  , n/2>
     To merge them we need only n/2 comparisons
     A recursion tree per class derives the result.
     (we just replace the node values eg n-1 by n/2.)
Q9. 26,
Justification: Use binary search 
with 2*lg(8192) = 2*13=26 comparisons (at most)
Q10. a.            T(n) = n lgn +16n . A=1,B=16, C=0
     b.            T(n) = 2n-4         D=E=0, F=2, G=-4
     c.            T(n) = 2 n lgn-2n   K=2, L=-2,M=0
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Date created     01/23/2025 @ 10:01
Date last edited 01/24/2025 @ 11:25