S.SAKTHI, LECTURER/IT

Transcription

1 NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DATA STRUCTURES AND ALGORITHMS QUESTION BANK PREPARED BY: S.SAKTHI-LECTURER/IT BRANCH/YEAR: EEE/II UNIT-I (LINEAR STRUCTURES) 1. What is meant by an abstract data type? An ADT is a set of operation. Abstract data types are mathematical abstractions.eg.objects such as list, set and graph along their operations can be viewed as ADT's. 2. What are the operations of ADT? Union, Intersection, size, complement and find are the various operations of ADT. 3. What is meant by list ADT? List ADT is a sequential storage structure. General list of the form a1, a2, a3.., an and the size of the list is 'n'. Any element in the list at the position I is defined to be ai, ai+1 the successor of ai and ai-1 is the predecessor of ai. 4. What are the various operations done under list ADT? Print list Insert Make empty Remove Next Previous Find kth 5. What are the different ways to implement list? Simple array implementation of list Linked list implementation of list 6. What are the advantages in the array implementation of list? a) Print list operation can be carried out at the linear time b) Fint Kth operation takes a constant time

2 7. What is a linked list? Linked list is a kind of series of data structures, which are not necessarily adjacent in memory. Each structure contain the element and a pointer to a record containing its successor. (OR) The linked list consists of a series of structures, which are not necessarily adjacent in memory. Each structure contains the element and a pointer to a structure containing its successor. We call this the next pointer. 8. What is a pointer? Pointer is a variable, which stores the address of the next element in the list. Pointer is basically a number. 9. What are different types of Linked List? Single linked list Double linked list Circular linked list 10. What are different types of Circular Linked List? Circular Single linked list Circular double linked list 11. List the basic operations carried out in a linked list? a. Creation of list b. Insertion of list c. Deletion of list d. Modification of list e. Traversal of List 12. Define double linked list? It is linear data structure which consists of two links or pointer fields Next pointer points to the address of the next(successor) node.previous pointer points to the address of the previous(predecessor) node. 13. What is a doubly linked list? In a simple linked list, there will be one pointer named as 'NEXT POINTER' to point the next element, where as in a doubly linked list, there will be two pointers one to point the next element and the other to point the previous element location.

3 14. Define double circularly linked list? In a doubly linked list, if the last node or pointer of the list, point to the first element of the list, then it is a circularly linked list. 15. Define Circular linked list? It is a list structure in which last node points to the first node, there is no null value. 16. What is the need for the header? Header of the linked list is the first element in the list and it stores the number of elements in the list. It points to the first data element of the list. 17. List three examples that uses linked list? Polynomial ADT Radix sort Multi lists 18. Give some examples for linear data structures? Stack Queue 19. What is a stack? Stack is a data structure in which both insertion and deletion occur at one end only. Stack is maintained with a single pointer to the top of the list of elements. The other name of stack is Last-in -First-out list. 20. Write postfix from of the expression A+B-C+D? A-B+C-D+ 21. How do you test for an empty queue? To test for an empty queue, we have to check whether READ=HEAD where REAR is a pointer pointing to the last node in a queue and HEAD is a pointer that pointer to the dummy header. In the case of array implementation of queue, the condition to be checked for an empty queue is READ<FRONT. 22. What are the postfix and prefix forms of the expression? A+B*(C-D)/(P-R) Postfix form: ABCD-*PR-/+ Prefix form: +A/*B-CD-PR

4 23. Explain the usage of stack in recursive algorithm implementation? In recursive algorithms, stack data structures is used to store the return address when a recursive call is Encountered and also to store the values of all the parameters essential to the current state of the procedure. 24. Write down the operations that can be done with queue data structure? Queue is a first - in -first out list. The operations that can be done with queue are addition and deletion. 25. What is a circular queue? The queue, which wraps around upon reaching the end of the array is called as circular queue. 26. Write down the definition of data structures? A data structure is a mathematical or logical way of organizing data in the memory that consider not only the items stored but also the relationship to each other and also it is characterized by accessing functions. 27. Give few examples for data structures? Stacks, Queue, Linked list, Trees, graphs 28. Define Algorithm? Algorithm is a solution to a problem independent of programming language. It consist of set of finite steps which, when carried out for a given set of inputs, produce the corresponding output and terminate in a finite time. 29. What are the features of an efficient algorithm? Free of ambiguity Efficient in execution time Concise and compact Completeness Definiteness Finiteness 30. List down any four applications of data structures? Compiler design Operating System Database Management system Network analysis

5 31. What is meant by an abstract data type(adt)? An ADT is a set of operation.a useful tool for specifying the logical properties of a datatype is the abstract data type.adt refers to the basic mathematical concept that defines the datatype. Eg.Objects such as list, set and graph along their operations can be viewed as ADT's. 32. What is a Stack? A Stack is an ordered collection of items into which new items may be inserted and from which items may be deleted at one end, called the top of the stack. The other name of stack is Last-in -First-out list. 33. What are the two operations of Stack? PUSH POP 34. Write postfix from of the expression A+B-C+D? A-B+C-D+ 35. What is a Queue? A Queue is an ordered collection of items from which items may be deleted at one end called the front of the queue and into which tems may be inserted at the other end called rear of the queue.queue is called as First in-first-out(fifo). 36. Mention applications of stack? a. Expression parsing b. Evaluation of Postfix c. Balancing parenthesis d. Tower of Hanoi e. Reversing a string 37. Deletion routine for linked lists: void delete( element_type x, LIST L ) { position p, tmp_cell; p = find_previous( x, L ); if( p->next!= NULL ) /* Implicit assumption of header use */ { /* x is found: delete it */ tmp_cell = p->next; p->next = tmp_cell->next; /* bypass the cell to be deleted */ free( tmp_cell ); } }

6 38. Insertion routine for linked lists void insert( element_type x, LIST L, position p ) { position tmp_cell; tmp_cell = (position) malloc( sizeof (struct node) ); if( tmp_cell == NULL ) fatal_error("out of space!!!"); else { tmp_cell->element = x; tmp_cell->next = p->next; p->next = tmp_cell; } } 39. Examples Of Linked List: The Polynomial ADT Multilists Radix sort 40. STACK: A stack is a list with the restriction that inserts and deletes can be performed in only one position, namely the endof the list called the top. The fundamental operations on a stack are push, which is equivalent to an insert, and pop,which deletes the most recently inserted element. Stacks are sometimes known as LIFO (last in, first out) lists. Stack model: input to a stack is by push, output is by pop 41. Applications of Stacks: Balancing Symbols Infix to Postfix Conversion Postfix Expressions

7 42. Routine to test whether a stack is empty--array implementation int is_empty( STACK S ) { return( S->top_of_stack == EMPTY_TOS ); } 43. Routine to create an empty stack--array implementation void make_null( STACK S ) { S->top_of_stack = EMPTY_TOS; } 44. Routine to push onto a stack--array implementation void push( element_type x, STACK S ) { if( is_full( S ) ) error("full stack"); else S->stack_array[ ++S->top_of_stack ] = x; } 45. Routine to return top of stack--array implementation element_type top( STACK S ) { if( is_empty( S ) ) error("empty stack"); else return S->stack_array[ S->top_of_stack ]; } 46. Routine to pop from a stack--array implementation void pop( STACK S ) { if( is_empty( S ) )

8 error("empty stack"); else S->top_of_stack--; } 47. QUEUE: The basic operations on a queue are enqueue, which inserts an element at the end of the list (called the rear), and dequeue, which deletes (and returns) the element at the start of the list (known as the front). Model of a queue 48. Applications of QUEUE: When jobs are submitted to a printer, they are arranged in order of arrival. Thus, essentially, jobs sent to a line printer are placed on a queue. Virtually every real-life line is (supposed to be) a queue. For instance, lines at ticket counters are queues, because service is first-come first-served. Another example concerns computer networks. There are many network setups of personal computers in which the disk is attached to one machine, known as the file server. Users on other machines are given access to files on a first-come first-served basis, so the data structure is a queue. Calls to large companies are generally placed on a queue when all operators are busy.in large universities, where resources are limited, students must sign a waiting list if all terminals are occupied. The student who has been at a terminal the longest is forced off first, and the student who has been waiting the longest is the next user to be allowed on. Applications of QUEUE: Real life line Calls to large companies Access to limited resources in Universities Accessing files from file serve Jobs submitted to printer

9 PART-B 1. Explain the linked list implementation of list ADT in Detail? Definition for linked list Figure for linked list Next pointer Header or dummy node Various operations Explanation Example figure Coding 2. Explain the cursor implementation of linked list? Definition for linked list Figure for linked list Next pointer Header or dummy node Various operations Explanation Example figure Coding 3. Explain the various applications of linked list? Polynomical ADT Operations Coding Figure Radix Sort Explanation Example Multilist Explanation Example figure 4. Explain the linked list implementation of stack ADT in detail? Definition for stack Stack model Figure Pointer-Top Operations Coding Example figure

10 5. Explain the array implementation of queue ADT in detail? Definition for stack Stack model Figure Pointer-FRONT, REAR Operations Coding Example figure 6. Explain singly linked list in detail. Definition, insert, deletion-routine, diagram 7. Explain doubly linked list in detail. Definition, insert, deletion-routine, diagram 8. Discuss the applications of stacks in brief. Balancing Symbols Infix to Postfix Conversion Postfix Expressions 9. Discuss various implementation of list in brief. Linked list implementation (stack, queue) Array implementation (stack, queue) Cursor implementation 10.Discuss briefly about array and linked list implementation for stack. 11. What are the various operations performed by singly linked list? 12. Write an algorithm for inserting and deleting an element from doubly linked list 13. What is a Stack? Explain with example. Definition of Stack Operations of Stack:PUSH and POP Example 14. Write a program to print out the elements of a singly linked list.

11 15. Given two sorted lists, L1 and L2, write a procedure to compute L1 L2 using only the basic list operations. 16. Given two sorted lists, L1 and L2, write a procedure to compute L1 L2 using only the basic list operations. 17. Write a program to evaluate a postfix expression. 18. Write the routines to implement queues using linked lists arrays 19. A deque is a data structure consisting of a list of items, on which the following operations are possible: push(x,d): Insert item x on the front end of deque d. pop(d): Remove the front item from deque d and return it. inject(x,d): Insert item x on the rear end of deque d. eject(d): Remove the rear item from deque d and return it. Write routines to support the deque that take O(1) time per operation.

12 UNIT-II (TREE STRUCTURES) 49. Define non-linear data structure? Data structure which is capable of expressing more complex relationship than that of physical adjacency is called non-linear data structure. 50. Define tree? A tree is a data structure, which represents hierarchical relationship between individual data items. 51. Define leaf? In a directed tree any node which has out degree o is called a terminal node or a leaf. 52. What is meant by directed tree? Directed tree is an acyclic diagraph which has one node called its root with indegree o whille all other nodes have indegree I. 53. What is a ordered tree? In a directed tree if the ordering of the nodes at each level is prescribed then such a tree is called ordered tree. 54. What is a Binary tree? A Binary tree is a finite set of elements that is either empty or is partitioned into three disjoint subsets.the first subset contains a single element called the root of the tree.the other two subsets are themselves binary trees called the left and right subtrees. 55. What are the applications of binary tree? Binary tree is used in data processing. a. File index schemes b. Hierarchical database management system 56. What is meant by traversing? Traversing a tree means processing it in such a way, that each node is visited only once. 57. What are the different types of traversing? The different types of traversing are a. Pre-order traversal-yields prefix from of expression. b. In-order traversal-yields infix form of expression. c. Post-order traversal-yields postfix from of expression.

13 58. What are the two methods of binary tree implementation? Two methods to implement a binary tree are, a. Linear representation. b. Linked representation 59. Define pre-order traversal? Pre-order traversal entails the following steps; a. Visit the root node b. Traverse the left subtree c. Traverse the right subtree 60. Define post-order traversal? Post order traversal entails the following steps; a. Traverse the left subtree b. Traverse the right subtree c. Visit the root node 61. Define in -order traversal? In-order traversal entails the following steps; a. Traverse the left subtree b. Visit the root node c. Traverse the right subtree 62. What is the length of the path in a tree? The length of the path is the number of edges on the path. In a tree there is exactly one path form the root to each node. 63. Define expression trees? The leaves of an expression tree are operands such as constants or variable names and the other nodes contain operators. 64. Define Strictly binary tree? If every nonleaf node in a binary tree has nonempty left and right subtrees,the tree is termed as a strictly binary tree. 65. Define complete binary tree? A complete binary tree of depth d is the strictly binary tree all of whose are at level d. 66. What is an almost complete binary tree? A binary tree of depth d is an almost complete binary tree if : Each leaf in the tree is either at level d or at level d-1

14 For any node nd in the tree with a right descendant at level d, all the left descendants of nd that are leaves are at level d. 67. TREE: A tree can be defined in several ways. One natural way to define a tree is recursively. A tree is a collection of nodes. The collection can be empty, which is sometimes denoted as A. Otherwise, a tree consists of a distinguished node r, called the root, and zero or more (sub)trees T1, T2,..., Tk, each of whose roots are connected by a directed edge to r.the root of each subtree is said to be a child of r, and r is the parent of each subtree root. Figure-1 shows a typical tree using the recursive definition. FIGURE-1 Generic tree From the recursive definition, we find that a tree is a collection of n nodes, one of which is the root, and n - 1edges. That there are n - 1 edges follows from the fact that each edge connects some node to its parent, and every node except the root has one parent A tree

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16 68. Find first child /next siblings for above figure: ANSWER:

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18 69. Expression Tree traversals Inorder traversal produces infix notation. This is a overly parenthesized notation. Print out the operator, then print put the left subtree inside parentheses,and then print out the right subtree inside parentheses. Postorder traversal produces postfix notation. Print out the left subtree, then print out the right subtree, and then printout the operator. Preorder traversal produces prefix notation. Print out the operator, then print out the right subtree, and then print out the left subtree. 70. Constructing an expression tree STEPS:

19 71. Binary Search Trees (BST) 72. Operations on BSTs

20 73. For the tree in Figure 4.63 : a. Which node is the root? b. Which nodes are leaves? 74. For each node in the tree of Figure 4.63 : a. Name the parent node. b. List the children. c. List the siblings. d. Compute the depth. e. Compute the height. 75. What is the depth of the tree in Figure 4.63? 76. (a)show the result of inserting 3, 1, 4, 6, 9, 2, 5, 7 into an initially empty binary search tree. (b)show the result of deleting the root.

21 77. Give the prefix, infix, and postfix expressions corresponding to the tree in Figure PART-B 1) Explain the different tree traversals with an application? 2) Define binary search tree? Explain the various operations with an example? 3) Write the function to insert, find and delete an element from a binary search tree. 4) What is a Binary tree? Explain Binary tree traversals in C? 5) Explain Representing lists as Binary tree? Write algorithm for finding K th element and deleting an element? 6) Discuss expression tree in brief. 7) Write an insert, delete, findmin, findmax, find routine for Binary search tree. 8) Write routines to implement the basic binary search tree operations.

22 UNIT-III (BALANCED SEARCH TREES AND INDEXING) 78. Define AVL tree AVL tree also called as height balanced tree.it is a height balanced tree in which every node will have a balancing factor of 1,0,1. Balancing factor of a node is given by the difference between theheight of the left sub tree and the height of the right sub tree. 79. Define Balancing factor Balancing factor of a node is given by the difference between the height of the left sub tree and the height of the right sub tree. BF=H l -H r where H l is the height of the left sub tree and H r is the height of the right sub tree 80. What are the various transformation performed in AVL tree? single rotation - single L rotation - single R rotation double rotation -LR rotation -RL rotation 81. What is meant by pivot node? The node to be inserted travel down the appropriate branch track along the way of the deepest level node on the branch that has a balance factor of +1 or -1 is called pivot node. 82. What is priority queue? A priority queue is a data structure that allows at least the following two operations: insert which does the obvious thing; and Deletemin, which finds,returns, and removes the minimum element in the priority queue. The Insert operation is the equivalent of Enqueue. Basic model of a priority queue

23 83. Applications of priority queues for scheduling purpose in operating system used for external sorting important for the implementation of greedy algorithm, which operate by repeatedly finding a minimum. 84. What are the main properties of a binary heap? Structure property Heap order property 85. Various implementation of binary heap/priority queue: Linked list, Binary search tree 86. Structure Property A heap is a binary tree that is completely filled, with the possible exception of the bottom level, which is filled from left to right. Such a tree is known as a complete binary tree A complete binary tree 87. Heap Order Property In a heap, for every node X, the key in the parent of X is smaller than (or equal to) the key in X, with the obvious exception of the root (which has no parent). In Figure: 2 the tree on the left is a heap, but the tree on the right is not (the dashed line shows the violation of heap order)

24 88. Basic Heap Operations: Insert Delete_min Decrease_key Increase_key Delete Build_heap Figure: 2 Two complete trees (only the left tree is a heap) 89. Insert HEAP: To insert an element x into the heap, we create a hole in the next available location, since otherwise the tree will not be complete. If x can be placed in the hole without violating heap order, then we do so and are done. Otherwise we slide the element that is in the hole's parent node into the hole, thus bubbling the hole up toward the root. We continue this process until x can be placed in the hole.

25 90. Delete_min HEAP: Delete_mins are handled in a similar manner as insertions. Finding the minimum is easy; the hard part is removing it. When the minimum is removed, a hole is created at the root. Since the heap now becomes one smaller, it follows that the last element x in the heap must move somewhere in the heap. If x can be placed in the hole, then we are done. This is unlikely, so we slide the smaller of the hole's children into the hole, thus pushing the hole down one level. We repeat this step until x can be placed in the hole. Thus, our action is to place x in its correct spot along a path from the root containing minimum children. In FIG:1 the left figure shows a heap prior to the delete_min. After 13 is removed, we must now try to place 31 in the heap. 31 cannot be placed in the hole, because this would violate heap order. Thus, we place the smaller child (14) in the hole, sliding the hole down one level (see Fig. 6.10). We repeat this again, placing 19 into the hole and creating a new hole one level deeper. We then place 26 in the hole and create a new hole on the bottom level. Finally, we are able to place 31 in the hole (Fig. 6.11). This general strategy is known as a percolate down. We use the same technique as in the insert routine to avoid the use of swaps in this routine. FIG:1 Creation of the hole at the root

26 91. What is the need for hashing? Hashing is used to perform insertions, deletions and find in constant average time. 92. Define hash function? Hash function takes an identifier and computes the address of that identifier in the hash table using some function. 93. List out the different types of hashing functions? The different types of hashing functions are, a. The division method b. The mind square method c. The folding method d. Multiplicative hashing e. Digit analysic 94. What are the problems in hashing? a. Collision b. Overflow 95. What are the problems in hashing? When two keys compute in to the same location or address in the hash table through any of the hashing function then it is termed collision. 96. General idea of hashing and what is the use of hashing function: A hash table similar to an array of some fixes size-containing keys. The keys specified here might be either integer or strings, the size of the table is taken as table size or the keys are mapped on to some number on the hash table from a range of 0 to table size

27 97. HASH FUNCTION: H(X) = X MOD TABLESIZE 98. Show the result of inserting 2, 1, 4, 5, 9, 3, 6, 7 into an initially empty AVL tree. 99. Open Hashing (Separate Chaining) is to keep a list of all elements that hash to the same value Eg: 0,1,4,25,16,9,49,36,64,81 H(X) = X MOD TABLESIZE Hash(x) = x mod 10. Open hashing has the disadvantage of requiring pointers Closed Hashing (Open Addressing) Closed hashing, also known as open addressing, is an alternative to resolving collisions with linked lists. In a closed hashing system, if a collision occurs, alternate cells are tried until an empty cell is found Three common collision resolution strategies of Closed Hashing (Open Addressing): Linear Probing Quadratic Probing Double Hashing

28 102. Linear Probing In linear probing, is a linear function of i, typically (i) = i. This amounts to trying cells sequentially (with wraparound) in search of an empty cell Quadratic Probing Quadratic probing is a collision resolution method that eliminates the primary clustering problem of linear pro bing. Quadratic probing is what you would expect-the collision function is quadratic. The popular choice is f (i) = i Double Hashing For double hashing, one popular choice is f(i) = i* h2(x). (ie) h1(x)=x mod tablesize h2(x) = R - (x mod R), with R a prime smaller than tablesize, 105. Primary clustering Primary clustering means that any key that hashes into the cluster will require several attempts to resolve the collision, and then it will add to the cluster Properties of B-tree: A B-tree of order m is a tree with the following structural properties: The root is either a leaf or has between 2 and m children. All nonleaf nodes (except the root) have between m/2 and m children. All leaves are at the same depth A B-tree of order 4 is more popularly known as a tree, and a B-tree of order 3 is known as a 2-3 tree Show the result of inserting the following keys into an initially empty 2-3 tree: 3, 1, 4, 5, 9, 2, 6, 8, 7, Show the result of inserting 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, and 2, one at a time, into an initially empty binary heap. PART-B 1. Define AVL trees? Explain the LL, RR, RL, LR case with an example? 2. Discuss various rotations of AVL tree in brief. 3. Discuss single rotation with its routine in brief. 4. Explain double rotation with its routine in detail. 5. Show the result of inserting the following words into the AVL tree: Madhushri, Kanchan, Anagha, Rohit, Neeta, Snehal, Pratibha, Lalitha, Mrudul, Kavitha, Apurva, Harsha.

29 6. Define priority queue? Explain the basic heap operation with an example? 7. Explain any two techniques to overcome hash collision? Separate chaining (open hash) a. Example b. Explanation c. Coding Open addressing (close addressing) Linear probing Quadratic probing Double hashing 8. What are the various methods involved to solve hashing function? 9. Discuss binary heap or priority queue in brief. 10. Write and test a program that performs the operations insert, delete_min, build_heap, find_min, decrease_key, delete, and increase_key in a binary heap. 11. Write note on B-tree or (2-3-4 tree, 2-3 tree) 12. Given input {4371, 1323, 6173, 4199, 4344, 9679, 1989} and a hash function h(x) = x (mod 10), show the resulting a. open hash table b. closed hash table using linear probing c. closed hash table using quadratic probing d. closed hash table with second hash function h2(x) = 7 - (x mod 7)

30 UNIT-IV (GRAPHS) 110. Define Graph? A graph G consist of a nonempty set V which is a set of nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to a set of pairs of elements of V. It can also be represented as G=(V, E) Define adjacent nodes? Any two nodes which are connected by an edge in a graph are called adjacent nodes. For example, if and edge xîe is associated with a pair of nodes (u,v) where u, v Î V, then we say that the edge x connects the nodes u and v What is a directed graph? A graph in which every edge is directed is called a directed graph What is a undirected graph? A graph in which every edge is undirected is called a directed graph What is a loop? An edge of a graph which connects to itself is called a loop or sling What is a simple graph? A simple graph is a graph, which has not more than one edge between a pair of nodes than such a graph is called a simple graph What is a weighted graph? A graph in which weights are assigned to every edge is called a weighted graph Define out degree of a graph? In a directed graph, for any node v, the number of edges which have v as their initial node is called the out degree of the node v Define indegree of a graph? In a directed graph, for any node v, the number of edges which have v as their terminal node is called the indegree of the node v Define path in a graph? The path in a graph is the route taken to reach terminal node from a starting node.

31 120. What is a simple path? A path in a diagram in which the edges are distinct is called a simple path. It is also called as edge simple What is a cycle or a circuit? A path which originates and ends in the same node is called a cycle or circuit What is an acyclic graph? A simple diagram which does not have any cycles is called an acyclic graph What is meant by strongly connected in a graph? An undirected graph is connected, if there is a path from every vertex to every other vertex. A directed graph with this property is called strongly connected When is a graph said to be weakly connected? When a directed graph is not strongly connected but the underlying graph is connected, then the graph is said to be weakly connected Name the different ways of representing a graph? a. Adjacency matrix b. Adjacency list 126. What is an undirected acyclic graph? When every edge in an acyclic graph is undirected, it is called an undirected acyclic graph. It is also called as undirected forest What are the two traversal strategies used in traversing a graph? a. Breadth first search b. Depth first search 128. What is a minimum spanning tree? A minimum spanning tree of an undirected graph G is a tree formed from graph edges that connects all the vertices of G at the lowest total cost What is a forest? A forest may be defined as an acyclic graph in which every node has one or no predecessors.a tree may be defined as a forest in which only a single node called root has no predecessors.

32 130. Define articulation points. If a graph is not biconnected,the vertices whose removal would disconnect the graph are known as articulation points Explain about Adjacency Matrix Adjacency matrix consists of a n*n matrix where n is the no. of vertices present In the graph,which consists of values either 0 or Explain about Adjacency linked list. It consists of a table with the no. of entries for each vertex for each entry a linked List is inititated for the vertices adjacent to the corresponding table entry What is a single souce shortest path problem? Given as an input,a weighted graph,g= and a distinguished vertex S as the source vertex.single source shortest path problem finds the shortest weighted path from s to every other vertex in G Explain about Unweighted shortest path. Single source shortest path finds the shortest path from the source to each and every vertex present in a unweighted graph.here no cost is associated with the edges connecting the vertices.always unit cost is associated with each edge Explain about Weighted shortest path Single source shortest path finds the shortest path from the source to each and Every vertex present In a weighted graph.in a weighted graph some cost is always Associated with the edges connecting the vertices What are the methods to solve minimum spanning tree? a) Prim s algorithm b) Kruskal s algorithm 137. Explain briefly about Prim s algorithm Prim s algorithm creates the spanning tree in various stages.at each stage,a node is picked as the root and an edge is added and thus the associated vertex along with it Define a depth first spanning tree. The tree that is formulated by depth first search on a graph is called as depth first spanning tree. The depth first spanning tree consists of tree edges and back edges What is a tree edge? Traversal from one vertex to the next vertex in a graph is called as a tree edge.

33 140. What is a back edge? The possibility of reaching an already marked vertex is indicated by a dashed line, in a graph is called as back edge 141. What are the conditions for a graph to become a tree? A graph is a tree if it has two properties. i. If it is a connected graph. ii. There should not be any cycles in the graph Define Acyclic graph. A graph with no cycles is called Acyclic graph.a directed graph with no Edges is called as a directed Acyclic graph (or) DAG.DAGS are used for compiler Optimization process A graph G = (V, E) consists of a set of vertices, V, and a set of edges, E. Each edge is a pair (v,w), where v,w V. Edges are sometimes referred to as arcs If the pair is ordered, then the graph is directed. Directed graphs are sometimes referred to as digraphs In an undirected graph with edge (v,w), and hence (w,v), w is adjacent to v and v is adjacent to w. Sometimes an edge has a third component, known as either a weight or a cost A cycle in a directed graph is a path of length at least 1 such that w1 = wn; this cycle is simple if the path is simple An undirected graph is connected if there is a path from every vertex to every other vertex. A directed graph with this property is called strongly connected 148. If a directed graph is not strongly connected, but the underlying graph (without direction to the arcs) is connected, then the graph is said to be weakly connected A complete graph is a graph in which there is an edge between every pair of vertices A topological sort is an ordering of vertices in a directed acyclic graph, such that if there is a path from vi to vj, then vj appears after vi in the ordering.

34 151. SINGLE-SOURCE SHORTEST-PATH PROBLEM: Given as input a weighted graph, G = (V, E), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G Prim's Algorithm One way to compute a minimum spanning tree is to grow the tree in successive stages. In each stage, one node is picked as the root, and we add an edge, and thus an associated vertex, to the tree Kruskal's Algorithm A second greedy strategy is continually to select the edges in order of smallest weight and accept an edge if it does not cause a cycle Applications of Depth-First Search Undirected Graphs Biconnectivity Euler Circuits Directed Graphs 155. Biconnectivity: A connected undirected graph is biconnected if there are no vertices whose removal disconnects the rest of the graph. PART-B 1. Explain the various representation of graph with example in detail? Adjacency matrix Figure Explanation Table Adjacency list Figure Explanation Table 2. Define topological sort? Explain with an example? Definition Explanation Example Table Coding

35 3. Explain Dijkstra's algorithm with an example? Explanation Example Graph Table Coding 4. Explain any one shortest path algorithms in detail Dijkstra's algorithm All pair shortest path algorithm 5. Explain Depth first and breadth first traversal. 6. Discuss minimum spanning tree in brief. 7. Explain Prim's algorithm with an example? Explanation Example Graph Table Coding 8. Explain Krushal's algorithm with an example? Explanation Example Graph Table Coding 9. Find a topological ordering for the graph in Figure -a Figure a

36 10. Find the shortest path from A to all other vertices for the graph in Figure Find the shortest unweighed path from B to all other vertices for the graph in Figure Figure Find a minimum spanning tree for the graph in Figure 9.82 using both Prim's and Kruskal's algorithms. Is this minimum spanning tree unique? Why? Figure Find all the articulation points in the graph in Figure Show the depth-first spanning tree and the values of num and low for each vertex. Figure 9.83

37 13. Discuss the application of depth first search in brief. Directed graph Biconnectivity Euler Circuits

38 UNIT-V PART-A 1. Greedy algorithm A greedy algorithm is any algorithm that follows the problem solving metaheuristic of making the locally optimal choice at each stage [1] with the hope of finding the global optimum. 2. Divide and Conquer Divide: Smaller problems are solved recursively (except, of course, base cases). Conquer: The solution to the original problem is then formed from the solutions to the subproblems. 3. Greedy Algorithm: Example: 1. Dijkstra s algorithm 2. Prim s Algorithm 3. Kruskal s Algorithm 4. Divide and Conquer algorithms Divide and Conquer algorithms break the problem into several subproblems that are similar to the original problem but smaller in size, solve the subproblems recursively, and then combine these solutions to create a solution to the original problem. 5. Divide and Conquer Example: Merge Sort algorithm, Quick sort 6. Traveling Salesperson Problem: Traveling salesperson problem is to find the shortest path in directed graph that starts at a given vertex, visits each vertex in the graph exactly once. Such a path is called an optimal tour. 7. BRANCH AND BOUND A B&B algorithm searches the complete space of solutions for a given problem for the best solution. Example: O/I Knapsack problem Traveling salesman problem

39 8. Define NP. NP stands for Non-deterministic Polynomial time. This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine. 9. What is P? P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine. Since it can solve in polynomial time, it can also be verified in polynomial time. Therefore P is a subset of NP. 10. What is NP-Complete? A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly (I e. in polynomial time) transformed into x. In other words: 1. x is in NP, and 2. Every problem in NP is reducible to x 11. Dynamic programming Dynamic programming is a method for efficiently solving a broad range of search and optimization problems which exhibit the characteristics of overlapping sub problems and optimal substructure. Example: All pairs shortest path, Optimal binary search tree 1. Explain greedy algorithm in detail. 1. Dijkstra s algorithm 2. Prim s Algorithm 3. Kruskal s Algorithm PART-B 2. Discuss divide and conquer algorithm in brief. Example: Merge Sort algorithm, Quick sort 3. Explain Dynamic programming in detail. All pairs shortest path, Optimal binary search tree 4. Write note on the following: a) Backtracking b) Branch and bound (hints: traveling salesman problem) c) Randomized algorithms d) NP-complete problems (hints: shortest path algorithms, traveling salesman problem)