QUESTION BANK FOR ANNA UNNIVERISTY SYLLABUS

Subject Code / Name: CS 2251 Design & Analysis of Algorithm

DAA 1 AMBIKA J KINGS COLLEGE ,THANJAVUR

UNIT-I PART-A

1. Define Big ‘Oh’ notation. (AU – 2011, 2012,2013)

The Big ‘Oh’ notation provides an upper bound for the function t. A function t(n) is said to be

in O(g(n)), if there exist some positive constant C and some non negative number No, such that ,

t(n)<=Cg(n), for all n>=no.

2. What is meant by linear search? (AU - 2012) ( AU - 2011)

Linear search, also known as sequential search, is a process that checks every element

in the list sequentially until the desired element is found. The computational complexity for

linear search is O(n), making it generally much less efficient than binary search (O(log n)). But

when list items can be arranged in order from greatest to least and the probabilities appear as

geometric distribution (f (x)=(1-p) x-1p, x=1,2), then linear search can have the potential to be

notably faster than binary search.

3. What is the properties of big-Oh notation. ( AU - 2011)

The mathematical properties of big oh. In particular, suppose we know that

and .

What can we say about the asymptotic behavior of the sum of and .

What can we say about the asymptotic behavior of the product of and .

How are and related when .

4. Using the step count method analyze the time complexity when 2 m × n matrices are added. (AU - 2011)

The best-case step count is the minimum number of steps that can be executed for the

given parameters.

The worst-case step count is the maximum number of steps that can be executed for the given

parameters.

The average step count is the average number of steps executed instances with the given

parameters.

5. What do you mean by an Algorithm? ( AU - 2013)



Ans: An algorithm is a finite set of instructions that, if followed, accomplishes a particular

task. In addition, all algorithms must satisfy the following criteria:

1) input

2) Output

3) Definiteness

4) Finiteness

5) Effectiveness.

6. What is the explicit formula for the nth Fibonacci number? (AU - 2007) The formula for the nth Fibonacci number is given by T(n)=1/√5фn-1/√5фn

7. Define Algorithm Validation.(AU - 2012)

It is the process of computing the correct answer for all possible legal inputs after the

algorithm is created or devised. The purpose of the validation is to assure that the algorithm

will work correctly independently of the programming languages. Once the validation is done

the program can be written and the second phase of the validation, which is referred to as

program proving or program verification, begins.

8. What is the Average case complexity of Linear search Algorithm? (AU - 2008)

Mathematically, the average case value has stated as A(n) = ∑P(I)T(I) i €Dn

9. Differentiate Time Complexity from Space complexity. (AU - 2010)

10. What is a Recurrence Equation? (AU - 2010)

It can be classified into homogeneous and inhomogeneous in the algebra of mathematics

there are easy ways to solve both homogeneous and inhomogeneous.

S.No Time Complexity

Space Complexity

1. Time complexity refers to the amount of time spent by the processor for the completion of the task.

space complexity refers to the amount of memory occupied by a specific process or task.

2. The total number of steps involved in a solution to solve a problem is the function of the size of the problem, which is the measure of that problem’s time complexity.

Space complexity is measured by using polynomial amounts of memory, with an infinite amount of time.

3. Quantities to estimate Time complexity estimation is based on execution time

Space complexity estimation is based on memory space



a0xn+a1xn-1+….+akxn-k=0

PART-B

1. Discuss in detail all the asymptotic notations with examples. (8)(AU - 2012)

2. With a suitable example, explain the method of solving recurrence equations. (8)(AU -

2012)

3. Define Asymptotic notations. Distinguish between Asymptotic notation and

conditional asymptotic notation. (10) ( AU - 2011)

4. Explain how the removing condition is done from the conditional Asymptotic notation

with an example. (6) ( AU - 2011)

5. Explain how analysis of linear search is done with a suitable illustration. (10) (AU - 2011)

6. Define recurrence equation and explain how solving recurrence equations are done.

(6) (AU - 2011)

7. Write the recursive and non-recursive versions of the factorial function.(8) (AU – 2011)

8. Examine how much time each function requires as ‘n’ becomes large. (8) (AU - 2011)

9. Explain asymptotic notations in detail. (8) (AU - 2011)

10. Write an algorithm for linear search and analyze the algorithm for its time complexity.(8)

(AU - 2011)

11. Explain briefly the Time Complexity estimation, space complexity estimation and the

tradeoff between Time and Space complexity. (6) (AU - 2010)

12. Write the linear search algorithm and analyze for its best, worst and average case

time complexity. (10) (AU - 2010)

13. Prove that for any two functions

(n f and ) (n g , we have )) ( ( ) ( n g n f θ = if and only if )) ( ( ) ( n g O n f = and )) ( ( ) n g n

f = . (6) (AU - 2010)

14. Explain how Time Complexity is calculated. Give an example. (AU - 2010)

15. Elaborate on Asymptotic Notations with examples. (AU - 2010)

UNIT II PART-A

1. What is the drawback of greedy algorithm? (AU - 2012)

Despite their simplicity, correct greedy algorithms can be subtle.

o It's easy to fool yourself into believing an incorrect greedy algorithm is correct.

o “I can't think of a counter-example, so there are none.”

Using greed is not an automatic.

It may not be the best solution and sometimes may even be the worst possible one.

Greedy algorithms don't work for some problems.

2. What is the time complexity of binary search? (AU - 2012)



In conclusion we are now able completely describe the computing time of binary search by giving

formulas that describe the best, average and worst cases.

Successful searches

Best Case-----O(1)

Average Case----O(logn)

Worst Case-----O(logn)

Unsuccessful searches

O(logn) best, average, worst

3. What greedy algorithms? ( AU - 2011)

A greedy algorithm obtains an optimal solution to a problem by making a sequence of choices.

At each decision point, the algorithm chooses the locally optimal solution. In other words,

when we are considering which choice to make, we make the choice that looks best in the

current situation, without considering results from sub problems.

4. What Knapsack problem? ( AU - 2011)

Given n items of known weights w1,w2...........wn and values v1,v2............vn and a knapsack of

capacity W, find the most valuable subset of the items that fit into the knapsack.(Assuming

all the weights and the knapsack's capacity are positive integers the item values do not

have to be integers.)

5. Write the Control abstraction for Divide-and conquer. (AU - 2013)

A control abstraction we mean a procedure whose flow of control is clear but whose primary

operations are by other procedures whose precise meanings are left undefined.

6. What do you mean by divide and conquer strategy?(AU - 2013)

Given a function to compute on ‘n’ inputs the divide-and-comquer strategy suggests

splitting the inputs in to’k’ distinct susbsets, 1k n, yielding ‘k’ subproblems. The

subproblems must be solved, and then a method must be found to combine subsolutions

into a solution of the whole. If the sub problems are still relatively large, then the divide-

and conquer strategy can possibly be reapplied.

7. State the general principle of greedy algorithm. (AU - 2010)

A greedy algorithm obtains an optimal solution to a problem by making a sequence

of choices. At each decision point, the algorithm chooses the locally optimal solution. In

other words, when we are considering which choice to make, we make the choice that

looks best in the current situation, without considering results from sub problems.

8. What is called Substitution Method? (AU - 2010)

The substitution method for solving recurrences consists of two steps:

1. Guess the form of the solution.

2. Use mathematical induction to find constants in the form and show that the solution works.



The inductive hypothesis is applied to smaller values, similar like recursive calls bring us closer to

the base case. The substitution method is powerful to establish lower or upper bounds on a

recurrence.

9. List out the drawbacks of the binary search algorithm.(AU - 2007)

Transfer of records from the physical table is very high during insertion and deletion

of the record.

Binary search is more complex compare to the linear search.

10. What is an Optimal Solution? (AU - 2010)

A feasible solution either maximizes or minimizes the given objective function is

called as optimal solution.

UNIT II

PART B

1. Explain divide – and – conquer method with merge sort algorithm. Give an example.

(16) (AU - 2012)

2. Write a recursive algorithm to determine the max and min from a given element and

explain. Derive the time complexity of this algorithm and compare it with a simple

brute force algorithm for finding max and min. (10) (AU - 2010)

3. Write a recursive algorithm to determine the max and min from a given element and

explain. Derive the time complexity of this algorithm and compare it with a simple

brute force algorithm for finding max and min. (10) (AU - 2010)

4. Explain how greedy method can be applied to solve the Knapsack problem.

5. What is divide and conquer strategy and explain the binary search with suitable

example problem. ( AU - 2011)

6. Distinguish between Quick sort and Merge sort, and arrange the following numbers in

increasing order using merge sort. (18, 29, 68, 32, 43,37, 87, 24, 47, 50)

(AU - 2011)

7. Using the divide and conquer approach to find the maximum and minimum in a set of

‘n’ elements. Also find the recurrence relation for the number of elements

compared and solve the same. (16)(AU - 2011)

8. For the following list of elements trace the recursive algorithm for finding max and

min and determine how many comparisons have been made.22, 13, –5, –8, 15, 60,

17, 31, 47. (6) (AU - 2010)



9. Write the container loading greedy algorithm and explain. Prove that this algorithm is

optimal. (8) (AU - 2010)

10. Suppose you have 6 containers whose weights are 50, 10, 30, 20, 60, 5 and a ship

whose capacity is 100. Use the above algorithm to find an optimal solution to this

instance of the container loading problem. (8) (AU - 2010)

11. With a suitable algorithm, explain the problem of finding the maximum and minimum

items in a set of n elements. (16)(AU - 2010)

12. Explain Merge Sort Problem using divide and conquer technique. Give an example.(16)

(AU - 2010)

UNIT III PART-A

1. State principle of optimality. (AU - 2012) , (AU - 2010)

"An optimal policy has the property that whatever the initial state and initial

decision are the remaining decision must constitute an optimal policy with regard to the

state resulting from the rest decision." If an optimal state P results in a state Q , this initial

state to this final state ,the portion of the original must be optimum. That is every part of

optimal policy is optimal.

2. Differentiate greedy method and dynamic programming. (AU - 2012) (AU - 2011)

Greedy method:

Greedy method focuses on expanding partially constructed solutions.

It provides many results such as feasible solution.

More efficient

Dynamic programming:

Focuses on principle of optimality.

It provides specific answers.

Less efficient

3. What is traveling salesperson problem? (AU - 2011)

Let g = (V, E) be a directed. The tour of G is a directed simple cycle that includes

every vertex in V. The cost of a tour is the sum of the cost of the edges on the tour. The

traveling salesperson problem to find a tour of minimum cost.

4. What do you mean by multistage graphs? (AU - 2011), (AU - 2010)

A Multistage graph G= (V,E) is a directed graph in which the vertices are partitioned

into k>=2 disjoint sets Vi, 1<i<k. In addition if <u,v> is an edge in E then u belongs to Vi



and V belongs to Vi+1 for some I, 1<i<K. The srts V1 and Vk are such that |v1|=|vk| =1.

The vertex s is the source and t the sink.

5. How the problems can be solved using Dynamic Programming?( AU - 2005)

Condition: An optimal sequence of decisions can be found by making many

decisions at a time and all the overlapping sub-instances are considered.

6. Write an algorithm to find the shortest path between all pairs of nodes. (AU - 2011) The shortest path problem can be defined for graphs whether undirected, directed,

or mixed. It is defined here for undirected graphs; for directed graphs the definition of

path requires that consecutive vertices be connected by an appropriate directed edge.

1. Assign to every node a tentative distance value:

2. Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisited

nodes called the unvisited set consisting of all the nodes.

3. For the current node, consider all of its unvisited neighbors and calculate

their tentative distances.

4. When we are done considering all of the neighbors of the current node, mark the

current node as visited and remove it from the unvisited set. A visited node will never

be checked again.

5. If the destination node has been marked visited or if the smallest tentative distance

among the nodes in the unvisited set is infinity, then stop. The algorithm has finished.

6. Select the unvisited node that is marked with the smallest tentative distance, and set

it as the new "current node" then go back to step 3.

7. Define Multigraph. (AU - 2013)

A multigraph G = (V,E,f ) consists of a non-empty

set V of vertices (or nodes), a set E (possibly empty) of

edges and a function f with domain E and codomain the

set of pairs in V.

Undirected multigraph (edges without own identity)

Directed multigraph (edges without own identity)

Directed multigraph (edges with own identity)

8. Compare divide and conquer with dynamic programming. (AU - 2010)

Divide And Conquer:

Given a function to compute on ‘n’ inputs the divide-and-conquer strategy suggests

splitting the inputs in to ’k’ distinct subsets, yielding ‘k’ sub problems. The sub problems

must be solved, and then a method must be found to combine sub solutions into a solution

of the whole. If the sub problems are still relatively large, then the divide-and conquer

strategy can possibly be reapplied.

Dynamic programming:

Focuses on principle of optimality.

It provides specific answers.

Less efficient



9. Define 0/1 knapsack problem.( AU - 2013) 0/1Knapsack Problem

•Suppose you are a thief, and you are nowin a jewelry shop (nobody is around !)

•You have a big knapsack that you have“borrowed”from some shop before

•Weight limit of knapsack:W

•There arenitems,I1,I2,...,In Ik has value vk, weight wks

Target: Get items with total value as large as possible without exceeding weight limit

10. Define Optimal Binary Search Tree. (AU - 2010)

If ‘q’ is always chosen such that ‘aq’ is the middle element (that is,

q=[(n+1)/2), then the resulting search algorithm is known as binary search. In conclusion we

are now able completely describe the computing time of binary search by giving formulas that

describe the best, average and worst cases.

UNIT III

PART-B 1. With a suitable example, explain all – pair shortest paths problem. (AU - 2012)

2. How is dynamic programming applied to solve the traveling salesperson problem?

Explain in detail with an example. (AU - 2012)

3. Explain the multistage graph problem with an example. (8) (AU - 2011)

4. Find an optimal solution to the knapsack instance n = 7, m= 15 (p1, p2, p3, ….p7) =

(10, 5, 15, 7, 6, 18, 3) and (w1, w2, w3, ... w7) (2, 3, 5, 7, 1, 4, 1) (8) (AU - 2011)

5. Describe binary search tree with three traversal patterns? Give suitable example

with neat diagram for all three traversal of binary search tree.(16) (AU - 2011)

6. Use function OBST to compute w(i, j), r(i, j), and c(i, j), 0<=i<j<=4, for the identifier

set ( ) 4 3 2 1 , , , a a a a = (cout, float, if, while) with p(1) = 1/20, p(2) = 1/5, p(3) =

1/10, p(4) = 1/20, q(0) = 1/5, q(1) = 1/10, q(2) = 1/5, q(3) = 1/20, and q(4) 1/20.

Using the r(i, j)’s, construct the optimal binary search tree.(16) (AU - 2011)

7. Using dynamic approach programming, solve the following graph using the

backward approach. (16)( AU - 2011)

8. Write and explain the algorithm to compute the all pairs source shortest path using

dynamic programming and prove that it is optimal (8) (AU - 2010)

9. For the following graph having four nodes represented by the matrix given below

determine the all pairs source shortest path (8) (AU - 2010)

10. Write the algorithm to compute the 0/1 knapsack problem using dynamic

programming and explain it. (8) (AU - 2010)

11. Solve the following instance of the 0/1, knapsack problem given the knapsack

capacity is W = 5 (8) (AU - 2010)



Items Weight Value

1 2 12

2 1 10

3 3 20

4 2 15

132 132 132 53099 3

12. Write down and explain the algorithm to solve all pairs shortest paths problem.

(16) (AU - 2010)

13. Explain how dynamic programming is applied to solve travelling salesperson

problem.(16)( AU - 2010)

UNIT IV PART-A

1. What is graph cloning? ( AU- 2011)

Cloning means to duplicate a graph along with data tables and results included on

that graph. Enter a new data set, and it will instantly be graphed the same way.

2. Define the chromatic number of a graph. (AU- 2011)

A coloring using at most k colors is called a (proper) k-coloring. The smallest

number of colors needed to color a graph G is called its chromatic number, and is often

denoted χ(G). Sometimes γ(G) is used, since χ(G) is also used to denote the Euler

characteristic of a graph. A graph that can be assigned a (proper) k-coloring is k-colorable,

and it is k-chromatic if its chromatic number is exactly k. A subset of vertices assigned to

the same color is called a color class, every such class forms an independent set.

3. Draw a graph with a cycle but with no Hamiltonian cycle. (AU- 2011)

4. State if backtracking always produces optimal solution.(AU – 2008)



Backtracking always produces optimal solution since backtracking is a systematic

way to go through all the possible configuration of a solution space for the problem

instances.

5. Define sum of subset problem. (AU-2013)

The subset sum problem is a good introduction to the NP-complete class of problems.

There are two reasons for this

It is a decision and not an optimization problem.

It has a very simple formal definition and problem statement.

6. Explain the idea behind backtracking. (AU-2010,2011,2012)

Backtracking is a technique used to solve problems with a large search space, that

Systematically tries and eliminates possibilities. A standard example of backtracking

would be going through a maze. At some point in a maze, you might have two options of

which direction to go:

Junction

------------------------ Portion A

|

Portion B

7. Define and solve the graph coloring problem. (AU - 2010)

The Graph coloring problem asks us to assign the smallest number of colors to

vertices of a graph so that no two adjacent vertices are the same color.

8. Define Hamiltonian Cycle. (AU - 2013)

A Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton

circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once

(except for the vertex that is both the start and end, which is visited twice). A graph that

contains a Hamiltonian cycle is called a Hamiltonian graph.

The problem of finding a Hamiltonian cycle or path in a graph is a special case of the

traveling salesman problem.

9. Differentiate Explicit and Implicit Constraints. (AU – 2010, 2012) 1) Explicit constraints:

Explicit constraints are rules that restrict each Xi to take values only from a given set.

Some examples are,

Xi0 or Si = {all non-negative real nos.}

Xi =0 or 1 or Si={0,1}.

LiXiUi or Si= {a: LiaUi}

All tupules that satisfy the explicit constraint define a possible solution space for I.

2) Implicit constraints:



The implicit constraint determines which of the tuples in the solution space I can

actually satisfy the criterion functions.

10. What is the difference between a Live Node and a Dead Node? (AU - 2010)

A node which has been generated and all of whose children have not yet been

generated is called as a live node.

Dead node is defined as a generated node, which is to be expanded further all of

whose children have been generated.

UNIT IV PART-B

1. Explain the algorithm for finding all m-colorings of a graph. (8) (AU - 2012)

2. Explain elaborately recursive backtracking algorithm? ( AU - 2011) (8)

3. What is Hamiltonian problem? Explain with an example using backtracking. (16)

( AU - 2011)

4. Write down and explain the procedure for tackling the 8 – queens problem using a

backtracking approach.(8) (AU - 2012)

5. How does backtracking work on the 8 Queens problem with suitable example? (8)

( AU - 2011)

6. Using backtracking, find the optimal solution to a knapsack problem for the

knapsack instance n = 8, m = 110, (p1, p2. ... p7) = (11, 21, 31, 33, 43, 53, 55, 65) and

(w1, w2,...,w7) = (1, 11, 21, 33, 43, 53, 55, 65). ( AU - 2011)

7. Write an algorithm for N QUEENS Problem and Trace it for n=6. (AU - 2011)

8. Write an algorithm to determine the Sum of Subsets for a given Sum and a Set of

numbers. Draw the tree representation to solve the subset sum problem given the

numbers set as {3, 5, 6, 7, 2} with the Sum = 15. Derive all the subsets. (8 + 8) (AU -

2010)

9. Write an algorithm to determine Hamiltonian cycle in a given graph using back

tracking. For the following graph determine the Hamiltonian cycle. (8 + 8) (AU -

2010)

10. Describe the backtracking solution to solve 8-Queens problem. (AU - 2010)

11. With an example, explain Graph Coloring Algorithm. (AU-2010)



UNIT V PART-A

1. What is an articulation point in a graph? (AU- 2012)

A vertex of a connected graph G is said to

be in articulation point, if its removal with all

edges incident to it breaks the graph into disjoint

pieces. An Articulation point in a connected

graph is a vertex that, if delete, would break the

graph into two or more pieces (connected

component).

Articulation points: A, H, G, J

2. What is the difference between BFS and DFS methods? (AU- 2012)

S.No DFS BFS

1. A depth first

traversal would visit

the nodes in this

order

A, B, D, C, E, F

A

/ \

B C

/ / \

D E F

A breadth first

traversal would

visit the node in

this order

A, B, C, D, E, F

A

/ \

B C

/ / \

D E F

2. Notice that you go all

the way down one

leg before moving

on.

Here we work all

the

way across each

level before going

down.

3. Depth first use a

stack. (The recursive

implementation uses

the call-stack...)

Breadth-first use a

queue.

3. What is spanning tree? Give an example. (AU- 2010,2011)

A minimum spanning tree (MST) or minimum

weight spanning tree is then a spanning tree with weight



less than or equal to the weight of every other spanning tree. More generally, any

undirected graph (not necessarily connected) has a minimum spanning forest, which is a

union of minimum spanning trees for its connected components.

4. What is NP Completeness? (AU - 2011)

In computational complexity theory, the complexity class NP-

complete(abbreviated NP-C or NPC) is a class of decision problems.

A decision problem is NP-complete if:

1. is in NP, and

2. Every problem in NP is reducible to in polynomial time.

can be shown to be in NP by demonstrating that a candidate solution to can be verified

in polynomial time.

5. Define a strongly connected digraph and give the minimum in degree of all the

nodes in the graph. (AU- 2011)

If G is a digraph, define a relation ~ on the

vertices by: a ~ b is there is both a path from a to b,

and a path from b to a. This is an equivalence relation.

The equivalence classes are called the strong

components of G.

G is strongly connected if it has just one strong

component.

6. Give the time complexity and space complexity of traveling salesperson problem. (AU-

2007) Time complexity is O (n2 2n).

Space complexity is O (n 2n).

7. Define NP Hard and NP Completeness. (AU- 2010)

• NP-complete problems are

defined as the hardest

problems in NP

• Most practical problems turn out to be either P or NP-

complete.

• Study NP-complete problems …

Theorem: If A is NP-Complete and A p B

B is NP-Hard

In addition, if B NP

B is NP-Complete

Proof: Assume that B P

Since A p B A P contradiction!

B is NP-Hard



8. State if backtracking always produces optimal solution (AU- 2008)

NO, backtracking always does not produce optimal solution.

9. What is a Biconnected Graph? (AU - 2010)

A graph with no articulation point called

biconnected. In other words, a graph is biconnected if

and only if any vertex is deleted, the graph remains

connected.

Biconnected components: {A, C, G, D, E, F} {G, J, L, B}、B、H、I、K

10. What is a FIFO branch-and-bound algorithm? (AU - 2010)

Branch and Bound is a state space search method in which

all the children of a node are generated before expanding any of

its children.

Live-node: A node that has not been expanded.

In the case of backtracking the cost criteria assumes

a last-in-first-out (LIFO) function, which can be realized with

a stack memory.

A first-in-first-out cost criteria implies the FIFO branch-and-bound algorithm, and it

can be realized with queue memory. A generalization to arbitrary cost criteria is the basis for

the priority branch-and-bound algorithm, and a priority queue memory can be employed to

realize the function.

UNIT V

PART-B

1. Discuss in detail about the biconnected components of a graph.(16)(AU - 2012)

2. Compare and contrast FIFO and LC branch – and – bound search techniques.(16)

3. Write a complete LC branch-and-bound algorithm for the job sequencing with

deadlines problem. Use the fixed tuple size formulation. (16) (AU-2011)

4. Write a non-deterministic algorithm to find whether a given graph contains a

Hamiltonian cycle. (16)

5. Write the Kruskal’s algorithm apply it to find a minimum spanning tree

for the following graph : (16)(AU - 2011)

6. Write short notes on NP-hard and NP-completeness. (8)(AU - 2011)

7. Explain with an algorithm as to how 0/1 knapsack problem is solved using branch

and bound technique. Apply branch and bound technique to solve the following 0/1

knapsack instance if W = 10. (8 + 8) (AU - 2010)

1

2 3 4 5

6 7 8 9



Items Weight Value

1 4 40

2 7 42

3 5 25

4 3 12

8. Write an algorithm and explain to determine Biconnected Components.

9. Prove the theorem that two Biconnected components can have at most

one vertex as common and this vertex is an articulation point. (10 + 6) (AU - 2010)

10. Explain in detail the Graph Traversals. (8)(AU - 2010)

11. With an example, explain how the branch-and-bound technique is used to solve 0/1

knapsack problem. (8)(AU - 2010)

UNIVERSITY QUESTION PAPERS

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

Fourth Semester

Computer Science and Engineering

CS2251 — DESIGN AND ANALYSIS OF ALGORITHMS

(Regulation 2008)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Differentiate Time Complexity from Space complexity.

2. What is a Recurrence Equation?

3. What is called Substitution Method?

4. What is an Optimal Solution?

5. Define Multistage Graphs.

6. Define Optimal Binary Search Tree.

7. Differentiate Explicit and Implicit Constraints.

8. What is the difference between a Live Node and a Dead Node?

9. What is a Biconnected Graph?

10. What is a FIFO branch-and-bound algorithm?

PART B — (5 × 16 = 80 Marks)

11. (a) Explain how Time Complexity is calculated. Give an example. Or



(b) Elaborate on Asymptotic Notations with examples.

12. (a) With a suitable algorithm, explain the problem of finding the maximum and minimum items

in a set of n elements. Or

(b) Explain Merge Sort Problem using divide and conquer technique. Give an example.

13. (a) Write down and explain the algorithm to solve all pairs shortest paths problem. Or

(b) Explain how dynamic programming is applied to solve travelling salesperson problem.

14. (a) Describe the backtracking solution to solve 8-Queens problem. Or

(b) With an example, explain Graph Coloring Algorithm.

15. (a) Explain in detail the Graph Traversals. Or

(b) With an example, explain how the branch-and-bound technique is used to solve 0/1 knapsack

problem.

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2011

Fourth Semester


CS 2251 — DESIGN AND ANALYSIS OF ALGORITHMS

(Regulation 2008)

Time : Three hours Maximum : 100 marks

Answer ALL questions

PART A — (10 × 2 = 20 marks)

1. Using the step count method analyze the time complexity when 2 m × n matrices are added.

2. An array has exactly n nodes. They are filled from the set {0, 1, 2,...,n-1, n}.

There are no duplicates in the list. Design an O(n) worst case time algorithm to find which one of

the elements from the above set is missing in the array.

3. Show the intermediate steps when the numbers 123, 23, 1, 43, 54, 36, 75, 34 are sorted using

merge sort.

4. Devise an algorithm to make a change for 1655 using the Greedy strategy. The coins available

are {1000, 500, 100, 50, 20, 10, 5}.

5. Write the difference between Greedy Method and Dynamic Programming.

6. Write an algorithm to find the shortest path between all pairs of nodes.

7. Define the chromatic number of a graph.

8. Draw a graph with a cycle but with no Hamiltonian cycle.

9. Define a strongly connected digraph and give the minimum in degree of all the nodes in the



graph.

10. Perform depth first and breadth first search on the following graph and find all the nodes

reachable from ‘a’.

PART B — (5 × 16 = 80 marks)

11. (a) Write the recursive and non-recursive versions of the factorial function. Examine how

much time each function requires as ‘n’ becomes large. Or

(b) (i) Explain asymptotic notations in detail. (8)

(ii) Write an algorithm for linear search and analyze the algorithm for its time complexity. (8)

12. (a) Write an algorithm to perform binary search on a sorted list of elements. Analyze the

algorithm for the best case, average case and worst case. Or

(b) Using the divide and conquer approach to find the maximum and minimum in a set of ‘n’

elements. Also find the recurrence relation for the number of elements compared and solve the

same.

13. (a) Use function OBST to compute w(i, j), r(i, j), and c(i, j), 0<=i<j<=4, for the identifier set ( ) 4

3 2 1 , , , a a a a = (cout, float, if, while) with p(1) = 1/20, p(2) = 1/5, p(3) = 1/10, p(4) = 1/20, q(0)

= 1/5, q(1) = 1/10, q(2) = 1/5, q(3) = 1/20, and q(4) 1/20. Using the r(i, j)’s, construct the

optimal binary search tree. Or

(b) Using dynamic approach programming, solve the following graph using the backward

approach.

14. (a) Using backtracking, find the optimal solution to a knapsack problem for the knapsack

instance n = 8, m = 110, (p1, p2. ... p7) = (11, 21, 31, 33, 43,53, 55, 65) and (w1, w2,...,w7) = (1, 11,

21, 33, 43, 53, 55, 65). Or

(b) Write an algorithm for N QUEENS Problem and Trace it for n=6.

15. (a) Write the Kruskal’s algorithm apply it to find a minimum spanning tree for the following

graph : Or

(b) Write short notes on NP-hard and NP-completeness.

B.E/B.Tech. DEGREE EXAMINATION, MAY/JUNE 2012

Fourth Semester


CS 2251/141401/CS 41/CS 1251/10144 CS 402/0802300013 – DESIGN AND ANALYSIS

OF ALGORITHMS

(Regulation 2008)

Time : Three Hours Maximum: 100 marks



Answer ALL questions.

PART A – ( 10 x 2 = 20 marks )

1. Define Big ‘Oh’ notation.

2. What is meant by linear search?

3. What is the drawback of greedy algorithm?

4. What is the time complexity of binary search?

5. State principle of optimality.

6. Differentiate greedy method and dynamic programming.

7. What is the difference between explicit and implicit constraints?

8. Define the basic principles of back tracking.

9. What is an articulation point in a graph?

10. What is the difference between BFS and DFS methods?

PART B – ( 5 x 16 = 80 marks )

1. (a) Discuss in detail all the asymptotic notations with examples. OR

(b) with a suitable example, explain the method of solving recurrence equations.

2. (a) Explain divide – and – conquer method with merge sort algorithm. Give an example.

OR

(b) Explain how greedy method can be applied to solve the Knapsack problem.

3. (a) With a suitable example, explain all – pair shortest paths problem. OR

(b) How is dynamic programming applied to solve the traveling salesperson problem? Explain

in detail with an example.

4. (a) Explain the algorithm for finding all m-colorings of a graph. OR

(b) Write down and explain the procedure for tackling the 8 – queens problem using a

backtracking approach.

5. (a) Discuss in detail about the biconnected components of a graph. OR

(b) Compare and contrast FIFO and LC branch – and – bound search techniques.

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011.

Fourth Semester



(Regulation 2008)




Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. What do you mean by linear search?

2. What is the properties of big-Oh notation.

3. What greedy algorithms?

4. What Knapsack problem?

5. What is traveling salesperson problem?

6. What do you mean by multistage graphs?

7. State the general backtracking method?

8. What is graph cloning?

9. What is spanning tree? Give an example.

10. What is NP Completeness?

PART B — (5 × 16 = 80 marks)

11. (a) (i) Define Asymptotic notations. Distinguish between Asymptotic notation and conditional

asymptotic notation. (10)

(ii) Explain how the removing condition is done from the conditional Asymptotic notation with an

example. (6) Or

(b) (i) Explain how analysis of linear search is done with a suitable illustration. (10)

(ii) Define recurrence equation and explain how solving recurrence equations are done. (6)

12. (a) What is divide and conquer strategy and explain the binary search with suitable example

problem. Or

(b) Distinguish between Quick sort and Merge sort, and arrange the following numbers in

increasing order using merge sort. (18, 29, 68, 32, 43,37, 87, 24, 47, 50)

13. (a) (i) Explain the multistage graph problem with an example. (8)

(ii) Find an optimal solution to the knapsack instance n = 7, m= 15 (p1, p2, p3, ….p7) = (10,

5,15, 7, 6, 18, 3) and (w1, w2, w3, ... w7) (2, 3, 5, 7, 1, 4, 1) (8) Or

(b) Describe binary search tree with three traversal patterns? Give suitable example with neat

diagram for all three traversal of binary search tree.(16)

14. (a) (i) How does backtracking work on the 8 Queens problem with suitable example? (8)

(ii) Explain elaborately recursive backtracking algorithm? (8) Or

(b) What is Hamiltonian problem? Explain with an example using backtracking. (16)

15. (a) Write a complete LC branch-and-bound algorithm for the job sequencing with deadlines

problem. Use the fixed tuple size formulation. (16) Or



(b) Write a non-deterministic algorithm to find whether a given graph contains a Hamiltonian

cycle. (16)





B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2007.

Fourth Semester



(Regulation 2008)


PART A (Answer ALL questions) 10* 2=20 1. What is meant by stepwise refinement? 2. How is the efficiency of an Algorithm defined? 3. Give the Smoothness rule. 4. What is Algorithm Visualisation? 5. State the time complexity of bubble sort Algorithm. 6. List out any two Drawbacks of binary search algorithm. 7. What is Mathematical modelling? 8. What is Prestructuring ? Give eg. 9. What is Heuristics? 10. What is the metric used to measure the accuracy of approximation of algorithms?

PART B 5*16=80

11. (a) (i) What are the important problem types focussed by the researchers? Explain any 2 with eg.

(ii)What is Empirical Analysis of an algorithm? Discuss its strength and weakness (or)

(b) (i) Discuss the Fundamentals of Analyse framework. (ii) Explain the various Asymptotic Notations used in Algorithm design.

12. (a)Discuss the general plan for analysing the efficiency of Non-recursive Algorithm. Write an algorithm to find the number of binary digits in the binary representation of a positive decimal integer and analyse its efficiency. (or) (b) What is the principle alternative to Mathematical analysis of an algorithm? Explain the steps in analysing the efficiency of this analysis with eg. 13. (a)Find the number of comparison made by the sentinel version of sequential search algorithm for (i) In worst case and (ii) In Average case. (or) (b) Give a suitable eg and explain the Depth First Search algorithm. 14. (a) (i) Define Heap.Explain the properties of Heap. (ii) With a simple eg Explain heap sort algorithm.(or) (b) Define Spanning tree. Discuss the design steps in Kruskal’s algorithm to construct minimum spanning tree with an eg. 15. (a) Explain subset sum problem and discuss the possible solution strategies using Backtracking. (or) (b) Discuss the solution for Knapsack problem using Branch and Bound Technique.



Two marks answers

1. In the early days of computing the dominant vehicle for specifying algorithm was a flowchart, a method of expressing an algorithm by a collection of connected geometric shapes containing descriptions of the algorithm’s steps

2. There are two kinds of efficiency: time efficiency and space efficiency. Time efficiency indicates how fast an algorithm in question runs; space efficiency deals with the extra space the algorithm requires.

3. Let T(n) be an eventually nondecreasing function and f(n)be a smooth function. If T(n) ε Ѳ(f(n)) for values of n that are powers of where b>=2, then T(n)εѲ(f(n))

4. Algorithm Visualisation is Visualisation of the abstractions that describe software. 5. Let f(n) be the number of comparisons required to sort the given n number using

bubble sort. Then f(n) = sum of comparisons required in each scan For i= 1 we required n-1 comparison For i=2 we required n-2 comparison For i=n-1 we required n comparison Therefore f(n) = (n-1)+(n-2)+...............+2+1

=(n-1)n/2=( n2-n)/2 =O(n2) (order of n2) Computing time = O(n2)

6. It is the answer we should have fully expected: Since the algorithm simply reduces the size of the remaining array by about half an each iteration, the number of such iteration needed to reduce the initial size n to the final size 1 has to be about log2 n

The logarithmic function grows so slowly that its values remain small even for every large values of n.

7. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system.

8 Prestructuring is the 2nd type of tech. That exploits space for time tradeoffs that uses extra space to fascilitate a faster and/or more flexible access to that data. 9. A heuristics is a common – sense rule drawn from experience rather than from a mathematically proved assertion. 10. The accuracy of approx alg Sa to a problem minimizing some function f,can be quantified by size of the relative error of the approximation. r(Sa) = f(Sa)/ f(S*) PART B ANSWERS 11 (a) (i) Important problem types are Sorting Searching String Processing Graph problem Combinatorial Problems Geometric Problems Numerical Problems(Refer page no 19 of Anany Levitin) (ii) A general plan for empirical analysis of algorithm efficiency:

1. Understand the experiment’s purpose 2. Decide on the efficiency metric M to be measured and the measurement unit 3. Decide on characteristics of the input sample 4. Prepare a program implementing the algorithm for the experimentation



5. Generate a sample of inputs 6. Run the algorithm on the sample’s inputs and record the data observed 7. Analyze the data obtained(Refer page no 85 from Anany Levitin)

11. (b) (i) Analysis Framework: -Measuring an input’s size -Unit for measuring running time - Order of growth -Worst-case, Best-case and Average –case Efficiencies (ii) Big Oh: The function f(n) = O(g(n)) iff there exist positive constants C and no such that f(n)C * g(n) for all n, n , n0. Omega: The function f(n) =W (g(n)) iff there exist positive constant C and no such that f(n) C * g(n) for all n, n n0.

Theta The function f(n) =(g(n)) iff there exist positive constant C1, C2, and no such that C1 g(n), f(n) C2 g(n) for all n, n n0. Little Oh The function f(n) = 0(g(n)) Iff Lim f(n) = 0 n - g(n) Little omega The function f(n) = (g(n)) Iff Lim g(n) = 0 n - f(n) 12.(a) (i)General Plan for Analyzing efficiency of Non recursive Algorithm: -Decide on a parameter indicating an input’s size -Identify the algorithm’s basic operation - Check whether the number of times the basic operation is executed depends only on the size of an input. - Set up a sum expressing the number of times the algorithm’s basic operation is executed. - Using standard formulas and rules of sum Manipulation (ii) Refer Page no 66 from Anany Levitin (b) Refer Page no 69 from Anany Levitin 13. (a) – Definition for sequential search- It compares given element with all the elements in an array - Algorithm for sequential search - Briefly explain about efficiency class for sequential search 13.(b) Algorithm for DFS:- Step-1: Set the starting point of traversal and push it inside the stack Step-2: Pop the stack and add the popped vertex to the list of visited vertices Step-3: Find the adjacent vertices for the most recently visited vertex( from the Adjacency Matrix) Step-4: Push these adjacent vertices inside the stack (in the increasing order of their depth) if they are not visited and not there in the stack already Step-5: Go to step-2 if the stack is not empty 14. (a) (i) Heap -A heap can be defined as a binary tree with keys assigned to its nodes provided the following two conditions are met 1. the binary tree is essentially complete 2. The Key at each node is greater than or equal to the keys at its children (ii) Refer Page no 219 from Anany Levitin (b) A minimum spanning tree (MST) or minimum weight spanning tree is then a spanning tree with weight less than or equal to the weight of every other spanning tree. More generally, any



undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components - Definition for minimum spanning tree - prim’s Algorithm for MST - Explain the graph in stepwise procedure by drawing graph and tabulation 15.(a) The subset sum problem is a good introduction to the NP-complete class of problems. There are two reasons for this It is a decision and not an optimization problem It has a very simple formal definition and problem statement. Although the subset sum problem is a decision problem, the cases when an approximate solution is sufficient have also been studied, in the field of approximations algorithms; 15. (b) (i)The term Branch-and-Bound refers to all the state space methods in which all children of the E-node are generated before any other live node can become the Enode. The searching techniques that are commonly used in Branch-and-Bound method. The searching techniques that are commonly used in Branch-and-Bound method are: i. LC search ii. 15 – puzzle problem iii. LC control abstraction iv. Bounding v. FIFO branch and bound vi. LC branch and bound (ii) Given n objects and a knapsack or bag. Object I has a weight wi and the knapsack has the capacity m. If a fraction xi 0<xi<1 of object i is placed in to knapsack then a profit of pixi earned. The objective is to obtain a filling of the knapsack that maximizes the total profit earned. Since the knapsack capacity is m, we require the total weight of all chosen objects to be almost m. ********************************** ALL THE BEST ************************************

http://en.wikipedia.org/wiki/Connected_component_%28graph_theory%29

http://en.wikipedia.org/wiki/Decision_problem

http://en.wikipedia.org/wiki/Optimization_problem

http://en.wikipedia.org/wiki/Approximation_algorithm

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