Alex. Gerbessiotis  03/07/2025
(c) All rights reserved.  MARCH   2025. 
----------------------------------------
Homework 2 solution key
-------------------------
1. (a)  K is larger. See also a Problem in Subject 2
1. (b)  10 = 5 (5-1)/2. Done in class (at least thrice).
1. (c)  3n-6 ; 
   For n   we do 0 comp; For n-1, 1 comp;
   For 1   we do 2 comp; For 2  , 3 comp
   For 3   we do 3 comp; ...
   For n-2 we do 3 comp;
   Thus for the first three keys of the input we do 3 comp
   For the remaining (n-3) keys  we do (n-3)3 comp
   Total is (n-3)3 +3 = 3n-9+3=3n-6.
1. (d) T(m)=Theta(mlgm)
1. (e) TRUE
1. (f) FALSE
1. (g) T(n)=Theta(n/lgn * lg(n/lgn)) = Theta (n)
1. (h) FALSE ; InsSort is Theta(n)
1. (i) T(n)=O(n) though technically running time is T(n)=Theta(n),
   this is not one of the three options given.


2.(a) T(n)=O((lgn)**2)
      W= X intersect Y intersect Z is not more than |Z|=lgn.
      Do a binary search of each Z[i] in X and also in Y.
      Total would be |Z| x O(lg|X|) + |Z|xO(lg|Y|)= O(lgn*lgn).
2.(b) T(n,m)=O(n*lgn)
      The Union-Find data structure can be built in O(nlgn+m)=O(nlgn)
      since m=32nlgn. The t queries whether bi and ci are in the
      same connected component require  2 FIND operations each
      for a total of 2 x t x Theta(1)= Theta(lgn). (t=lgn.)
      Running time is dominated by the building of U-F.


3.(a) K=0 L=4
      Build a MIN heap X  in <=2n comparisons.
      Build a MAX heap Y  in <=2n comparisons.
      Perform on X, 5 RemoveMin ops to find 1st, 3rd, 5th, 7th
      (discard 2nd, 4th, 6th keys) smallest keys.
      This takes 7*O(lgn) time i.e. O(lgn).
      Perform on Y, 5 RemoveMax ops to find 1st, 3rd, 5th, 7th
      (discard 2nd, 4th, 6th keys) largest keys.
      This takes 7*O(lgn) time i.e. O(lgn).
      Total is  0 * n *lgn + 4 *n + o(n)

4.(a) T(n,m)=Theta(n*n)
4.(b) T(n,m)=Theta(n)
4.(c) T(n,m)=Theta(n)
4.(d) T(n,m)=Theta(n+m)
4.(e) T(n,m)=Theta(n+m)
--------------------------------------------------------------------
Date created     03/07/2025 @ 17:50
Date last edited 03/07/2025 @ 18:26