Algorithmic Strategies: Book Back Questions and Answers

Computer Science : Algorithmic Strategies

Evaluation

Part – I

Choose the best answer:         (1 Marks)

1   .The word comes from the name of a Persian mathematician Abu Ja’far Mohammed ibn-i Musa al Khowarizmi is called?

(A) Flowchart      

(B) Flow       

(C) Algorithm       

(D) Syntax

2.  From the following sorting algorithms which algorithm needs the minimum number of swaps?

(A) Bubble sort

(B) Quick sort

(C) Merge sort

(D) Selection sort

3. Two main measures for the efficiency of an algorithm are 

(A) Processor and memory      

(B) Complexity and capacity

(C) Time and space 

(D) Data and space

4. The complexity of linear search algorithm is            

(A) O(n)

(B) O(log n)

(C) O(n2)

(D) O(n log n)

5.  From the following sorting algorithms which has the lowest worst case complexity?

(A) Bubble sort 

(B) Quick sort 

(C) Merge sort 

(D) Selection sort

6. Which of the following is not a stable sorting algorithm?

(A) Insertion sort 

(B) Selection sort 

(C) Bubble sort 

(D) Merge sort 

7. Time complexity of bubble sort in best case is

(A) θ (n)      

(B) θ (nlogn)

(C) θ (n2)

(D) θ (n(logn) 2)

8. The Θ notation in asymptotic evaluation represents

(A) Base case

(B) Average case

(C) Worst case

(D) NULL case

9.  If a problem can be broken into subproblems which are reused several times, the problem possesses which property?

(A) Overlapping subproblems

(B) Optimal substructure

(C) Memoization

(D) Greedy

10. In dynamic programming, the technique of storing the previously calculated values is called ?

(A) Saving value property

(B) Storing value property

(C) Memoization

(D) Mapping

Part – II

Answer the following questions (2 Marks)

1. What is an Algorithm?

Ans. An algorithm is a finite set of instructions to accomplish a particular task. It is a step-by-step procedure for solving a given problem.

2. Define Pseudo code.

Ans. (i) Pseudo code is an informal high level description of the operations principle of a computer program or other algorithm.

(ii) It uses the structural conventions of a normal programming language, but is intended for human reading rather than machine reading.

3. Who is an Algorist?

Ans. (i) Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits.

(ii) One who practices algorism is known as an algorist.

4. What is Sorting?

Ans. Sorting is any process of arranging information or data in an ordered sequence either in ascending or descending order.

5. What is searching? Write its types.

Ans. Searching is designed to check for an element or retrieve an element from any data structure where it is store(d)

Types:

(i) Linear Search

(ii) Binary Search.

Part – III

Answer the following questions       (3 Marks)

1. List the characteristics of an algorithm.

Ans.

(i) Input

(ii) Output

(iii) Finiteness

(iv) Definiteness

(v) Effectiveness

(vi) Correctness

(vii) Simplicity

(viii) Unambiguous

(ix) Feasibility

(x) Portable

(xi) Independent

2. Discuss about Algorithmic complexity and its types.

Ans. The complexity of an algorithm f (n) gives the running time and/or the storage space required by the algorithm in terms of n as the size of input data.

(i) Time Complexity: The Time complexity of an algorithm is given by the number of steps taken by the algorithm to complete the process.

(ii) Space Complexity : Space complexity of an algorithm is the amount of memory required to run to its completion.

3. What are the factors that influence time and space complexity.

Ans. (i) Time Factor -Time is measured by counting the number of key operations like comparisons in the sorting algorithm.

(ii) Space Factor - Space is measured by the maximum memory space required by the algorithm.

4. Write a note on Asymptotic notation.

Ans. Asymptotic Notations are languages that uses meaningful statements about time and space complexity. Hie following three asymptotic notations are mostly used to represent time complexity of algorithms:

(i) Big O: Big O is often used to describe the worst-case of an algorithm.

(ii) Big Ω: Big Omega is the reverse Big O, if Bi O is used to describe the upper bound (worst - case) of a asymptotic function, Big Omega is used to describe the lower bound (best-case).

(iii) Big Θ: When an algorithm has a complexity with lower bound = upper bound, say that an algorithm has a complexity O (n log n) and Ω (n log n), its actually has the complexity Θ (n log n), which means the running time of that algorithm always falls in n log n in the best-case and worst-case.

5. What do you understand by Dynamic programming?

Ans. (i) Dynamic programming is an algorithmic design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions.

(ii) Dynamic programming approach is similar to divide and conquer. The given problem is divided into smaller and yet smaller possible sub-problems.

(iii) Dynamic programming is used whenever problems can be divided into similar sub-problems. So that their results can be re-used to complete the process.

(iv) Dynamic programming approaches are used to find the solution in optimized way. For every inner sub problem, dynamic algorithm will try to check the results of the previously solved sub-problems. The solutions of overlapped sub-problems are combined in order to get the better solution.

Part – IV

Answer the following questions       (5 Marks)

1. Explain the characteristics of an algorithm.

Ans.

Input: Zero or more quantities to be supplied.

Output: Al least one quantity is produced.

Finiteness: Algorithms must terminate after Unites number of steps.

Definiteness: all operations should be well defined. For example operations involving division by zero or taking square root for negative number are unacceptable.

Effectiveness: Every instruction must be carried out effectively.

Correctness: The algorithms should be error free.

Simplicity: East to implement.

Unambiguous: Algorithm should be clear and unambiguous. Each of its steps and their inputs/outputs should be clear and must lead to only one meaning.

Feasibility: Should be feasible with the available resources.

Portable: An algorithm should be generic, independent of any programming language or an operating system able to handle all range of inputs.

Independent: An algorithm should have step-by-step directions, which should be independent of any programming code.

2. Discuss about Linear search algorithm.

Ans. (i) Linear search also called sequential search is a sequential method for finding a particular value in a list.

(ii) This method checks the search element with each element in sequence until the desired element is found or the list is exhausted. In this searching algorithm, list need not be ordered.

Pseudo code:

(i) Traverse the array using for loop

(ii) In every iteration, compare the target search key value with the current value of the list.

(iii) If the values match, display the current index and value of the array.If the values do not match, move on to the next array element.

(iii) If no match is found, display the search element not found.

To search the number 25 in the array given below, linear search will go step by step in a sequential order starting from the first element in the given array if the search element is found that index is returned otherwise the search is continued till the last index of the array. In this example number 25 is found at index number 3.

Example 1:

Input: values[] = {5, 34, 65, 12, 77, 35}

target = 77

Output: 4

Example 2:

Input: values[] = {101, 392, 1, 54, 32, 22, 90, 93}

target = 200

Output: -1 (not found)

3. What is Binary search? Discuss with example.

Ans. Binary search: Binary search also called half¬interval search algorithm. It finds the position of a search element within a sorted array. The binary search algorithm can be done as divide- and-conquer search algorithm and executes in logarithmic time.

Pseudo code for Binary search :

Start with the middle element:

(i) If the search element is equal to the middle element of the array i.e., the middle value = number of elements in array/2, then return the index of the middle element.

(ii) If not, then compare the middle element with the search value,

(iii) If the search element is greater than the number in the middle index, then select the elements to the right side of the middle index, and go to Step-1.

(iv) If the search element is less than the number in the middle index, then select the elements to the left side of the middle index, and start with Step-1.

(v) When a match is found, display success message with the index of the element matched.

(vi) If no match is found Tor all comparisons, then display unsuccessful message.

Binary Search Working principles:

(i) List of elements in an array must be sorted first for Binary search. The following example describes the step by step operation of binary search.

(ii) Consider the following array of elements, the array is being sorted so it enables to do the binary search algorithm. Let us assume that the search element is 60 and we need to search the location or index of search element 60 using binary search.

(iii) First, we find index of middle element of the array by using this formula :

mid = low + (high - low) / 2

(iv) Here it is, 0 + (9 - 0 ) / 2 = 4 (fractional part ignored). So, 4 is the mid value of the array.

(v) Now compare the search element with the value stored at mid value location 4. The value stored at location or index 4 is 50, which is not match with search element. As the search value 60 is greater than 50.

(vi) Now we change our low to mid + 1 and find the new mid value again using the formula.

low to mid + 1

mid = low + (high - low) / 2

(vii) Our new mid is 7 now. We compare the value stored at location 7 with our target value 31.

(viii) The value stored at location or index 7 is not a match with search element, rather it is more than what we are looking for. So, the search element must be in the lower part from the current mid value location

(ix) The search element still not found. Hence, we calculated the mid again by using the formula.

high = mid -1

mid = low + (high - low)/2

Now the mid value is 5.

(x) Now we compare the value stored at location 5 with our search element. We found that it is a match.

(xi) We can conclude that the search element 60 is found at location or index 5. For example if we take the search element as 95, For this value this binary search algorithm return unsuccessful result.

4. Explain the Bubble sort algorithm with example.

Ans. Bubble sort algorithm:

(i)  Bubble sort is a simple sorting algorithm. The algorithm starts at the beginning of the list of values stored in an array. It compares each pair of adjacent elements and swaps them if they are in the unsorted order.

(ii) This comparison and passed to be continued until no swaps are needed, which indicates that the list of values stored in an array is sorted. The algorithm is a comparison sort, is named for the way smaller elements "bubble" to the top of the list.

(iii) Although the algorithm is simple, it is too slow and less efficient when compared to insertion sort and other sorting methods.

(iv)  Assume list is an array of n elements. The swap function swaps the values of the given array elements.

Pseudo code:

(i) Start with the first element i.e., index = 0, compare the current element with the next element of the array.

(ii) If the current element is greater than the next element of the array, swap them.

(iii) If the current element is less than the next or right side of the element, move to the next element. Go to Step 1 and repeat until end of the index is reached.

(iv) Let's consider an array with values {15, 11, 16, 12, 14, 13} Below, we have a pictorial representation of how bubble sort will sort the given array.

 (v) The above pictorial example is for iteration-1. Similarly, remairfing iteration can be done. The final iteration will give the sorted array. At the end of all the iterations we will get the sorted values in an array as given below :

5. Explain the concept of Dynamic programming with suitable example.

(i) Dynamic programming is an algorithmic design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions.

(ii) Dynamic programming approach is similar to divide and conquer. The given problem is divided into smaller and yet smaller possible sub-problems.

(iii) Dynamic programming is used whenever problems can be divided into similar sub¬problems. so that their results can be re¬used to complete the process.

(iv) Dynamic programming approaches are used to find the solution in optimized way. For every inner sub problem, dynamic algorithm will try to check the results of the previously solved sub-problems.

(v) The solutions of overlapped sub-problems are combined in order to get the better solution.

Steps to do Dynamic programming :

(i) The given problem will be divided into smaller overlapping sub-problems.

(ii) An optimum solution for the given problem can be achieved by using result of smaller sub-problem.

(iii) Dynamic algorithms uses Memoization.

Fibonacci Series - An example:

(i) Fibonacci series generates the subsequent number by adding two previous numbers. Fibonacci series starts from two numbers - Fib 0 & Fib 1. The initial values of Fib 0 & Fib 1 can be taken as 0 and 1.

(ii) Fibonacci series satisfies the following conditions :

Fibn = Fib_n-l . + Fib_n-2

(iii) Hence, a Fibonacci series for the n value 8 can look like this

Fib₈ = 0 1 1 23 58 13

Fibonacci Iterative Algorithm with Dynamic programming approach : The following exam¬ple shows a simple Dynamic programming approach for the generation of Fibonacci series.

Initialize f0=0, fl =1

step-1: Print the initial values of Fibonacci f0 and fl

step-2: Calculate fibanocci fib ← f0 + fl

step-3: Assign f0 ← fl, fl ← fib

step-4: Print the next consecutive value of fibanocci fib

step-5: Goto step-2 and repeat until the specified number of terms generated

For example if we generate fibobnacci series upto 10 digits, the algorithm will generate the series as shown below:

The Fibonacci series is : 0 1 1 2 3 5 8 13 21 34 55.