Data Structures and Algorithms
What is data structure?
A data structure is a way of organizing data that considers not only the items stored, but also their relationship to each other. Advance knowledge about the relationship between data items allows designing of efficient algorithms for the manipulation of data.
List out the areas in which data structures are applied extensively?
Ø Compiler Design,
Ø Operating System,
Ø Database Management System,
Ø Statistical analysis package,
Ø Numerical Analysis,
Ø Artificial Intelligence,
What are the major data structures used in the following areas : RDBMS, Network data model & Hierarchical data model.
Ø RDBMS – Array (i.e. Array of structures)
Ø Network data model – Graph
Ø Hierarchical data model – Trees
If you are using C language to implement the heterogeneous linked list, what pointer type will you use?
The heterogeneous linked list contains different data types in its nodes and we need a link, pointer to connect them. It is not possible to use ordinary pointers for this. So we go for void pointer. Void pointer is capable of storing pointer to any type as it is a generic pointer type.
Minimum number of queues needed to implement a priority queue?
Two. One queue is used for actual storing of data and another for storing priorities.
What is the data structure used to perform recursion?
Stack. Because of its LIFO (Last In First Out) property it remembers its ‘caller’; so the function knows where to return when the function has to return. Recursion makes use of system stack for storing the return addresses of the function calls.
Every recursive function has its equivalent iterative (non-recursive) function. Even when such equivalent iterative procedures are written, explicit stack is to be used.
What are the notations used in Evaluation of Arithmetic Expressions using prefix and postfix forms?
Polish and Reverse Polish notations.
Convert the expression ((A + B) * C – (D – E) ^ (F + G)) to equivalent Prefix and Postfix notations.
^ - * +ABC - DE + FGPostfix Notation:
AB + C * DE - - FG + ^
Sorting is not possible by using which of the following methods?
Using insertion we can perform insertion sort, using selection we can perform selection sort, using exchange we can perform the bubble sort (and other similar sorting methods). But no sorting method can be done just using deletion.
A binary tree with 20 nodes has null branches.
Let us take a tree with 5 nodes (n=5)
It will have only 6 (ie,5+1) null branches. In general, a binary tree with n nodes has exactly n+1 null nodes.
What are the methods available in storing sequential files ?
Ø Straight merging,
Ø Natural merging,
Ø Polyphase sort,
Ø Distribution of Initial runs.
How many different trees are possible with 10 nodes ?
For example, consider a tree with 3 nodes(n=3), it will have the maximum combination of 5 different (ie, 23 - 3 = 5) trees.
i ii iii iv v
If there are n nodes, there exist 2n-n different trees.
List out a few of the applications of tree data-structure in compilers/compiler design.
Ø The manipulation of Arithmetic expression,
Ø Symbol Table construction,
Ø Syntax analysis.
List out a few of the applications that make use of Multilinked Structures.
Ø Sparse matrix,
Ø Index generation.
In tree construction which is the suitable efficient data structure?
(a ) Array
(b) Linked list
Answer: (b) Linked list
What type of the algorithm is used in solving the 8 Queens problem?
In an AVL tree, what is the condition for balancing to be done?
The ‘pivotal value’(or the ‘Height factor’) is greater than 1 or less than–1.
What is the bucket size, when overlapping and collision occur at the same time?
One. If there is only one entry possible in the bucket, when the collision occurs, there is no way to accommodate the colliding value. This results in the overlapping of values.
Traverse the given tree using Inorder, Preorder and Postorder traversals.
Ø Inorder : D H B E A F C I G J
Ø Preorder: A B D H E C F G I J
Ø Postorder: H D E B F I J G C A
There are 8, 15, 13, 14 nodes were there in 4 different trees. Which of them could have formed a full binary tree?
There are 2n-1 nodes in a full binary tree.
By the method of elimination:
Full binary trees contain odd number of nodes. So there cannot be full binary trees with 8 or 14 nodes, so rejected. With 13 nodes you can form a complete binary tree but not a full binary tree. So the correct answer is 15.
Full and Complete binary trees are different. All full binary trees are complete binary trees but not vice versa.
In the given binary tree, using array, at which location can you store the node 4?
Answer: At location 6
where LCn means Left Child of node n and RCn means Right Child of node n
Sort the given values using Quick Sort?
Sorting takes place from the pivot value, which is the first value of the given elements, this is marked bold. The values at the left pointer and right pointer are indicated using L and R respectively.
Since pivot is not yet changed the same process is continued after interchanging the values at L and R positions
When the L and R pointers cross each other the pivot value is interchanged with the value at right pointer. If the pivot is changed it means that the pivot has occupied its original position in the sorted order (shown in bold italics) and hence two different arrays are formed, one from start of the original array to the pivot position-1 and the other from pivot position+1 to end.
In the next pass we get the sorted form of the array.
For the given graph, draw the DFS and BFS?
Ø BFS: A X G H P E M Y J
Ø DFS: A X H P E Y M J G
Classify the Hashing Functions based on the various methods by which the key value is found.
Ø Direct method,
Ø Subtraction method,
Ø Modulo-Division method,
Ø Digit-Extraction method,
Ø Mid-Square method,
Ø Folding method,
Ø Pseudo-random method.
What are the types of Collision Resolution Techniques and the methods used in each of the type?
Ø Open addressing (closed hashing).
The methods used include:
Ø Closed addressing (open hashing)
The methods used include:
Linked list, Binary tree…
In RDBMS, what is the efficient data structure used in the internal storage representation?
B+ tree. Because in B+ tree, all the data are stored only in leaf nodes, that makes searching easier. This corresponds to the records that shall be stored in leaf nodes.
Draw the B-tree of order 3 created by inserting the following data arriving in sequence – 92 24 6 7 11 8 22 4 5 16 19 20 78
Of the following tree structure, which is efficient considering space and time complexities?
(a) Incomplete Binary Tree
(b) Complete Binary Tree
(c) Full Binary Tree
Answer: (b) Complete Binary Tree.
By the method of elimination:
Full binary tree loses its nature when operations of insertions and deletions are done. For incomplete binary trees, extra storage is required and overhead of NULL node checking takes place. So complete binary tree is the better one since the property of complete binary tree is maintained even after operations like additions and deletions are done on it.
What is a spanning Tree?
A spanning tree is a tree associated with a network. All the nodes of the graph appear on the tree once. A minimal spanning tree is a spanning tree organized so that the total edge weight between nodes is minimized.
Does the minimal spanning tree of a graph give the shortest distance between any 2 specified nodes?
Minimal spanning tree assures that the total weight of the tree is kept at its minimum. But it doesn’t mean that the distance between any two nodes involved in the minimum-spanning tree is minimum.
Convert the given graph with weighted edges to minimal spanning tree.
the equivalent minimal spanning tree is:
Which is the simplest file structure?
Answer: (a) Sequential
Is linked-list a linear or non-linear data structure?
According to Access strategies Linked list is a linear one.
According to Storage, Linked List is a Non-linear one.
Draw a binary Tree for the expression :
A * B - (C + D) * (P / Q)
For the following COBOL code, draw the Binary tree?
03 FIRST_NAME PIC X(10).
03 LAST_NAME PIC X(10).
03 FIRST_SEM PIC XX.
03 SECOND_SEM PIC XX.
Section II – Algorithms
What is an ‘algorithm’?
An algorithm consists of a finite set of steps that may require one or more operations. These operations should be definite and effective. An algorithm should produce one or more output’s and may have zero or more inputs.
This consists of five distinct areas:
1. to device algorithms
2. to validate the algorithms
3. to express the algorithms
4. to analyse the algorithms
5. to test the programs for the algorithms
What is a ‘computational procedure’?
An algorithm that does not terminate is called ‘computational procedure’. Example for such ‘computational procedure’ is an ‘operating system’.
Define - ‘recursive algorithm’.
An algorithm is said to be recursive if the same algorithm is invoked in the body of the algorithm. Recursive algorithms can be divided into direct recursive and indirect recursive algorithms.
Direct recursive: An algorithm that calls itself.
Indirect Recursive: An algorithm ‘A’ is said to be indirect recursive if it calls another algorithm which in-turn calls algorithm ‘A’.
How can you classify performance analysis?
‘Performance analysis’ can be classified as:
i. priori analysis
ii. posteriori analysis
The bounds of algorithms’ computing time are obtained by formulating a function.
Testing the actual computation of space and time are recorded while the algorithm is executing.
Define - ‘Big- O’.
For the function f(n)
iff there exist positive constants c and d such that:
for all n,n>=d.
This is defined to be the worst-time complexity of the function f(n).
3n+2 <=4n for all n>=2.
Give various computing times and their meaning.
Few of the important computing times are:
Computing Time Meaning
O(1) : constant computing time
O(n) : linear computing time
O(n*n) : quadratic computing time
O(n*n*n) : cubic computing time
O(2*2*2*2*..............*n) : exponential computing time
Give the most important ‘basic designs‘ of algorithms.
There are five important basic designs for algorithms. They are:
i. Divide and conquer,
ii. The greedy method,
iii. Dynamic programming,
v. Branch and bound.
How does ‘divide and conquer’ algorithms work?
For a function to compute on n inputs the divide and conquer strategy suggests the inputs into a k distinct subsets, 1<k<=n, yielding k sub-problems. These sub-problems must be solved and then a method must be found to combine the sub-solutions into a solution of the whole.
An example for this approach is ‘binary search’ algorithm. The time complexity of binary search algorithm is O(log n).
What is Greedy Method?
The greedy method suggests that one can devise an algorithm that works in stages, considering one input at a time. At each stage, a decision is made regarding whether a particular input is an optimal solution. An example for solution using greedy method is ‘knapsack problem’.
What is Dynamic Programming?
Dynamic Programming is an algorithm design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions. An example for algorithm using dynamic programming is ‘multistage graphs’.
What are the time complexities for the following algorithms?
Binary search : O(logn)
Finding maximum and minimum for a given set of numbers
Merge Sort : O(nlogn)
Insertion Sort : O(n*n);
Quick Sort : O(nlogn)
Selection Sort : O(n)
What is the difference between Merge Sort and Quick sort?
Both Merge-sort and Quick-sort have same time complexity i.e. O(nlogn). In merge sort the file a[1:n] was divided at its midpoint into sub-arrays which are independently sorted and later merged. Whereas, in quick sort the division into two sub-arrays is made so that the sorted sub-arrays do not need to be merged latter.
Is there any optimum solution for Matrix multiplication?
Yes. Divide and conquer method suggests Strassen’s matrix multiplication method to be used. If we follow this method, the time complexity is O(n*n*n……..*2.81) times rather O(n*n*n*………*3) times.
Define ‘minimum cost spanning method’.
Let G=(V,E) be an undirected connected graph. A sub-graph t =(V, E’) of G is a spanning tree of G if and only if t is a tree.
To find out minimum cost spanning method we have following method’s;
Prim’s Algorithm : O(n*n)
Kruskal’s Algorithm : O(e loge)
Define ’articulation points’.
A Vertex V in a connected graph G is an articulation point if and only if the deletion of vertex V together will all edges incident for disconnects the graph into two or more non-empty Components.
Define ‘biconnected graph’.
A graph G is biconnected if and only if it contains no articulation points.
What are ‘explicit’ and ‘implicit’ constraints?
‘Explicit constraints’ are rules that restrict each xi to take on values only from a given set. ‘Implicit constraints’ are rules that determine which of the tuples in the solution space of i satisfy the criterion function.
Give ‘Cookies – theorem’.
Satisfiability is in P iff P=NP
where P is the set of all decision problems solvable by deterministic algorithms in polynomial time and NP is the set of all decision problems solvable by non-deterministic algorithms in a polynomial-time.