Short question and answer for Data Structure and Analysis Of Algorithms

Short question and answer for Data Structure and Analysis Of Algorithms 

UNIT I – COMPLEXITY ANALYSIS & ELEMENTARY DATA STRUCTURES

1. What do asymptotic notation means?

             Asymptotic notations are terminology that is introduced to enable us to make meaningful statements about the time and space complexity of an algorithm. The different notations are  

Big – Oh notation

Omega notation  

Theta notation.

2. What are the basic asymptotic efficiency classes?

The various basic efficiency classes are

Constant : 1

Logarithmic : log n

Logarithmic : log n

Linear  : n

N-log-n : nlog n

Quadratic : n²

Cubic  : n³

 Exponential : 2ⁿ

Factorial  : n!

3. Define O-notation?

A function t(n) is said to be in O(g(n)), denoted by t(n)

O(g(n)), if t(n) is bounded above by some constant multiple of g(n) for all large n, i.e., if there exists some positive constant c and some nonnegative integer

n0 such that  T(n) ” cg(n) for all n • n0

4. What are exponential growth functions?

The functions 2ⁿ and n! are exponential growth functions, because these two functions grow so fast that their values become astronomically large even for rather smaller values of n. 

5. What is worst-case efficiency?

The worst-case efficiency of an algorithm is its efficiency for the worst-case input of size n, which is an input or inputs of size n for which the algorithm runs the longest among all possible inputs of that size. 

6. What is best-case efficiency?

The best-case efficiency of an algorithm is its efficiency for the best-case input of size n, which is an input or inputs for which the algorithm runs the fastest among all possible inputs of that size.  

7. What is average case efficiency?

The average case efficiency of an algorithm is its efficiency for an average case input of size n. It provides information about an algorithm behavior on a “typical” or “random” input.  

8. What is amortized efficiency?

In some situations a single operation can be expensive, but the total time for the entire sequence of n such operations is always significantly better that the worst case efficiency of that single operation multiplied by n. this is called amortized efficiency.

9. What is the formula used in Euclid’s algorithm for finding the greatest common divisor of two numbers?

Euclid’s algorithm is based on repeatedly applying the equality  Gcd(m,n)=gcd(n,m mod n) until m mod n is equal to 0, since gcd(m,0)=m.  

10. Define articulation points. 

If a graph is not biconnected,the vertices whose removal would disconnect the graph are known as articulation points.

11. Give some example of NP complete problems.

i. Hamiltonian circuit.

ii. Travelling salesmen problems

iii. Longest path problems

iv. Bin packing

v. Knapsack problem

vi. Graph colouring problem

12. What is the recurrence relation to find out the number of multiplications and the initial condition for finding the n-th factorial number?

The recurrence relation and initial condition for the number of multiplications is

  M(n)=M(n-1)+1 for n>0  M(0)=0

13. What is the basic operation in the Tower of Hanoi problem and give the recurrence relation for the number of moves?

The moving of disks is considered the basic operation in the Tower of Hanoi problem and the recurrence relation for the number of moves is given as  M(n)=2M(n)+1 for n>1  M(1)=1

14. Who introduced the Fibonacci numbers and how can it be defined by a simple recurrence?

Leonardo Fibonacci introduced the fibonacci numbers in 1202 as a solution to a problem about the size of rabbit population. It can be defined by the simple recurrence   F(n)=F(n-1)+F(n-2) for n>1 And two initial conditions  F(0)=0 and F(1)=1

15. What is the explicit formula for the nth Fibonacci number?

The formula for the nth Fibonacci number is given by 

F(n)= 1/5(Φn - Φn)

 Where 

Φ =(1+5)/2  

Φ =(1-5)/2

16. Define Data Structures

Data Structures is defined as the way of organizing all data items that consider not

only the elements stored but also stores the relationship between the elements.

17. Define Linked Lists

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 theNext Pointer. The last cell’sNext pointer points to NULL.

18. State the different types of linked lists

The different types of linked list include singly linked list, doubly linked list and

circular linked list.

11. List the basic operations carried out in a linked list

The basic operations carried out in a linked list include:

• Creation of a list

• Insertion of a node

• Deletion of a node

• Modification of a node

• Traversal of the list

19. List out the advantages of using a linked list

• It is not necessary to specify the number of elements in a linked list during its

declaration

• Linked list can grow and shrink in size depending upon the insertion and deletion

that occurs in the list

• Insertions and deletions at any place in a list can be handled easily and efficiently

• A linked list does not waste any memory space 

20. List out the applications of a linked list

Some of the important applications of linked lists are manipulation of

polynomials, sparse matrices, stacks and queues.

21. State the difference between arrays and linked lists

Arrays	Linked Lists
Size of an array is fixed	Size of a list is variable
It is necessary to specify the number of elements during declaration.	It is not necessary to specify the number of elements during declaration
Insertions and deletions are somewhat difficult	Insertions and deletions are carried out easily
It occupies less memory than a linked list for the same number of elements	It occupies more memory

22. What do you mean by balanced trees?

Balanced trees have the structure of binary trees and obey binary search tree properties. Apart from these properties, they have some special constraints, which differ from one data structure to another. However, these constraints are aimed only at reducing the height of the tree, because this factor determines the time complexity.

Eg: AVL trees, Splay trees. 

23. What is the use of threaded binary tree?

In threaded binary tree, the NULL pointers are replaced by some addresses. The left pointer of the node points to its predecessor and the right pointer of the node points to its successor. 

                                                                       

24.Construction of expression trees?                          

1.convert the given infix expression into postfix notation  

2. Create a stack and read each character of the expression and push into the stack, if   operands are encountered.   

3.when an operator is encountered pop 2 values from the stack. From a tree using the operator.

25. Why you need a data structure?

A data structure helps you to understand the relationship of one data element with the other and organize it within the memory. Sometimes the organization might be simple and can be very clearly visioned. Eg) List of names of months in a year –Linear Data Structure, List of historical places in the world- Non-Linear Data Structure. A data structure helps you to analyze the data, store it and organize it in a logical and mathematical manner. 

UNIT II – HEAP STRUCTURES

1. Define Heap.

A heap is a partially ordered data structure, and can be defined as a binary tree assigned to its nodes, one key per node, provided the following two conditions are met

i.The tree’s shape requirement-The binary tree is essentially complete, that is all the leaves are full except possibly the last level, where only some rightmost leaves will be missing.

ii.The parental dominance requirement-The key at each node is greater that or equal to the keys of its children 

2. What are the properties of binary heap?

i) Structure Property

ii) Heap Order Property

3. What is a min-heap?

 A min-heap is a mirror image of the heap structure. It is a complete binary tree in which every element is less than or equal to its children. So the root of the min-heap contains the smallest element.  

4. what  is maxheap?

              If we want the elements in the more typical increasing sorted order,we can change the ordering property so that the parent has a larger key than the child.it is called max heap

5. What do you mean by structure property in a heap?

A heap is a binary tree that is completely filled with the possible exception at the bottom level, which is filled from left to right. Such a tree is known as a complete binary tree.

6. What do you mean by 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 exception of the root (which has no parent).

7. What are the applications of priority queues?

• The selection problem

• Event simulation

8. What is the main use of heap?

Heaps are especially suitable for implementing priority queues. Priority queue is a set of items with orderable characteristic called an item’s priority, with the following operations

i. Finding an item with the highest priority

ii.Deleting an item with highest priority

iii.Adding a new item to the set

9. Give three properties of heaps?

The properties of heap are 

      There exists exactly one essentially complete binary tree with ‘n’ nodes. Its height is equal to log2n

      The root of the heap is always the largest element

      A node of a heap considered with all its descendants is also a heap 

10. Give the main property of a heap that is implemented as an array.

A heap can be implemented as an array by recording its elements in the top-down, left-to-right fashion. It is convenient to store the heap’s elements in positions 1 through n of such an array. In such a representation

i. The parental node keys will be in the first n/2 positions of the array, while the leaf keys will occupy the last n/2 positions

ii.The children of a key in the array’s parental position ‘i’ (1≤i≤n/2) will be in positions 2i and 2i+1and correspondingly, the parent of the key in position ‘i’(2≤i≤n) will be in position i/2. 

11. What are the two alternatives that are used to construct a heap?

The two alternatives to construct a heap are 

      Bottom-up heap construction

      Top-down heap construction

12. What is the algorithm to delete the root’s key from the heap?

ALGORITHM

i.Exchange the root’s key with the last key K of the heap

ii.Decrease the heap’s size by one

iii.“Heapify” the smaller tree by sifting K down the tree exactly in the same way as bottom-up heap construction. Verify the parental dominance for K: if it holds stop the process, if not swap K with the larger of its children and repeat this operation until the parental dominance holds for K in its new position.

13. What do you mean by the term “Percolate up”?

To insert an element, we have to create a hole in the next available heap location. Inserting an element in the hole would sometimes violate the heap order property, so we have to slide down the parent into the hole. This strategy is continued until the correct location for the new element is found. This general strategy is known as a percolate up; the new element is percolated up the heap until the correct location is found.

14. What do you mean by the term “Percolate down”?

When the minimum element 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. This general strategy is known as percolate down. 

15. What is meant by Implicit Heaps?

        A particularly simple and beautiful implementation of the heap structure is the implicit heap. Data is simply put into an array and the childrens of element x[i] are defined to be the elements x[2i] and x[2i+1]. Thus the parent of the element c can be found at c/2 (integer division).

16.What is a skew heap? 

A heap data structure that is stored in a binary tree (not necessarily complete and balanced). The insertion and deletion of elements in the tree come from merging two skew heaps together in such a way that the heap property is preserved and the tree does not degrade to a linear tree.

17. How to merging 2 skew heaps?

Suppose we are merging a heap containing the elements 2, 5, and 7 with a heap containing the elements 4 and 6.

                    7  <- merge ->    6

                   / \                     /

                  5   2                             4

 i.   Identify the root, thus 7 becomes the new root and the left

   subtree of the heap with root 7 becomes the right subtree of

   the other heap:

                        7          2 <- merge -> 6   (to form the left subtree

                         \                           /     of the new skew heap)

                          5                        4

  ii.

                                           7

                                          / \

                                         6   5

                                          \

                                           4   <- merge -> 2

iii.

                                           7

                                          / \

                                         6   5

                                        / \

                                       2   4 

18. Define Fabinacci Heap.

The Fibonacci heap is a data structure that supports all the basic heap operations in O(1) amortized time, with the exception of delete_min and delete, which take O (log n) amortized time. It immediately follows that the heap operations in Dijkstra's algorithm will require a total of O(|E| + |V| log |V|) time.

19. What is Lazy merging?
 Two heaps are merged only when it is required to do so. This is similar to lazy deletion. For lazy merging, merges are cheap, but because lazy merging does not actually combine trees, the delete_min operation could encounter lots of trees, making that operation expensive. Any one delete_min could take linear time, but it is always possible to charge the time to previous merge operations. In particular, an expensive delete_min must have been preceded by a large number of unduly cheap merges, which have been able to store up extra potential. 

20. Write the amortized analysis of Lazy Binomial Queues

To carry out the amortized analysis of lazy binomial queues, we will use the same potential function that was used for standard binomial queues. Thus, the potential of a lazy binomial queue is the number of trees.

21. Binomial heaps:

A set of binomial trees satisfying the following:

1.  Each binomial tree in H is heap-ordered:

            the key of a node is greater than or equal to the key of its parent

2. There is at most one binomial tree in H whose root has a given degree

22.Write the Dijkstra’s shortest path algorithm

Let G = (V,E) be a weighted (weights are non-negative) undirected graph, let s Î V. Want to find the distance (length of the shortest path), d(s,v) from s to every other vertex.

UNIT III – SEARCH STRUCTERS

1. What is binary search?

Binary search is a remarkably efficient algorithm for searching in a sorted array. It works by comparing a search key K with the arrays middle element A[m]. if they match the algorithm stops; otherwise the same operation is repeated recursively for the first half of the array if K < A[m] and the second half if K > A[m].

                                        K  

A[0]………A[m-1] A[m] A[m+1]………A[n-1]

                      search here if K<A[m]                           search here if K>A[m]

2. What is a binary tree extension and what is its use?

 The binary tree extension can be drawn by replacing the empty subtrees by special nodes in a binary tree. The extra nodes shown as little squares are called external & the original nodes shown as little circles called internal. The extension of a empty binary tree is a single external node. The binary tree extension helps in analysis of tree algorithms.

3. What are the classic traversals of a binary tree?

 The classic traversals are as follows

i. Preorder traversal: the root is visited before left & right subtrees 

ii. Inorder traversal: the root is visited after visiting left subtree and before visiting right subtree

iii. Postorder traversal: the root is visited after visiting the left and right subtrees

4. Mention an algorithm to find out the height of a binary tree. 

ALGORITHM

Height(T) //Compares recursively the height of a binary tree

//Input: A binary tree T //Output: The height of T

            if T = Φ

return –1

else

return max{Height(TL),

 Height(TR)}+1

5. What are binary search trees and what is it mainly used for?

 Binary search trees is one of the principal data structures for implementing dictionaries. It is a binary tree whose nodes contain elements of a set of orderable items, one element per node, so that all elements in the left subtree are smaller than the element in the subtree’s root and all elements in the right subtree are greater than it.

6. Define AVL trees and who was it invented by? 

An AVL tree is a binary search tree in which the balance factor of every node, which is defined as the difference between the heights of the node’s left and right subtrees, is either 0 or +1 or –1. the height of an empty subtree is defined as –1. AVL trees were invented in 1962 by two Russian scientists, G.M.Adelson-Velsky and E.M.Landis, after whom the data struture is named. 

7.  Define AVL Tree. 

An empty tree is height balanced. If T is a non-empty binary tree with TL and 

TR as its left and right subtrees, then T is height balanced if

i) TL and TR are height balanced and

ii) │hL - hR│≤ 1

Where hL and hR are the heights of TL and TR respectively.

8. What are the various transformation performed in AVL tree?  

                   1.single rotation   - single L rotation  - single R rotation 

                   2.double rotation  -LR rotation          -RL rotation 

9. What are the categories of AVL rotations?

Let A be the nearest ancestor of the newly inserted nod which has the balancing factor ±2. Then the rotations can be classified into the following four categories:

Left-Left: The newly inserted node is in the left subtree of the left child of A.

Right-Right: The newly inserted node is in the right subtree of the right child of  A.

Left-Right: The newly inserted node is in the right subtree of the left child of A.

Right-Left: The newly inserted node is in the left subtree of the right child of A.

10. What do you mean by balance factor of a node in AVL tree?

The height of left subtree minus height of right subtree is called balance factor of a node in AVL tree.The balance factor may be either 0 or +1 or -1.The height of an empty tree is -1.

11. Write about the efficiency of AVL trees?

As with any search tree , the critical characteristic is the tree’s height. The tree’s height is bounded above and below by logarithmic functions. The height ‘h’ of any AVL tree with ‘n’ nodes satisfies the inequalities

log2 n ≤ h < 1.4405 log2(n+2) – 1.3277

The inequalities imply that the operations of searching and insertion are θ(log n) in the worst case. The operation of key deletion in an AVL tree is more difficult than insertion, but it turns out to have the same efficiency class as insertion i.e., logarithmic

12. What is the minimum number of nodes in an AVL tree of height h? 

The minimum number of nodes S(h), in an AVL tree of height h is given

by S(h)=S(h-1)+S(h-2)+1. For h=0, S(h)=1. 

13. What are 2-3 trees and who invented them? 

A 2-3 tree is a tree that can have nodes of two kinds:2-nodes and 3-nodes. A 2-node contains a single key K and has two children, the left child serves as the root of a subtree whose keys are less than K and the right child serves as the root of a subtree with keys greater than K.  A 3-node contains two ordered keys K1 & K2 (K1<K2). The leftmost child serves as the root of a subtree with keys less than K1, the middle child serves as the root of a subtree with keys between K1 & K2 and the rightmost child serves as the root of a subtree with keys greater than K2. The last requirement of 2-3 trees is that all its leaves must be on the same level, a 2-3 tree is always height balanced. 2-3 trees were introduced by John Hopcroft in 1970.

14. What do you mean by 2-3-4 tree? 

A B-tree of order 4 is called 2-3-4 tree. A B-tree of order 4 is a tree that is not

binary with the following structural properties:

• The root is either a leaf or has between 2 and 4 children.

• All non-leaf nodes (except the root) have between 2 and 4 children.

• All leaves are at the same depth.

14. Define B-tree of order M. 

A B-tree of order M is a tree that is not binary with the following structural 

properties:

• The root is either a leaf or has between 2 and M children.

• All non-leaf nodes (except the root) have between ┌M/2┐ and M children.

• All leaves are at the same depth.

15. What are the applications of B-tree? 

• Database implementation

• Indexing on non primary key fields 

16. Definition of a red-black tree

A red-black tree is a binary search tree which has the following red-black properties:

Every node is either red or black. Every leaf (NULL) is black. If a node is red, then both its children are black. Every simple path from a node to a descendant leaf contains the same number of black nodes.

	A basic red-black tree
	Basic red-black tree with the sentinel nodes added. Implementations of the red-black tree algorithms will usually include the sentinel nodes as a convenient means of flagging that you have reached a leaf node. They are the NULL black nodes of property 2.

17. Define splay tree. 

A splay tree is a binary search tree in which restructuring is done using a scheme called splay. The splay is a heuristic method which moves a given vertex v to the root of the splay tree using a sequence of rotations.

18. What is the idea behind splaying? 

Splaying reduces the total accessing time if the most frequently accessed node is moved towards the root. It does not require to maintain any information regarding the height or balance factor and hence saves space and simplifies the code to some extent.

19. List the types of rotations available in Splay tree. 

Let us assume that the splay is performed at vertex v, whose parent and

grandparent are p and g respectively. Then, the three rotations are named as:

i. Zig: If p is the root and v is the left child of p, then left-left rotation at p would

suffice. This case always terminates the splay as v reaches the root after this

rotation.

ii. Zig-Zig: If p is not the root, p is the left child and v is also a left child, then a left-

left rotation at g followed by a left-left rotation at p, brings v as an ancestor of g

as well as p.

iii. Zig-Zag: If p is not the root, p is the left child and v is a right child, perform a

left-right rotation at g and bring v as an ancestor of p as well as g.

20. Define brute force string matching. 

The brute force string matching has a given string of n characters called the text and a string of m characters called the pattern, find a substring of the text that matches the pattern. And find the index I of the leftmost character of the first matching substring in the text.

21. What are the advantages of brute force technique? 

The various advantages of brute force technique are

i. Brute force applicable to a very wide variety of problems. It is used for many elementary but important algorithmic tasks 

ii.For some important problems this approach yields reasonable algorithms of at least some practical value with no limitation on instance size

iii.The expense to design a more efficient algorithm may be unjustifiable if only a few instances of problems need to be solved and a brute force algorithm can solve those instances with acceptable speed

iv. Even if inefficient in general it can still be used for solving small-size instances of a problem

v. It can serve as a yardstick with which to judge more efficient alternatives for solving a problem

22. What are the properties of binary heap? 

i) Structure Property

ii) Heap Order Property 

UNIT IV – GREEDY & DIVIDE AND CONQUER

1. Give the general plan for divide-and-conquer algorithms.

The general plan is as follows 

i.A problems instance is divided into several smaller instances of the same problem, ideally about the same size

ii.The smaller instances are solved, typically recursively

iii.If necessary the solutions obtained are combined to get the solution of the original problem

2. State the Master theorem and its use.

If f(n) εθ(nd) where d ≥ 0 in recurrence equation T(n) = aT(n/b)+f(n), then 

    

θ(nd)  if a<b^d

  ^T(n)

  

θ(ndlog n) if a=b^d

     

θ(nlogba) if a>b^d

The efficiency analysis of many divide-and-conquer algorithms are greatly simplified by the use of Master theorem.

3. What is the general divide-and-conquer recurrence relation?

 An instance of size ‘n’ can be divided into several instances of size n/b, with ‘a’ of them needing to be solved. Assuming that size ‘n’ is a power of ‘b’, to simplify the  analysis, the following recurrence for the running time is obtained:  T(n) = aT(n/b)+f(n)  Where f(n) is a function that accounts for the time spent on dividing the problem into smaller ones and on combining their solutions.

4. What is decrease and conquer approach and mention its variations?

 The decrease and conquer technique based on exploiting the relationship between a solution to a given instance of a problem and a

solution to a smaller instance of the same problem. The three major variations are

      Decrease by a constant

      Decrease by a constant-factor

      Variable size decrease

5. What is a tree edge and back edge?

 In the depth first search forest, whenever a new unvisited vertex is reached for the first time, it is attached as a child to the vertex from which it is being reached. Such an edge is called tree edge because the set of all such edges forms a forest. The algorithm encounters an edge leading to a previously visited vertex other than its immediate predecessor. Such an edge is called a back edge because it connects a vertex to its ancestor, other than the parent, in the depth first search forest.

 6. What is a tree edge and cross edge?

 In the breadth first search forest, whenever a new unvisited vertex is reached for the first time, it is attached as a child to the vertex from which it is being reached. Such an edge is called tree edge. If an edge is leading to a previously visited vertex other than its immediate predecessor, that edge is noted as cross edge.

7. What is transform and conquer technique?

The group of design techniques that are based on the idea of transformation is called transform and conquer technique because the methods work as two stage procedures. First in the transformation stage, the

problem’s instance is modified to be more amenable (agreeable) to the solution. Then in the second or conquering stage, it is solved.

8. What is greedy technique?

 Greedy technique suggests a greedy grab of the best alternative available in the hope that a sequence of locally optimal choices will yield a globally optimal solution to the entire problem. The choice must be made as follows

i.Feasible : It has to satisfy the problem’s constraints

ii.Locally optimal : It has to be the best local choice among all feasible choices available on that step.

iii.Irrevocable : Once made, it cannot be changed on a subsequent step of the algorithm  

9. What is a state space tree?

The processing of backtracking is implemented by constructing a tree of choices being made. This is called the state-space tree. Its root represents a initial state before the search for a solution begins. The nodes of the first level in the tree represent the choices made for the first component of the solution, the nodes in the second level represent the choices for the second component and so on.

 10. What is a promising node in the state-space tree?

A node in a state-space tree is said to be promising if it corresponds to a partially constructed solution that may still lead to a complete solution. 

11. What is a non-promising node in the state-space tree?

A node in a state-space tree is said to be promising if it corresponds to a partially constructed solution that may still lead to a complete solution; otherwise it is called non-promising. 

12. What do leaves in the state space tree represent?

 Leaves in the state-space tree represent either non-promising dead ends or complete solutions found by the algorithm.

13. What is the manner in which the state-space tree for a backtracking algorithm is constructed?

In the majority of cases, a state-space tree for backtracking algorithm is constructed in the manner of depth-first search. If the current node is promising, its child is generated by adding the first remaining legitimate option for the next component of a solution, and the processing moves to this child. If the current node turns out to be non-promising, the algorithm backtracks to the node’s parent to consider the next possible solution to the problem, it either stops or backtracks to continue searching for other possible solutions.

14. What is a feasible solution and what is an optimal solution?

In optimization problems, a feasible solution is a point in the problem’s search space that satisfies all the problem’s constraints, while an optimal solution is a feasible solution with the best value of the objective function.

15. Define Divide and Conquer algorithm? 

Divide and Conquer algorithm is based on dividing the problem to be solved into several, smaller sub instances, solving them independently and then combining the sub instances solutions so as to yield a solution for the original instance.

16. Mention some application of Divide and Conquer algorithm? 

a. Quick Sort  b. Merge Sort  c. Binary search

17. What are the two stages for heap sort?

Stage 1 : Construction of heap Stage 2 : Root deletion N-1 times 

18. What is divide and conquer strategy ?

In divide and conquer strategy the given  problem is divided into smaller                                                  

Problems and solved  recursively. The conquering phase consists of patching together the answers .  Divide – and – conquer is a very  powerful use of recursion that we will see many times.

19. What do you mean by separate chaining?

Separate chaining is a collision resolution technique to keep the list of all elements that hash to the same value. This is called separate chaining because each hash table element is a separate chain (linked list). Each linked list contains all the elements whose keys hash to the same index.

20. Write the advantage  and Disadvantages of separate chaining.

Adv:

• More number of elements can be inserted as it uses linked lists.

Dis Adv.

• The elements are evenly distributed. Some elements may have more

elements and some may not have anything.

• It requires pointers. This leads to slow the algorithm down a bit because of
the time required to allocate new cells, and also essentially requires the
implementation of a second data structure.

21. What do you mean by open addressing?

Open addressing is a collision resolving strategy in which, if collision occurs alternative cells are tried until an empty cell is found. The cells h0(x), h1(x), h2(x),…. are tried in succession, where hi(x)=(Hash(x)+F(i))mod Tablesize with F(0)=0. The function F is the collision resolution strategy.

22. What do you mean by Probing?

Probing is the process of getting next available hash table array cell.

23. What do you mean by linear probing?

Linear probing is an open addressing collision resolution strategy in which F is a linear function of i, F(i)=i. This amounts to trying sequentially in search of an empty cell. If the table is big enough, a free cell can always be found, but the time to do so can get quite large.

24. What do you mean by primary clustering?

In linear probing collision resolution strategy, even if the table is relatively

empty, blocks of occupied cells start forming. This effect is known as primary

clustering means that any key hashes into the cluster will require several attempts

to resolve the collision and then it will add to the cluster.

25. What do you mean by quadratic probing?

Quadratic probing is an open addressing collision resolution strategy in which F(i)=i2. There is no guarantee of finding an empty cell once the table gets half full if the table size is not prime. This is because at most half of the table can be used as alternative locations to resolve collisions.

26. What do you mean by secondary clustering?

Although quadratic probing eliminates primary clustering, elements that hash to the same position will probe the same alternative cells. This is known as secondary clustering.

27. List the limitations of linear probing.

• Time taken for finding the next available cell is large.

• In linear probing, we come across a problem known as clustering.

28. Mention one advantage and disadvantage of using quadratic probing.

Advantage: The problem of primary clustering is eliminated.

Disadvantage: There is no guarantee of finding an unoccupied cell once the table

is nearly half full.

UNIT V –DYNAMIC PROGRAMMING AND BACKTRACKING

1. 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).

2. What is meant by strongly and Weekly 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 a directed graph is not strongly connected but the underlying graph is

connected, then the graph is said to be weakly connected.

3. List the two important key points of depth first search.

i) If path exists from one node to another node, walk across the edge – exploring

the edge.

ii) If path does not exist from one specific node to any other node, return to the

previous node where we have been before – backtracking.

4. What do you mean by breadth first search (BFS)?

BFS performs simultaneous explorations starting from a common point and
spreading out independently.

5. Differentiate BFS and DFS.

No.	DFS	BFS
1.	Backtracking is possible from a dead end	Backtracking is not possible
2.	Vertices from which exploration is incomplete are processed in a LIFO order	The vertices to be explored are organized as a FIFO queue
3.	Search is done in one particular Direction	The vertices in the same level are maintained parallely

6. What do you mean by articulation point?

If a graph is not biconnected, the vertices whose removal would disconnect the

graph are known as articulation points.

7. What is a state space tree?

The processing of backtracking is implemented by constructing a tree of choices being made. This is called the state-space tree. Its root represents a initial state before the search for a solution begins. The nodes of the first level in the tree represent the choices made for the first component of the solution, the nodes in the second level represent the choices for the second component and so on.

 8. What is a promising node in the state-space tree?

A node in a state-space tree is said to be promising if it corresponds to a partially constructed solution that may still lead to a complete solution.  

9. What is a non-promising node in the state-space tree?

A node in a state-space tree is said to be promising if it corresponds to a partially constructed solution that may still lead to a complete solution; otherwise it is called non-promising. 

10. What do leaves in the state space tree represent?

 Leaves in the state-space tree represent either non-promising dead ends or complete solutions found by the algorithm.

11. What is dynamic programming and who discovered it?

 Dynamic programming is a technique for solving problems with overlapping subproblems. These subproblems arise from a recurrence relating a solution to a given problem with solutions to its smaller subproblems only once and recording the results in a table from which the solution to the original problem is obtained. It was invented by a prominent U.S Mathematician, Richard Bellman in the 1950s.

12. What is backtracking?

Backtracking constructs solutions one component at a time and such partially constructed solutions are evaluated as follows

i. If a partially constructed solution can be developed further without

violating the problem’s constraints, it is done by taking the first remaining legitimate option for the next component.

ii.If there is no legitimate option for the next component, no alternatives for the remaining component need to be considered. In this case, the algorithm backtracks to replace the last component of the partially constructed solution with its next option.  

13. What is the manner in which the state-space tree for a backtracking algorithm is constructed?

In the majority of cases, a state-space tree for backtracking algorithm is constructed in the manner of depth-first search. If the current node is promising, its child is generated by adding the first remaining legitimate option for the next component of a solution, and the processing moves to this child. If the current node turns out to be non-promising, the algorithm backtracks to the node’s parent to consider the next possible solution to the problem, it either stops or backtracks to continue searching for other possible solutions.

14. How can the output of a backtracking algorithm be thought of?

The output of a backtracking algorithm can be thought of as an n-tuple (x1, …xn) where each coordinate xi is an element of some finite linearly ordered set Si. If such a tuple (x1, …xi) is not a solution, the algorithm finds the next element in Si+1 that is consistent with the values of (x1, …xi) and the problem’s constraints and adds it to the tuple as its (I+1)st coordinate. If such an element does not exist, the algorithm backtracks to consider the next value of xi, and so on.  

15. Give a template for a generic backtracking algorithm.

 ALGORITHM

Backtrack

(X[1..i]) //Gives a template of a generic backtracking algorithm //Input X[1..i] specifies the first I promising components of a solution  //Output All the tuples representing the problem’s solution if X[1..i] is a solution write X[1..i] else  for each element x[Si+1 ]consistent with X[1..i] and the constraints do   X[i+1]  x  

Backtrack(X[1..i+1]

16. Write a recursive algorithm for solving Tower of Hanoi problem. ALGORITHM

To move n>1 disks from peg1 to peg3, with peg2 as auxiliary, first move recursively n-1 disks from peg1 to peg2 with peg3 as auxiliary.

      Then move the largest disk directly from peg1 to peg3

      Finally move recursively n-1 disks from peg2 to peg3 with peg1 as auxiliary

      If n=1 simply move the single disk from source peg to destination peg.

17. What is the basic operation in the Tower of Hanoi problem and give the recurrence relation for the number of moves?

The moving of disks is considered the basic operation in the Tower of Hanoi problem and the recurrence relation for the number of moves is given as  M(n)=2M(n)+1 for n>1  M(1)=1

18. What is n-queens problem?

 The problem is to place ‘n’ queens on an n-by-n chessboard so that no two queens attack each other by being in the same row or in the column or in the same diagonal.

19Define the Hamiltonian circuit.

The Hamiltonian is defined as a cycle that passes through all the vertices of the graph exactly once. It is named after the Irish mathematician Sir William Rowan Hamilton (1805-1865).It is a sequence of n+1 adjacent vertices  vi0, vi1,……, vin-1, vi0

where the first vertex of the sequence is same as the last one while all the other n-1 vertices are distinct.

20. What is the method used to find the solution in n-queen problem by symmetry?

The board of the n-queens problem has several symmetries so that some solutions can be obtained by other reflections. Placements in the last n/2 columns need not be considered, because any solution with the first queen in square (1,i), n/2≤i≤n can be obtained by reflection from a solution with the first queen in square (1,n-i+1)

21. What are the additional features required in branch-and-bound when compared to backtracking?

Compared to backtracking, branch-and-bound requires:

i.A way to provide, for every node of a state space tree, a bound on the best value of the objective function on any solution that can be obtained by adding further components to the partial solution represented by the node.

ii.The value of the best solution seen so far

22. What is knapsack problem?

Given n items of known weights wi and values vi, i=1,2,…,n, and a knapsack of capacity W, find the most valuable subset of the items that fit the knapsack. It is convenient to order the items of a given instance in descending order by their value-to-weight ratios. Then the first item gives the best payoff per weight unit and the last one gives the worst payoff per weight unit. 

23. Give the formula used to find the upper bound for knapsack problem.

 A simple way to find the upper bound ‘ub’ is to add ‘v’, the total value of the items already selected, the product of the remaining capacity of the knapsack W-w and the best per unit payoff among the remaining items, which is v

i+1/wi+1  ub = v + (W-w)( vi+1/wi+1) 

24. What is the traveling salesman problem?

The problem can be modeled as a weighted graph, with the graph’s vertices representing the cities and the edge weights specifying the distances. Then the problem can be stated as finding the shortest Hamiltonian circuit of the graph, where the Hamiltonian is defined as a cycle that passes through all the vertices of the graph exactly once.

25. What are the strengths of backtracking and branch-and-bound?

 The strengths are as follows

i.It is typically applied to difficult combinatorial problems for which no efficient algorithm for finding exact solution possibly exist

ii.It holds hope for solving some instances of nontrivial sizes in an acceptable amount of time

iii.Even if it does not eliminate any elements of a problem’s state space and ends up generating all its elements, it provides a specific technique for doing so, which can be of some value.