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CS502 Assignment No. 02 Solution & Discussion Due Date: Dec 07, 2015

CS502 - Fundamentals of Algorithms Assignment No. 02 Solution Fall 2015 Due Date Dec 07, 2015

Assignment No. 02
Semester: Fall 2015

CS502: Fundamentals of Algorithms

Due Date:07/12/2015

Instructions

Please read the following instructions carefully before submitting assignment:

 

It should be clear that your assignment will not get any credit (zero marks) if:

  • The assignment is submitted after due date.
  • The submitted assignment is other than .doc file.
  • The submitted assignment does NOT open or file is corrupted.
  • The assignment is copied (from other student or ditto copy from any other source).

 

Objective

 

The objective of this assignment is to enable students:

 

  • Write and solve recurrence relations of recursive algorithms using iteration method
  • Design algorithm using Divide and conquer approach

 

Submission

 

You are required to submit your solution through LMS as MS Word document.

 

For any query about the assignment, contact at CS502@vu.edu.pk

                                                             GOOD LUCK

 

Question 1:

 

Consider the following recursive algorithm for computing the sum of the first n squares:

Sum(n) = 12 + 22 + . . . + n2.

 

Algorithm: SUM(n)

if n = 1 return 1

else return SUM(n − 1) + n ∗ n

 

Write recurrence relation for above algorithm and solve it using Iteration Method.

 

Question 2:

 

In Divide and conquer strategy, three main steps are performed:

 

  1. 1.      Divide: Divides the problem into a small number of pieces
  2. 2.      Conquer: Solves each piece by applying divide and conquer to it recursively
  3. 3.      Combine: Combines/merges the pieces together into a global solution.

 

Write an algorithm to find minimum number from a given array of size ‘n’ using divide and conquer approach.

 

 

Lectures Covered:  This assignment covers first 15 Lectures.

Deadline:             Your assignment must be uploaded/submitted at or before 07 Dec, 2015. 

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Replies to This Discussion

mila sol ??? sahi wala ... btaa do koi 

kon sa solution correct ha...............

Which solution is right one ?

idea 

First of all, if n == 1 you should probably return 1. And yes, this recursive function computes 1 + 2^3 + 3^3 + ... + n^3. How do we know that?

Well, take an example like n = 5;

  • R(5) returns R(4) + 5^3
  • R(4) returns R(3) + 4^3
  • ....
  • R(2) returns R(1) + 2^3
  • R(1) returns 1

If you add them up => R(5) returns 5^3 + 4^3 + .. + 2^3 + 1.

 

EXPLANATION:

Denoting S(n) the sum of the first n cubes, S(n) must be a polynomial of the fourth degree in n, let

S(n) = an^4+bn³+cn²+dn.

This is because

1) S(0)= 0, so there is no independent term,

2) When computing S(n)-S(n-1), which must equal n³, you get a polynomial of the third degree, by cancellation of the quartic term:

S(n)-S(n-1) = a(n^4-(n-1)^4)+b(n³-(n-1)³)+c(n²-(n-1)²)+d(n-(n-1)).

Developing and simplifying,

a(4n³-6n²+4n-1)+b(3n²-3n+1)+c(2n-1)+d = n³.

Let us identify the coefficients:

n³:  4a        =1 
n²: -6a+3b     =0 
n:   4a-3b+2c  =0 
1:   -a +b -c+d=0
Solving this triangular system is straigthforward:

a=1/4 
b=1/2 
c=1/4 
d=0.

and finally

S(n) = (n^4+2n³+n²)/4 = n²(n+1)²/4.

 Check iteration method of power 2 k=log n

T(n) = 2kT (n/(2k)) + (n+n+n+....+n)

= n + n log n

see on page 31


Lecture number 8 the solution s given 

EXAMPLE FROM BOOK:

plz guide us ya working to nahe karne 

T(n) = 2kT(n/(2k)) + (n + n + n + · · · + n) | {z }
k times
= 2kT(n/(2k)) + kn
= 2(logn)T(n/(2(logn))) + (log n)n
= 2(logn)T(n/n) + (log n)n
= nT(1) + n log n = n + n log n

Q 2 ka solution???

rreplz

koi ALLAH ka banda correct solution bta day pleaseeee .........

barii bari mehrbani hugi or bari duain milay gi ....

ALLAH usay JANNAT UL FIRDOUS mn ala MUQAM ata karay ....... 

Kuch khuda ka khof kro yaar aik to assignment ka kehtay ho..ooper say jannat main bheji ja rhay ho...agr sub ooper chaly gay to assignment kon bnay ga.

Hahaha Honey Bunney hosla rakho tens mat lo mil jy ga solution abi to daidh ganta pra hy last 10 minut tk dhundo

fir wohi bat a gai na is baar b me ne pichli baar b kaha tha yar k double minded na kiya karo. yaar plz aik sulotion py discation kiya karen sab apna apna solotion dety hean or baki preshan ho jaty hean. ab koi b next agy ho k ye ni keh raha k ye thek he or ye galat he 

Assignment m kerna kya hey ?? :P 

achar bana kar Market main Supply kar dena hy....

assignment solve kar k VULMS par post kia jata hy aur Dua ki jati hy k ya Allah assignment copy ki huei hy, tu kam sy kam aadhy marks to mil hi jaen.....

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