CSL 201 Data Structures Mid-Semester Exam minutes

Transcription

1 CL 201 Data tructures Mid-emester Exam minutes Name: Roll Number: Please read the following instructions carefully This is a closed book, closed notes exam. Calculators are allowed. However laptops or mobile phones are not allowed. Use the space provided after every question for writing your answer. You will be given additional sheets for rough work. Please attach the additional sheet(s) along with this booklet. Be precise and concise in your answers. Include explanations, derivations, and examples when appropriate. This can fetch you partial scores even if the final answer is incorrect. Please write legibly There are 5 questions worth a total of 50 points. Work efficiently. ome questions are easier than others. Try to answer the easier ones before you get bogged down by the harder ones. Do not panic. Question # Max. core core Total 50 1

2 1 One pointers (1 x 6 = 6 points) 1.1 Consider a circular doubly linked list with 11 nodes. How many links do we have to change for deleting a node other than the head node? 1.2 What is the maximum difference in the height of two leaf nodes in a min heap? 1.3 What is the minimum and maximum height of a binary tree? 1.4 Given two hash tables containing the same set of n elements, one that uses chaining and one that uses linear probing, What is the complexity of finding the minimum element from both the hash tables? 1.5 What is the complexity of selection sort and heap sort? 1.6 Under what assumptions can a unique tree be constructed from the pre and post order traversals of the tree? 2 Two pointers (2 x 6 = 12 points) 2.1 We have an array based stack implementation that increments the array size by 5 elements every time the array becomes full. tarting with an empty stack, what is the cost of 16 pushes? 2.2 how that if f 1 (n) = O(g 1 (n)) and f 2 (n) = O(g 2 (n)), then f 1 (n) f 2 (n) is O(g 1 (n) g 2 (n)). 2

3 2.3 A Θ(n 2 ) algorithm always takes longer to run than a Θ(log n) algorithm. True or False, Explain. 2.4 If f(n) = O(g(n)) and g(n) = O(f(n)) then f(n) = g(n). True or False, Explain. 2.5 Compute the Big O complexity for the following algorithm. 1 p u b l i c s t a t i c i n t fun ( i n t number ) { 2 i f ( number < 2) 3 r e t u r n number ; 4 i n t sum = 0 ; 5 f o r ( i n t i =1; i <number ; i =2) 6 sum += i ; 7 r e t u r n fun ( number 1) + sum ; 8 } 2.6 Given three heaps, each containing n elements. What is the complexity of an efficient algorithm for constructing a single heap containing all the 3n elements? 3

4 3 Four pointers (4 x 3 = 12 points) 3.1 Given an array A of n elements with the following property: There exists an index i [0, n 1] such that A[0] < A[1] <... < A[i 1] < A[i] > A[i + 1] >... > A[n]. All elements in the array before the index i are in increasing order and after index i are in decreasing order. Describe an efficient algorithm that can find such an i. 3.2 Describe a recursive algorithm that will check if an array A of integers contains an integer A[i] that is the sum of two integers that appear earlier in A, that is, such that A[i] = A[j] + A[k], for j, k < i. 3.3 Consider strings constructed using only the following four characters (, ), [, and ]. Character ( matches with only ) and [ matches with only ]. A string comprising of these characters is 4

5 complete if every character is followed by its matching character or a substring that is complete followed by its matching character. For example (), [], ([]), []([]()) are all examples of complete strings, while ([)] and [][(]) are examples of incomplete strings. Write the pseudocode for a function that determines (if it exists) the string of the patterns that when appended to the original string makes it complete. The function should return null if such a substring does not exist. Here are some input and output examples to help you out. You might find the following two functions to be useful while writing the pseudocode. Input Output (( )) ([( )]) ([) null [()(] null 1 boolean i s B e g i n ( char x ) { 2 i f ( x == ( ) ( x == [ ) 3 r e t u r n t r u e 4 e l s e 5 r e t u r n f a l s e 6 } 7 char matchingend ( char x ) { 8 i f ( x == ( ) 9 r e t u r n ) 10 e l s e 11 r e t u r n ] 12 } 5

6 4 ix pointers (6 x 2 = 12 points) 4.1 A binary tree is a tree in which each node has 0, 1, or 2 children. Given 3 nodes, there are exactly 5 binary trees that can be constructed using these nodes, namely, Figure 1: binary trees with three nodes Write a method int nbtrees(int n) that returns the number of binary trees that can be constructed with n nodes, where n 0 What is the complexity of the method? 6

7 4.2 uppose we wish to implement a hash table where collisions are resolved by chaining. However we want to store the data in the slots of the table itself, rather than creating new lists. All empty slots in the table are marked with a boolean flag and are linked together to form a free list. Write the pseudocode for put(key, value) and remove(key) methods that performs these operations in expected constant time. Assume that the operations of a singly or doubly linked list are available. 7

8 5 Eight pointer (8 x 1 = 8 points) 5.1 A binary tree has three kinds of nodes - Goal (G), imple () and Complex nodes (C). Write the pseudocode for a function that finds a path from the root node of the binary tree to any goal node that passes through minimum number of complex nodes. The function should return the goal node that has been reached, return null if there are no goal nodes. Following are few examples for your reference (8 points) C C G C G C G G C Input Output G G (either one) null 8