CS502- Fundamental of Algorithm Solved Subjective

QNo.1  What is heap and what is heap order? (Mark2)

Answer:-

The heap is the section of computer memory where all the variables created or initialized at runtime are stored.  The heap order property: in a

(min) heap, for every node X, the key in the parent is smaller than or equal to the key in X.

Ref:  Handouts Page no. 40

QNo.2  Quick sort such that sort the array in to non-increasing order? (Mark2).

Quick sorting, an array A[1..n] of n  numbers We are to reorder these elements into increasing (or decreasing) order.  More generally, A is an array of objects and we sort them based on one of the attributes - the key value. The key value need not be a number. It can be any object from a totally ordered domain. Totally ordered domain means that for any two elements of the domain, x and y, either x < y, x = y or x > y.

Ref:  Handouts Page no. 40 

QNo.3  Draw the cost table for chain multiplication problem with initial states(Mark3).

 (A1)(A2A3A4 . . .An)

or (A1A2)(A3A4 . . .An)

or (A1A2A3)(A4 . . .An)

or (A1A2A3A4 . . .An−1)(An)

Ref:  Handouts Page no. 90

QNo.4  we can avoid unnecessary repetitions for recursive calls? (Mark3)

We can avoid these unnecessary repetitions by writing down the results of recursive calls and looking them up again if we need them later. This process is called memorization.

Ref:  Handouts Page no. 49

Vukwl- Virtual Education Solution

Worst case for edit distance algorithm? What is the simple change that can change the worst case time ? 5 marks

Analysis of DP edit distance

There are  entries in the matrix. Each entry E(i,j) takes  time to compute. The total running is  Recursion clearly leads to the same repetitive call pattern that we saw in Fibonnaci sequence. To avoid this, we will  use the DP approach. We will build the solutionb bottom-up. We will use the base case E(0,j) to fill first row and the base case E(I,0) to fill first column. We will fill the remaining E matrix row by row.

Ref:  Handouts Page no. 14

 Describe an efficient algorithm to find the median of a set of 106 integers; it is known that there are fewer than 100 distinct integers in the set

Step1:Start

Step2:Find the 100 distinct numbers among 10^6 integers.

Step3:Sort the 100 distinct numbers

Step4:Count the distinct numbers

Step5: if count is odd,middle number is the median

Step6:if count is even,add the middle two numbers then divide by 2,the result is the median Step5:End

number.

Ref:  Handouts Page no. 34

Vukwl- Virtual Education Solution

What is the formula for generating Catalan numbers?

Equation (22) is a recurrence relation.

C_(n+1) = C_n * [2(2n+1)] / (n+2)

we have the values of n in one column and the values of C_n in another, then to put this formula in Excel, on the (n+1)-th row just replace C_n and n with the appropriate cells from the previous row.

Ref:  Handouts Page no. 85

What are Catalan numbers? Give the formula.

Catalan numbers form a sequence of natural numbers that occur in variouscounting problems, often involving recursively defined objects

Formula is C(n) = 2n Cn / (n+1).

Ref:  Handouts Page no. 85
 

 Q-Write a pseudo code Fibonacci With memorization? -- (3)

Sol
MEMOFIB(n)
1 if (n < 2)
2 then return n
3 if (F[n] is undefined)
4 then F[n] MEMOFIB(n − 1) + MEMOFIB(n − 2)
5 return F[n]

Ref:  Handouts Page no. 12, 74

Q – Write Down the steps of Dynamic programming (5)

Dynamic programming is essentially recursion without repetition. Developing a dynamic programming
algorithm generally involves two separate steps:

• Formulate problem recursively. Write down a formula for the whole problem as a simple combination of answers to smaller sub problems.
• Build solution to recurrence from bottom up. Write an algorithm that starts with base cases and works its way up to the final solution.

Dynamic programming algorithms need to store the results of intermediate sub problems. This is often but not always done with some kind of table. We will now cover a number of examples of problems in which the solution is based on dynamic programming strategy.

Ref:  Handouts Page no. 74 – 77

How we build heap

We build a max heap out of the given array of numbers A[1..n]. We repeatedly extract the maximum item from the heap. Once the max item is removed, we are left with a hole at the root.

Ref:  Handouts Page no. 41

Write Pseudo code for KNAPSACK algorithm? 5 marks

Solution:

KNAPSACK (n, W)

1 for w=0,W

2 do V[0,w]ß0

3 for i=0,n

4 do V[i,0]ß0

5   for w=0,W

6   do if(wi w&  +V[i-1,w- ]>V[i-1,w])

7                      then V[I,w]ß +V[i-1,w]

8                      else V[i,w]ßV[i-1,w]

The time complexity is clearly O(n,W), it must be cautioned that as n and W get large, both time and space complexity become significant.

Ref:  Handouts Page no. 96

Spelling correction in edit distance? 3 marks

A better way to display this editing process is to place the words above the other:

S  D  I  M  D  M

 M A  -   T  H  S

  A  -   R  T  -  S

Vukwl- Virtual Education Solution

THE FIRST WORD HAS AGAP FOR EVERY INSERTION (1)AND THE SECOND WORD HAS A GAP FOR EVERY DELETION (D). MATHES (M) DO NOT COUNT. THE EDIT TRANSCRIPT IS DEFINED AS A STRING OVER THE ALPHABETM,S,I,d THAT DESCRIBES A TRANSFORMATION OF ONE STRING INTO OTHER. FOR EXAMPLE

S D I M D M

1+1+1+0+1+0+=4
Ref:  Handouts Page no. 77

Differentiate b/w Bubble sort, insertion sort and selection sort? 3 marks

Bubble sort: scan the array. Whenever two consecutive items are found that are out of order, swap them. Repeat until all consecutive items are in order.

Insertion sort: assume that A[1…i-1] have already been sorted. Insert A[i] into its proper position in this sub array. Create this position by shifting all larger elements to the right.

Selection sort:

Assume that A[1..i-1] contain the i-1 smallest elements in sorted order. Find the smallest in A[i….n] swap it with A[i].

Ref:  Handouts Page no. 54

Write down the steps of dynamic programming strategy.   (2 marks)

Developing a dynamic programming algorithm generally involves two separate steps:

1_formulate problem recursively.

Write down a formula for the whole problem as a simple combination of answers to smaller sub problems.

2_ Build solution to recurrence from bottom up:

Ref:  Handouts Page no. 75

Vukwl- Virtual Education Solution

Solve the recursion problem. (5marks.)

Recursion clearly leads to the same repetitive call pattern that we saw in Fibonnaci sequence. To avoid this, we will  use the DP approach. We will build the solutionb bottom-up. We will use the base case E(0,j) to fill first row and the base case E(I,0) to fill first column. We will fill the remaining E matrix row by row.

If we trace through the recursive calls to MemoFib, we find that array F[] gets filled from bottom up…i.e., first  F[2], then F[3], and so on, upto F[n]. we can replace recursion with a simple for-loop that just fills up the array F[] in that order.

                                                                      
•       We are given an array of n elements of x1 , x2 ,x3 , ,,,,xn,
suggest best sorting algorithm of order On.  (5 marks).

The main shortcoming of counting sort is that it is useful for small integers, i.e., 1…k where k is small. If this were a million more, the size of the rank array would also be a million. Radix sort provides a nice work around this limitation by sorting numbers one digit at a time