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VELAMMAL COLLEGE OF ENGINEERING AND TECHNOLOGY
DEPARTMENT OF INFORMATION TECHNOLOGY
QUESTION BANK
SUBJECT : CS6402 - Design and Analysis of Algorithms DATE:28.03.18
SEM / YEAR: IV Sem/ II Year BATCH: 2016 – 2020
Part – A (10 * 2 = 20)
1. Define Notion of Algorithm. (Remember) CO1
An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.
2. Write down the properties of asymptotic notations. (Remember) CO1
Symmetry
f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))
Transpose Symmetry
f(n) = O(g(n)) if and only if g(n) = Ω(f(n)) f(n) = o(g(n)) if and only if g(n) = ω(f(n))
3. Define convex hull by Brute Force. (Remember) CO2
The convex hull of a set S of points is the smallest convex set containing S. (The smallest requirement means that the convex hull of S must be a subset of any convex set containing S).
4. What are the variations of merge sort? (Remember) CO2
First, thealgorithm can be implemented bottom up by merging pairs of the array’s elements,then merging the sorted pairs, and so on. This avoids the time and space overhead of using a stack to handle recursive calls.Second, we can divide a list to be sorted in more than two parts, sort each recursively, and then merge them together. This scheme, which is particularly useful for sorting files residing on secondary memory devices, is called multiway mergesort.
5. What is meant by principle of optimality? (Remember) CO3
The principle of optimality is the basic principle of dynamic programming, which was developed by Richard Bellman: that an optimal path has the property that whatever the initial conditions and control variables (choices) over some initial period, the control (or decision variables) chosen over the remaining period must be
optimal for the remaining problem, with the state resulting from the early decisions taken to be the initial condition.
6. Define fringe vertex. (Remember) CO3
Vertices adjacent to tree vertices but not yet visited
7. Define the Iterative improvement technique. (Remember) CO4The objective function is the function that problem seeks to maximize or minimize. Iterative improvement is frequently used in numerical problems, for example root finding or finding the
maximum of a function. We will concentrate on iterative improve to graph problems.
8. Define Blocking pair. (Remember) CO4
A pair (m, w) is called a blocking pair for a marriage matching, M, if both m and w prefer each other more than
there mate in the marriage, M.
9. Explain whether backtracking algorithm always produces optimal. (Understand) CO5
Yes, the idea of the backtracking can be further enhanced by evaluation the quality of partially
constructed solution.
10. How is the accuracy of the approximation algorithms measured? Remember) CO5
The accuracy ratio
Part – B (5 * 13 = 65)
1. i)Suppose W satisfies the following recurrence equation and base case
(where c is a constant): W(n) = c.n+W(n/2) and W(1) =1.
What is the asymptotic order of W(n)(Apply) Ans: W(n) = O(log n) 5 marks
CO1
(ii) Make use of two queues implement a stack. Identify the running time of the stack operations.
(Apply) CO1
Explanation with neat sketch 8 marks
OR
2. What are the properties of Asymptotic notations and derive it. (Understand) CO1
Properties of asymptotic notations 4 marks
Proof of each property 9 marks1. Transitive
If f(n) = Θ(g(n)) and g(n) = Θ(h(n)), then f(n) = Θ(h(n)) If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n)) If f(n) = o(g(n)) and g(n) = o(h(n)), then f(n) = o(h(n)) If f(n) = Ω(g(n)) and g(n) = Ω(h(n)), then f(n) = Ω(h(n)) If f(n) = ω(g(n)) and g(n) = ω(h(n)), then f(n) = ω(h(n))
2. Reflexivity f(n) = Θ(f(n)) f(n) = O(f(n)) f(n) = Ω(f(n))
3. Symmetry f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))
4. Transpose Symmetry f(n) = O(g(n)) if and only if g(n) = Ω(f(n)) f(n) = o(g(n)) if and only if g(n) = ω(f(n))
5. Some other properties of asymptotic notations are as follows: If f (n) is O(h(n)) and g(n) is O(h(n)), then f (n) + g(n) is O(h(n)). The function loga n is O(logb n) for any positive numbers a and b ≠ 1. loga n is O(lg n) for any positive a ≠ 1, where lg n = log2 n.3. Apply quick sort to sort the list E, X, A, M, P, L, E in alphabetical order. Draw the tree of the
recursive calls made. (Apply) CO2
Quick sort Procedure 2 marks
Explanation with workout problem 7 marks
Tree of Recursive Call 4 marks
OR
4. There are 4 people who need to be assigned to execute 4 jobs(one person per job) and the problem is to find an assignment with the minimum total cost. The assignment costs is given below, solve the assignment problem by exhaustive search. (Apply) CO2
Job1 Job2 Job3 Job4Person1 9 2 7 8Person 2 6 4 3 7Person 3 5 8 1 8Person 4 7 6 9 4
5. Solve the all-pairs shortest path problem for the digraph with the following weight matrix. (Apply) CO3
OR6. Consider the following graph. Explain the list of edges in the MST in the order that Prim’s
algorithm inserts them. Start Prim’s algorithm from Vertex A. (Apply) CO3
Prim’s algorithm explained with problem solving 10 marks
Path calculation 3 marks
7. Consider an instance of the stable marriage problem given by the ranking matrix. (Apply) CO4
For each of its marriage matching’s, indicate whether it is stable or unstable matching’s, specify a blocking pair. For the stable matching, indicate whether they are man-optimal, woman-optimal or neither. (Assume that the Greek and English letters denote the men and women respectively)
OR
8. Test for the following theorem: “The value of a maximum flow in a network is equal to the capacity of its minimum cut”. (Analyze) CO4
Theorem Proof explanation 8 amrksProof (solvable problem) 5 marks
9. Solve the following instance of the Knapsack problem by branch and bound algorithm. Knapsack capacity W=10 using Approximation Algorithm. (Analyze) CO5
Value to weight ratio 2 marks
State Space Tree 10 marks
Result Set 1 mark
OR
10. Apply backtracking to the problem of finding a Hamiltonian circuit in the following graph. (Apply) CO5 Hamiltonian Circuit -- > State Space tree 8 marksHamiltonian Path finding with node numbering 3 marksExplanation 2 marks
Part – C (1 * 15 = 15)
1. Solve the following recurrence relations:(Apply) CO1
a) x(n) = x(n-1) + 5 for n>1; x(1) = 0b) x(n) = 3x(n-1) for n>1; x(1) = 4c) x(n) = x(n-1) + n for n>0; x(1) = 0d) x(n) = x(n/2) + n for n>1; x(1) = 1 (solve for n=2k)e) x(n) = x(n/3) + 1 for n>1; x(1) = 1(solve for n=2k)
Each one carries 3 marks
OR
2. Apply the bottom-up dynamic programming algorithm to the following instance of the knapsack problem: (Apply) CO3
Course Incharge Module Coordinator Verified ByMr. Suresh Babu P, AP III/IT Mrs.D.Anandhavalli HoD/IT