Data Structures and Programming Spring 2014, Midterm Exam.

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1 Data Structures and Programming Spring 2014, Midterm Exam. 1. (10 pts) Order te following functions 2.2 n, log(n 10 ), , 25n log(n), 1.1 n, 2n 5.5, 4 log(n), 2 10, n 1.02, 5n 5, 76n, 8n 5 + 5n 2 by teir asymptotic growt rate, in non-decreasing order. Indicate by circling tose functions tat are big- Teta (i.e., Θ(..)) of eac oter = 2 10 < log(n 10 ) = 4 log(n) < 76n < 25n log(n) < n 1.02 < 5n 5 = 8n 5 + 5n 2 < 2n 5.5 < 1.1 n < 2.2 n 2. (8 pts) Give te best asymptotic ( big-o, i.e., O()) caracterization of te best and worst case time complexity of te algoritm Count(A, B, n). Explain in detail ow you computed te complexity. Note tat denotes te assignment statement. Algoritm Count(A, B, n) Input: Arrays A and B of size n. A, B store positive integers. i 0 sum 0 wile i < n do if A[i] < n ten for j 0 to A[i] do sum sum + B[j] i i + 1 return sum Best case: O(n) wen if A[i] < n always false; Worst case: O(n 2 ) wen wen if A[i] < n always true 3. (10 pts) Consider te use of te exclusive-or encoding metod to represent doubly linked lists as discussed in class. Given a node X, let Link(X) be te exclusive-or encoding kept in node X, i.e., Link(X) is te outcome of taking te exclusive-or of te addresses of X s predecessor and successor along te list. Consider a doubly linked list consisting of te following: X 1 X 2 X 3 X 4 X 5 X 6. Suppose pointers P and Q point to nodes X 3 and X 4, respectively. Explain te updates needed to delete X 3 and X 4, making te new list X 1 X 2 X 5 X 6. Note tat after te deletion, P and Q point to X 2 and X 5, respectively. Use additional temporary pointer(s), if needed. You may use te notation x y exp 1 exp 2 defined in class to describe your algoritm. Before Address Link( Link(P ) Q) Link(P ) Q P Q Link(Q) P Link(Link(Q) P ) P Q Node X 1 X 2 X 3 X 4 X 5 X 6 After Address P new Link(P ) Q Q new Link(Q) P Node X 1 X 2 X 5 X 6 Link (Link( Link(P ) Q) (Link(Link(Q) P )) P ) ( Link(Q) P ) Q) ( Link(P ) Q) Hence, te operations are P Q Link(Link(P) Q) Link(Link(Q) P ) Link(P) Q Link(Q) P (Link(Link(P) Q) P ) ( Link(Q) P ) (Link(Link(Q) P )) Q) ( Link(P ) Q) 1

2 4. (10 pts) Suppose tat we ave numbers between 1 and 1000 in a binary searc tree and want to searc for te number 363. For eac of te following sequences, can it be te sequence of nodes examined? Yes or No? No explanations are needed. (a) 2, 252, 401, 398, 330, 344, 397, 363 (b) 924, 220, 911, 244, 898, 258, 362, 363 (c) 925, 202, 911, 240, 912, 245, 363 (d) 2, 399, 387, 219, 266, 382, 381, 278, 363 (e) 935, 278, 347, 621, 299, 392, 358, 363 Solutions: (c) and (e): NO. 5. (12 pts) Consider te following AVL tree. Perform operation delete(1) and rebalance te tree if necessary. You ave to use exactly te same algoritm as in class. Sow all te intermediate trees. Solution Figure 1: An AVL tree 2

3 6. (10 pts) (a) (4 pts) Define treaded binary trees. A binary tree is treaded by making all rigt cild pointers tat would normally be null point to te inorder successor of te node, and all left cild pointers tat would normally be null point to te inorder predecessor of te node. (b) (6 pts) Design a non-recursive algoritm to do inorder traversal of a treaded binary tree. Explain your algoritm in Cinese or Englis. Also use an example to sow ow your algoritm works. 3

4 7. (10 pts) Draw te expression tree and write te prefix and postfix forms for te following expression. Solution (A + B) D + (E/(F + A D)) + C. + + C * / + D E + A B F * A D Prefix: + + * + A B D / E + F * A D C Postfix: A B + D * E F A D * + / + C + 8. (15 pts) (a) (10 pts) Draw te Splay tree tat results from inserting te keys 4, 9, 3, 7, 5, 6 in tat order into an initially empty Splay tree. Sow all te intermediate trees (i.e., draw te tree after eac insertion is carried out). Also find te potential for eac intermediate tree using te potential function discussed in class (i.e., potential of a tree T is node x in T log(size(x)), were size(x) is te number of nodes in te subtree rooted at x. Solution Potential of te final tree is log(size(x 3 )) + log(size(x 4 )) + log(size(x 5 )) + log(size(x 6 )) + log(size(x 7 )) + log(size(x 9 )) = = 5 (b) (5 pts) Suppose we want to insert 10 numbers ({1, 2, 10}) into an initially empty Splay tree. Wat will be te order of insertions wat will result in a splay tree of eigt 10 (i.e., becoming a list). Wy? Te keys of value 10, 9,.., 4, 3, 2, 1 are inserted in tis order in a splay tree. Te resulting tree is:

5 9. (10 pts) A red-black binary searc tree (BST) is called a left-leaning red-black BST if red links lean left (i.e., eac red node is te left cild of its parent). See Figure 2 for an example. Figure 3 summarizes te rebalancing rules for left-leaning red-black trees w.r.t. insertion. Suppose we insert key 99 into te tree in Figure 2. Sow ow to rebalance te tree. Sow your derivation in sufficient detail red link left rotate rigt rotate flip colors Figure 2: A left-leaning red-black BST Figure 3: Rebalancing rules To insert 99, we do two color flips, followed by one left rotation. Here is a drawing of te resulting red-black BST (5 pts) Let A be an array of N integers. Give an algoritm to output all te distinct numbers in A in ascending order. Your algoritm must finis in O(N log T ) time, were T is te number of distinct integers in A. For example, given A = {35, 10, 59, 17, 61, 17, 52, 10}, your algoritm sould output 10, 17, 35, 52, 59, 61. Describe your algoritm in Englis or Cinese, and explain wy your algoritm takes O(N logt ) time. Scan te array and maintain all te distinct numbers in a binary tree. After reading te next element, perform dictionary searc to ceck if it already exists in te tree, and output it if not (in wic case we insert te element in te tree). As te tree indexes at most T elements at any moment, te total cost is O(NlogT ). 5