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)).

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1 Code No: R Set No (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)). 2. (a) List some of the relative advantages and disadvantages of the partition algorithm (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 spanning tree algorithm, in total, if the input is a complete undirected graph with n vertices and v i is the start vertex. (b) Design a linear-time algorithm for solving the single source shortest path algorithm for directed a cyclic graphs represented by their adjacency linked lists. [6+10] 4. (a) Solve the following 0/1 Knapsack problem using dynamic programming m=6, n=3, (w 1, w 2, w 3 )=(2,3,3), (p 1, p 2, p 3 )=(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 directed graph. 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 Bound methods. (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]

2 Code No: R Set No (a) Give a monte-carlo algorithm for finding the majority element in an array. (b) Show that f(n)=8n+128=0(n 2 ). [8+8] 2. (a) Suppose a binary tree has leaves l 1 l 2...l m at depths d 1, d 2...d m respectively prove that m 2 di 1 and determine when the equality is true. i=1 (b) Write and explain the control abstraction algorithm of divide and conquer. [8+8] 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 directed graph. 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 Bound methods. (b) Describe the TSP in Branch and Bound. [8+8] 1 of 2

3 Code No: R Set No (a) Explain the classes of NP-hard and NP-complete. (b) Describe clique decision problem and write the algorithm for the same. [8+8] 2 of 2

4 Code No: R Set No (a) Define omega notation. Explain the terms involved in it. Give an example. (b) Show that f 1 (n) f 2 (n) = 0(g 1 (n) g 2 (n) wheref 1 (n) = 0(g 1 (n) and f 2 (n) = 0(g 2 (n)). 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 is 0(n 2 2 n ) and space complexity is O(n2 n ). (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 analyze its 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 and Bound. (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]

5 Code No: R Set No (a) Define omega notation. Explain the terms involved in it. Give an example. (b) Show that f 1 (n) f 2 (n) = 0(g 1 (n) g 2 (n) wheref 1 (n) = 0(g 1 (n) and f 2 (n) = 0(g 2 (n)). 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 programming (c) A B C D A B C D (a) Explain the properties of depth-first search. [8+8] (b) Write a non-recursive algorithm of Post-order traversal of a tree and also analyze 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 complexity. [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]