## Design and Analysis of Algorithms - Jun 2013

### Computer Science Engg. (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** What is an algorithm? What are the properties of an algorithm? Explain with an example.(8 marks)
**1 (b)** Explain brute force method for algorithm design and analysis. Explain the brute force string matching algorithm with its efficiency.(8 marks)
**1 (c)** Express using asymptotic notation i) n! ii) 6×2^{n}+n^{2}(4 marks)
**2 (a)** Explain divide and conquer technique. Write the algorithm for binary search and find average case efficiency.(10 marks)
**2 (b)** What is stable algorithm? Is quick sort stable? Explain with example.(6 marks)
**2 (c)** Give an algorithm for merge sort.(4 marks)
**3 (a)** Explain the concept of greedy technique for Prim's algorithm. Obtain minimum cost spanning tree for the graph below Prim's algorithm.

(9 marks)
**3 (b)** Solve the following single source shortest path problem assuming vertex 5 as the source.

(9 marks)
**3 (c)** Define the following: i) Optimal solution; ii) Feasible solution(2 marks)
**4 (a)** Using Floyd's algorithm solve the all pair shortest problem for the graph whose weight matrix is given below: $$ \begin{bmatrix} 0 &\infty &3 &\infty \\2 &0 &\infty &\infty \\\infty &7 &0 &1 \\6 &\infty &\infty &0 \end{bmatrix} $$(7 marks)
**4 (b)** Using dynamic programming, solve the following knapsack instance.

N=4. M=5

(W_{1}, W_{2}, W_{3}, W_{4})= (2, 1, 3, 2)

( P_{1}, P_{2}, P_{3}, P_{4})=(12, 10, 20, 15).(5 marks)
**4 (c)** Outline an exhaustive search algorithm to solve traveling salesman problem.(8 marks)
**5 (a)** Write and explain DFS and BFS algorithm with example.(8 marks)
**5 (b)** Obtain topologies sorting for the given diagram using source removal method.

(5 marks)
**5 (c)** Explain Horspool's string matching algorithm for a text that comprises letter and space (denoted by hyphen) i.e "JIM-SAW-ME-IN-BARBER-SHOP" with pattern "BARBER". Explain its working along with a neat table and algorithm to shift table.(7 marks)
**6 (a)** Define the following:

i) Class P

ii) Class NP

iii) NP complete problem

iv) NP hard problem(8 marks)
**6 (b)** Write the decision tree to sort the elements using selection sort and find the lower bound.(8 marks)
**6 (c)** What is numeric analysis?(2 marks)
**6 (d)** Brief overflow and underflow in numeric analysis algorithms.(2 marks)
**7 (a)** What is back tracking? Apply back tracking problem to solve the instance of the sum of subset problem: S={3,5,6,7} and d=15.(7 marks)
**7 (b)** With the help of a state space tree, solve the following instance of the knapsack problem by the branch and bound algorithm.

Item | Weight | Value |

1 2 3 4 | 4 7 5 3 | 40 42 25 12 |

Knapsack | Capacity | W=10 |

**7 (c)**Explain how backtracking is used for solving 4-Queen's problem. Show the state space table.(7 marks)

**8 (a)**What is prefix computation problem? Give the algorithm for prefix computation which uses

i) n processors; ii) n/log n, processors.

Obtain the time complexities of these algorithms.(10 marks)

**8 (b)**What is super linear speed up? Obtain the maximum speed up where P=10 and various values of f=0.5, 0.1, 0.001.(5 marks)

**8 (c)**What are the different ways resolving read and write conflicts?(5 marks)