Analysis and Design of Algorithms 2a. - Vtustudentsparadise

I of2USN06cs43Fourth Semester B.E. Degree Examination, May/June 2010Analysis and Design of AlgorithmsTime: 3 hrs.Note: Answer any FIVE full questions, selectingot least TWO questions from each part.Max. Marks:100la.2a.b.c.b.c-PART _ ACompare the orders of growth of log, (n) and 16-. What is your conclusion? (06 Marks)Define O-notation. If fr(n) e O(e,(r)) and fr(n) e O(gr(t )), prove thatfr (n) + f, (n) e O(max{g, (r), g, (r,)}).(06 Marks)Given a positive decimal integer n, write a recursive algorithm which computes the numberof binary digits in the binary representation of n. Write the corresponding recurrence relationand solve it.(08 Marks)Explain the algorithm for selection sort. If A is ai array of size n, obtain an expression forthe number of key comparisons.(06 Marks)Using bubble sort algorithm, arrange the letters of the word 'QUESTION' in alphabeticalorder.(06 Marks)Show how divide and conquer technique can be used to compute the product of two n-digitintegers. If n is a power of 2, obtain a recurrence relation for M(n), the number ofmultiplications and solve it.(0E Marks)3b.c.What are the three major variations of decrease and conqueran example.Sort the letters of the word "EXAMPLE" in alphabetical orderDescribe the Johnson Trotter algorithm for generatingpermutations of {3, 5, 7} using the following:i) Bottom up minimal change algorithmii) Johnson Trotter algorithm.technique? Explain each with(06 Marksiusing insertion ron.,ouMarks)permutations. Generate all(0E Marks)4 a. Write an algorithm for DFS. With an example, explain horv this algorithm can be used tosolve topological sorting problem.(I0lllarks)b. Using quick sort, arrange the lefters of the word "QUICKSORT" in alphabetical order.Show all the steps elearly and draw the tree of the recursive calls made. (10 Marks)PART - B5 a. Define a2-3 tree. For any 2-3 tree of height h consisting of n nodes, prove the following:logr(n + l) - I h < logr(n + 1)-< - 1b. Describe the algorithm for heap sort.c. Show how Horspool's algorithm can be usedtext. Consider all the four cases.(06 Marks)(06 Marks)to search for the pattern BARBER in a given(08 Marks)

2 of?06cs,6a. Apply Warshall's algorithm tofollowing adjacency matrix:(oll, olo ob.[ooUsing Floyd's algorithm, solveweight matrix is given below:Using Kruskal's algorithm,given below:06cO@Jobtainfind the transitive closure00\I 0lo tl0o)the all-pairs shortest path2@l032oo0 4a208@@Joo@@ 0a minimumof the graph defined by t(05 Marproblem for the graph whr(10 Marcost spanning tree for the graph Fig.Q((05 MarFig.Q6(c)7 a. Construct a Huffrnan code for the following data: (06 MaCharacter A B C D EProbabilitv 0.4 0.1 0.2 0.15 0.15Decode the text whose encoding is 100010111001010 using the above Huffinan code.b. Write short notes on P, NP and NP-complete problems. (06 Mac. Explain how backtracking is used for solving 4 -8a. Solve the fibllowine insknce ofem lgngItem I 2 3 4Weieht 4 7 5 3Value $40 $42 $2s $i2b.The capacity of the knapsack iJ$i: l0.queens problem. Show the state space tr(0E Mabranch and bound algorithm:(08 MaWhen does collision occur in hashing? What are the different mechanisms used to resrcollisions?(04 MaWhat are decision trees? Explain how decision trees are used in sorting algorithms.,^ _ _ (08 Ma***,1.*