# 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

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Q1) xxxxx xx xxxxxxxx the xxxxx test, xxxx xxxx of the following xx xxxxx

xx xxx xxxxxxxxx xxxxx n(n-1)/2 xxx xxxxx n^2 grow asymptotically xx xxxxx rate.

Limit xxxxxxxxx x Limit xxxxxxxxxxxxx

n x> xxx x x> ∞

x = Limit (n-1)/2n

x x> xxx

x Limit xxx - xxxx

n -> xxx

= xxx - xxx x Limit 1/n

n x> ∞

= xxx x 1/2 * 0

x 1/2 x xxxxxxxx

As Limit f(n)/g(n) = xxxxxxxxx xxxx and xxxx xxxx grow asymptotically at xxxxx xxxx

-----------------------------------------------------------------

xx The xxxxxxxxx f(n)=log x xxxx xxxxxxxxxxxxxx at xxxxxx rate xxxx xxxxxxx

xxxxx f(n)/g(n) x xxxxx xxxxxxxx = xxxxx xxxxxxx xx I’Hopital’s xxxx

n x> xxx n -> ∞ x -> ∞

x x

xx xxx xxxxxxxxx = 0, f(n) xxxx asymptotically at xxxxxx rate than g(n)

x -> xxx

-------------------------------------------------------------------

xxx Show that log xxxx = Θ (nlog xxx

Log(n!) = xxx 1 + xxx x x Log 3 + xxxxxxxx Log x

Taking upper xxxxx

<=Log n x Log x x ………………..+ xxx x

= x Log n

= xxx xxx n)

xxxxxxx = xxx x + xxx x x xxx x + xxxxxxxx Log x

Taking xxxxx xxxxx

x Log 1 x xxxxxx Log n/2 x xxxxxxxxx xxx n

>= xxx xxx + …….+ Log x (taking only xxx xxxx

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