algorithm
Q1)
Show, by applying the limit test, that each of the following is true.
a) The functions f(n)= n(n-1)/2 and g(n)= n^2 grow asymptotically at equal rate.
b) The functions f(n)=log n grow asymptotically at slower rate than g(n)=n.
Q2)
Show that log (n!) = Θ (nlog n);
Q3)
Design an algorithm that uses comparisons to select the largest and the
second largest of n elements. Find the time complexity of your algroithm
(expressed using the big-O notation).
Q4)
Given an a binary array or list of n elements, where each element is either
a 0 or 1, we would like to arrange the elements so that all of those that
are equal to 0's appear first followed by all the elements that are equal
to 1's.
a) Write an algroithm or a function that uses comparisons to arrange the
elements as given above. Do not use any extra arrays in your algorithm.
b) Find the time, T(n), needed by your algorithm in the worst-case and
then express it using the big-O notation.
c) Find the time, T(n), needed by your algorithm in the best-case and
then express it using the big-Ω notation.
d) Find the time, T(n), needed by your algorithm in the average-case
and express it using the big-Θ notation.
Answered all questions
body preview (529 words)
xxx xxxxx by applying the limit xxxxx that xxxx xx the xxxxxxxxx xx xxxxx
xx xxx xxxxxxxxx xxxxx n(n-1)/2 xxx g(n)= n^2 xxxx xxxxxxxxxxxxxx at equal rate.
xxxxx xxxxxxxxx = Limit n(n-1)/2)/n^2
x - ∞ n xx ∞
x = xxxxx xxxxxxxx
n - xxx
= Limit 1/2 x 1/2n
x x xx xxx
= 1/2 x 1/2 * xxxxx xxx
n xx ∞
= xxx x xxx * 0
= 1/2 x xxxxxxxx
xx Limit xxxxxxxxx = xxxxxxxxx F(n) xxx g(n) xxxx xxxx xxxxxxxxxxxxxx xx xxxxx xxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
b) xxx functions f(n)=log n grow xxxxxxxxxxxxxx xx slower rate than xxxxxxx
xxxxx f(n)/g(n) x Limit xxxxxxxx x xxxxx xxxxxxx By I’Hopital’s xxxx
x xx xxx x xx ∞ x xx ∞
= x
xx Lim f(n)/g(n) = 0, xxxx xxxx asymptotically at slower rate than g(n)
n xx ∞
-------------------------------------------------------------------
Q2) Show that log (n!) x Θ (nlog n);
xxxxxxx x Log x + Log 2 x Log 3 + xxxxxxxx Log x
Taking xxxxx xxxxx
<=Log n + Log n + ………………..+ Log n
= x xxx x
x O(n xxx xx
xxxxxxx = xxx x + xxx 2 x Log x + …….+ Log x
Taking lower bound
= xxx x + …..+ xxx xxx x ……..+ log x
= Log xxx + …….+ Log x (taking xxxx xxx half xx xxxxxxx
x Log xxx x xxxxxxxxx Log n/2
x xxx xxx n/2
x Ω xx xxx xx
As Log(n!) upper bound xxx lower bound are xxxx xx Log(n!)= xxxx log n )
-----------------------------------------------------------
Q3)Design an algorithm xxxx uses comparisons xx select xxx largest and the second xxxxxxx xx n
- - - more text follows - - -
Buy this answer  | Try it before you buy it |