CS PhD Qualifying Exam, Summer 2008Data Struct & Algorithms (CS 610)Answer ALL 5 problems, each worth 20 points.Note: Be concise in describing your algorithms. If you use a known algorithm, then you can use itas a black-box (subroutine) without going into details of how that known algorithm works. Avoiduseless rambling about known algorithms, as that is viewed negatively!Definition (Little oh):We sayT(n) iso(f(n)), or sayT(n) isasymptotically lowerthanf(n),if:limn→∞T(n)f(n)= 0.For example, logniso(n), butn/2 is noto(n).1. You are given twosortedsequencesAandBof lengthnandn2respectively. We want tofind a sorted sequenceCthat contains those elements that arecommonto bothAandB.For example, ifA= (2,5,9) andB= (1,2,3,4,6,7,8,9,10) thenC= (2,9).Describe anasymptotically best algorithm for this task. Analyze its worst-case time complexity.2. We define a Binary-Search-Tree (BST) withnnodes to bebalancedif its height isO(logn).For example, AVL trees and red-blank trees are balanced.We have two arbitrary (not necessarily balanced) binary-search-treesT1andT2withn1andn2nodes,n=n1+n2. Assume alln