Sr05220502 Design and Analysis of Algorithms

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Code No: R05220502 Set No. 1

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007DESIGN AND ANALYSIS OF ALGORITHMS

(Computer Science & Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) Explain the asymptotic notations used in algorithm analysis.

(b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)). [10+6]

2. (a) List some of the relative advantages and disadvantages of the partition algo-rithm

(b) Write the Quicksort algorithm? Analize the time complexity in worst case.[6+10]

3. (a) How many comparisons of edge weights will be done by the minimum spanningtree algorithm, in total, if the input is a complete undirected graph with nvertices and vi is the start vertex.

(b) Design a linear-time algorithm for solving the single source shortest path algo-rithm for directed a cyclic graphs represented by their adjacency linked lists.

[6+10]

4. (a) Solve the following 0/1 Knapsack problem using dynamic programmingm=6, n=3, (w1, w2, w3)=(2,3,3), ( p1, p2, p3)=(1,2,4)

(b) Write an algorithm of all pairs shortest path problem. [8+8]

5. (a) Write a pseudocode for finding the strongly connected components of directedgraph. Also analyze its time complexity.

(b) Explain the Inorder traversal of a tree with an example. [8+8]

6. (a) Describe graph coloring problem and its time complexity.

(b) Write an algorithm of 8-queens problem using backtracking. [8+8]

7. (a) What is Bounding? Explain how these bound are useful in Branch and Boundmethods.

(b) Describe the TSP in Branch and Bound. [8+8]

8. (a) Explain about cook’s theorem.

(b) Explain the strategy to prove that a problem is NP hard. [8+8]

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1. (a) Give a monte-carlo algorithm for finding the majority element in an array.

(b) Show that f(n)=8n+128=0(n2). [8+8]

2. (a) Suppose a binary tree has leaves ℓ1ℓ2.......lm at depths d1, d2....dm respectively

prove thatm

i=1

2−di ≤ 1 and determine when the equality is true.

(b) Write and explain the control abstraction algorithm of divide and conquer.[8+8]

3. (a) What is spanning tree? Explain the prim’s algorithm with an example.

(b) Explain the terms Feasible solution, optimal solution and objective function.[10+6]

4. (a) Find one problem for which the principle of optimality does not hold. Explain

why the principle does not hold.(b) Find the shortest path between all pairs of nodes in the following graph.

(Figure 4b) [8+8]

Figure 4b

5. (a) Write a pseudocode for finding the strongly connected components of directedgraph. Also analyze its time complexity.

(b) Explain the Inorder traversal of a tree with an example. [8+8]

6. (a) Write an algorithm of finding all m-colorings of a graph.

(b) Describe the 4-queens problem using backtracking. [8+8]

7. (a) What is Bounding? Explain how these bound are useful in Branch and Boundmethods.

(b) Describe the TSP in Branch and Bound. [8+8]

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8. (a) Explain the classes of NP-hard and NP-complete.

(b) Describe clique decision problem and write the algorithm for the same. [8+8]

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1. (a) Define omega notation. Explain the terms involved in it. Give an example.

(b) Show that f 1(n)×f 2(n) = 0(g1(n)×g2(n) wheref 1(n) = 0(g1(n) and f 2(n) =0(g2(n)). [10+6]

2. (a) Write and explain the control abstraction for Divide and conquer.

(b) Suggest refinements to mergesort to make it in-place. [8+8]



4. (a) For the Travelling sales person algorithm show that the time complexity is0(n22n) and space complexity is O(n2n).

(b) Write an algorithm of matrix chain multiplication. [8+8]

5. (a) Explain the properties of strongly connected components.

(b) Write a non-recursive algorithm of In-order traversal of a tree and also analyzeits time complexity. [6+10]

6. Describe Backtracking technique to m-coloring graph. Explain with an example.[16]

7. (a) Explain the method of reduction to solve TSP problem using Branch andBound.

(b) Explain the principles of FIFO Branch and Bound. [8+8]

8. (a) Explain the classes of P and NP.

(b) Write a nondeterministic Knapsack algorithm. [8+8]

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1. (a) Define omega notation. Explain the terms involved in it. Give an example.

(b) Show that f 1(n)×f 2(n) = 0(g1(n)×g2(n) wheref 1(n) = 0(g1(n) and f 2(n) =0(g2(n)). [10+6]

2. (a) Write and explain the control abstraction for Divide and conquer.

(b) Suggest refinements to mergesort to make it in-place. [8+8]



4. (a) Explain the OBST algorithm.

(b) Find the shortest tour of a TSP for following instance using dynamic program-

ming

(c)

A B C D

A

B

C

D

∞ 12 5 711 ∞ 13 64 9 ∞ 18

10 3 2 ∞

[8+8]

5. (a) Explain the properties of depth-first search.

(b) Write a non-recursive algorithm of Post-order traversal of a tree and alsoanalyze its time complexity. [6+10]

6. (a) Draw the state space tree for m coloring when n=3 and m=3

(b) Write a recursive backtracking algorithm. [8+8]

7. Write the LCBB algorithm for the 0/1 Knapsack problem. Also Analyse its com-plexity. [16]

8. (a) Explain the classes of NP-hard and NP-complete.

(b) Describe clique decision problem and write the algorithm for the same. [8+8]

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Sr05220502 Design and Analysis of Algorithms