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Quantum Notes For B.tech 1st Year

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Enigineering Physics Quantum PDF for B.Tech 1 ST  Year

UNIT-1 RELATIVISTIC MECHANICS

FRAME OF REFERENCE, INERTIAL & NON-INERTIAL FRAMES, GALILEAN

TRANSFORMATIONS, MICHELSON- MORLEY EXPERIMENT, POSTULATES OF

SPECIAL THEORY OF RELATIVITY, LORENTZ TRANSFORMATIONS, LENGTH

CONTRACTION, TIME DILATION, VELOCITY ADDITION THEOREM, VARIATION OF

MASS WITH VELOCITY, EINSTEIN’S M ES ENERGY RELATION, RELATIVISTIC

RELATION BETWEEN ENERGY AND MOM.NTUM, MASSLESS PARTICLE.

UNIT-2: ELECTROMAGNETIC FIELD THEORY

CONTINUITY EQUATION FOR CURRENT DENSITY, DISPLACEMENT CURRENT,

MODIFYING EQUATION FOR THE CURL OF MAGNETIC FIELD TO SATISFY

CONTINUITY EQUATION, MAXWELL’S EQUATIONS IN VACUUM AND IN NON

CONDUCTING MEDIUM, ENERGY IN AN ELECTROMAGNETIC FIELD, POYNTING

VECTOR AND POYNTING THEOREM, PLANE ELECTROMAGNETIC WAVES IN

VACUUM AND THEIR TRANSVERSE NATURE. RELATION BETWEEN ELECTRIC AND

MAGNETIC FIELDS OF AN ELECTROMAGNETIC WAVE, ENERGY AND MOMENTUM

CARRIED BY ELECTROMAGNETIC WAVES, RESULTANT PRESSURE, SKIN DEPTH.

UNIT-3 : QUANTUM MECHANICS

BLACK BODY RADIATION, STEFAN’S LAW, WIEN’S LAW, RAYLEIGH-JEANS LAW

AND PLANCK’S LAW, WAVE PARTICLE DUALITY, MATTER WAVES, TIME

DEPENDENT AND TIME-INDEPENDENT SCHRODINGER WAVE EQUATION, BOR

INTERPRETATION OF WAVE FUNCTION, SOLUTION TO STATIONARY STATE

SCHRODINGER WAVE EQUATION FOR ONE-DIMENSIONAL PARTICLE IN A BOX,

COMPTON EFFECT.

UNIT-4 : WAVE OPTICS

COHERENT SOURCES, INTERFERENCE IN UNIFORM AND WEDGE SHAPED THIN

FILMS, NECESSITY OF EXTENDED SOURCES, NEWTON’S RINGS AND ITS

APPLICATIONS. FRAUNHOFFER DIFFRACTION AT SINGLE SLIT AND AT DOUBLE SLIT,

ABSENT SPECTRA, DIFFRACTION GRATING SPECTRA WITH GRATING, DISPERSIVE

POWER, RESOLVING POWER OF GRATING, RAYLEIGH’S CRITERION OF RESOLUTION.

RESOLVING POWER OF GRATING.

UNIT-5: FIBER OPTICS AND LASER

FIBRE OPTICS: INTRODUCTION TO FIBRE OPTICS, ACCEPTANCE ANGLE,

NUMERICAL APERTURE, NORMALIZED FREQUENCY, CLASSIFICATION OF FIBRE,

ATTENUATION AND DISPERSION IN OPTICAL FIBRES,

LASER: ABSORPTION OF RADIATION, SPONTANEOUS AND STIMULATED

EMISSION OF RADIATION, EINSTEIN’ COEFFICIENTS, POPULATION INVERSION.

VARIOUS LEVELS OF LASER, RUBY LASER, HE-NE LASER, LASER APPLICATIONS.

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Table of Contents

Enigineering Chemistry Quantum PDF for B.Tech 1 ST Year

UNIT-I: MOLECULAR ORBITAL THEORY

MOLECULAR ORBITAL THEORY AND ITS APPLICATIONS TO HOMO-NUCLEAR

DIATOMIC MOLECULES. BAND THEORY OF SOLIDS. LIQUID CRYSTALS AND ITS

APPLICATIONS. POINT DEFECTS IN SOLIDS. STRUCTURE AND APPLICATIONS OF

GRAPHITE AND FULLERENES. CONCEPTS OF NANO-MATERIALS AND ITS

APPLICATIONS.

UNIT-II: POLYMERS AND ORGANOMETALLICS

POLYMERS: BASIC CONCEPTS OF POLYMER- BLENDS AND COMPOSITES.

CONDUCTING AND BIODEGRADABLEPOLYMERS. PREPARATIONS AND

APPLICATIONS OF SOME INDUSTRIALLY IMPORTANT POLYMERS(BUNA N,

BUNA S, NEOPRENE, NYLON 6, NYLON 6,6 , TERYLENE). GENERAL METHODS

OF SYNTHESIS OF ORGANOMETALLIC COMPOUND (GRIGNARD REAGENT) AND

THEIR APPLICATIONS IN POLYMERIZATION.

UNIT-III : ELECTROCHEMISTRY

ELECTROCHEMISTRY: GALVANIC CEL, ELECTRODE POTENTIAL, LEAD STORAGE

BATTERY. CORROSION, CAUSES AND AS PREVENTION. SETTING AND HARDENING

OF CEMENT, APPLICATIONS OF CEMENT. PLASTER OF PARIS. LUBRICANTS-

CLASSIFICATION, MECHANISM AND APPLICATIONS.

UNIT-IV : WATER TREATMENT

HARDNESS OF WATER. DISADVANTAGE OF HARD WATER. BOILER TROUBLES,

TECHNIQUES FOR WATER SOFTENING; LIME-SODA, ZEOLITE, ION EXCHANGE RESIN,

REVERSE OSMOSIS. PHASE RULE AND ITS APPLICATION TO WATER SYSTEM.

UNIT-V : FUELS AND SPECTRAL TECHNIQUES

FUELS: CLASSIFICATION OF FUELS. ANALYSIS OF COAL. DETERMINATION OF

CALORIFIC VALUES (BOMB CALORIMETER & DULONG’S METHOD). BIOGAS.

ELEMENTARY IDEAS AND SIMPLE APPLICATIONS OF UV, VISIBLE, IR AND

HINMR SPECTRAL TECHNIQUES.

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Enigineering Maths-I Quantum Notes for B.Tech 1 ST  Year

UNIT-1 : DIFFERENTIAL CALCULUS – I

SUCCESSIVE DIFFERENTIATION, LEIBNIZ’S THEOREM, LIMIT, CONTINUITY

AND DIFFERENTIABILITY OF FUNCTIONS OF SEVERAL VARIABLES, PARTIAL

DERIVATIVES, EULER’, THEOREM FOR HOMOGENEOUS FUNCTIONS, TOTAL

DERIVATIVES, CHANGEBF VARIABLES, CURVE TRACING: CARTESIAN AND POLAR

COORDINATES,

UNIT-II : DIFFERENTIAL CALCULUS – II

TAYLOR’S AND MACLAURIN’S THEOREM, EXPANSION OF FUNCTION OF SEVERAL

VARIABLES, JACOBIAN, APPROXIMATION OF ERRORS, EXTREMA OF FUNCTIONS

OF SEVERAL VARIABLES, LAGRANGE’S METHOD OF MULTIPLIERS (SIMPLE

APPLICATIONS).

UNIT-III : MATRIX ALGEBRA

TYPES OF MATRICES, INVERSE OF A MATRIX BY ELEMENTARY

TRANSFORMATIONS, RANK OF A MATRIX (ECHELON & NORMAL FORM), LINEAR

DEPENDENCE, CONSISTENCY OF LINEAR SYSTEM OF EQUATIONS AND THEIR

SOLUTION, CHARACTERISTIC EQUATION, EIGEN VALUES AND EIGEN VECTORS,

(CAYLEY-HAMILION THEOREM, DIAGONALIZATION, COMPLEXAND UNITARY

MATRICES AND ITS PROPERTIES.

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UNIT-IV: MULTIPLE INTEGRALS

DOUBLE AND TRIPLE INTEGRALS, CHANGE OF ORDER OF INTEGRATION, CHANGE

OF VARIABLES, APPLICATION OF INTEGRATION TO LENGTHS, SURFACE AREAS

AND VOLUMES -CARTESIAN AND POLAR COORDINATES. BETA AND GAMMA

FUNCTIONS, DIRICHLET’S INTEGRAL AND ITS APPLICATIONS.

UNIT-V : VECTOR CALCULUS

POINT FUNCTION, GRADIENT, DIVERGENCE AND CURL OF A VECTOR AND THEIR

PHYSICAL INTERPRETATIONS, VECTOR IDENTITIES, TANGENT AND NORMAL,

DIRECTIONAL DERIVATIVES. LINE, SURFACE AND VOLUME INTEGRALS,

APPLICATIONS OF GREEN’S, STOKE’S AND GAUSS DIVERGENCE THEOREMS

(WITHOUT PROOF).

Basic Electrical Engineering Quantum Notes for B.Tech 1 ST  Year

UNIT-1 : DC CIRCUITS

AT J ELECTRICAL FIREUIT DLEFMENTS (R, L AND C), CONCEPT OF ACTIVE AND PASSIVE*

DEMENTS, VOLTAGE AND CURRENT SOURCES, CONCEPT OF LINEARITY AND LINEAR

-NETORK, UNILATERAL, AND BILATERAL, ELEMENTS, KIRCHHOFF’S LAWS, LOOP AND

NODAL METHODS OF ANALYSIS, STAR-DELTA TRANSFORMATION, SUPERPOSITION

(C% – (NEORÉM THEVENIN THEOREM, NORTON THEOREM

UNIT-2 : STEADY-STATE ANALYSIS OF 1-& AC CIRCUITS

SINUSCIDALLY VARYING VOLTÄGE AND CURRENT.

ANALYSIS OF SINGLE PHASE AC CIRCUITS CONSISTING OF R, L. C, RL, RC. RICINU

COMBINATIONS (SERIES AND PARALLEL), APPARENT, ACTIVE & REACTIVE POWER,

POWER FACTOR, POWER FACTOR IMPROVEMENT CONCEPT OF RESONANCE IN SERIES

DE PARALLEL CIRCUITS, BANDWIDTH AND QUALITY FACTOR. THREE PHASE BALANCED

CIRCUITS, VOLTAGE AND CURRENT RELATIONS IN STAR AND DELTA CONNECTIONS

UNIT-3 : TRANSFORMERS

MAGNETIC MATERIALS, BI CHARACTERISTICS, IDEAL AND PRACTICAL TRANSLORMER,

EQUIVALENT CIRRUIT, LOSSES IN TRANSFORMERS, REGULATION AND EFFICIENCY, AUTO .

TRANSFORMER AND THREE-PHASE TRANSFORMER CONNECTIONS.

UNIT-4 : ELECTRICAL MACHINES

DC MACHINES: PRINDPLE & CONSTRUCTION, TYPES, EMF EQUATION OF GENERATOR

AND TORQUE EQUATION OF MOTOR, *PPLICATIONS OF DC MOTORS (SIMPLE NUMERICAL

/ PROBLEMS)

THREE PHASE INDUCTION MOTOR. PRINCIPLE & CORSTRUCTION, TYPES, SLIP-TORQUE

(CITE. CHARACTERISTIOS, APPLICATIONS (NUMERICAL PROBLEMS RELATED TO SLIP ONLY)

SINGLE PHASE INDUCTION MOTOR: PRINDPLE OF OPERATION AND INTRODUCTION F749

METHODS OF STARTING, APPLICATIONS

THREE PHASE SYNCHRONOUS MACHINES PRINCIPLE OF OPERATION @F ALTERNATOR AND

SYNCHRONOUS MOTOR AND THEIR APPLICATIONS.

UNIT-S : ELECTRICAL INSTALLATIONS

 (AR_ COMPONENT OF LT SWILCHGEAR: SWITCH FUSE UNIT (SFU), MCB, ELCB, MOCB

TYPES OF WIRES AND CABLES, IMPORTANCE OF EARTHING TYPES OF BATTERLES,

IMPORTANT CHARACTERISTICS FOR BALTERIES ELEMENTARY CALCULATIONS FOR ENERGY .L.E

CONSUMPTION AND SAVINGS, BATTERY BACKUP.

Programming for Problem Solving Quantum Notes

UNIT-1 : INTRODUCTION TO PROGRAMMING

INTRODUCTION TO COMPONENTS OF A COMPUTER SYSTEM: MEMORY,

PROCESSOR. 1/0 DEVICES, STORAGE, OPERATING SYSTEM, CONCEPT OF

ASSEMBLER, COMPILER, INTERPRETER, LOADER AND LINKER.

IDEA OF ALGORITHM: REPRESENTATION OF ALGORITHM, FLOWCHART, PSEUDO

CODE WITH EXAMPLES, FROM ALGORITHMS TO PROGRAMS, SOURCE CODE

PROGRAMMING BASICS: STRUCTURE OF C PROGRAM, WRITING AND

EXECUTING THE FIRST C. PROGRAM, SYNTAX AND LOGICAL ERRORS IN

COMPILATION, OBJECT AND EXECUTABLE CODE. COMPONENTS OF C

LANGUAGE, STANDARD 1/0 IN C, FUNDAMENTAL DATA TYPES, YARIABLES

AND MEMORY LOCATIONS, STORAGE CLASSES.

UNIT-2: ARITHMETIC EXPRESSIONS

ARITHMETIC EXPRESSIONS AND PRECEDENCE:OPERATORS AND EXPRESSION USING

NUMERIC AND RELATIONAL OPERATORS, MIXED OPERANDS, TYPE CONVERSION

LOGICIT OPERATORS, BIT OPERATIONS, APSIGNMENT OPERATOR, OPERATOR

PRECEDENCE AND ASSOCIATIVITY,

CONDITIONAL BRANCHING2APPLYING IF AND SWITCH STATEMENTS, NESTING

IF AND ELSE, USE OF BREAK AND DEFAULT WITH SWITCH.

UNIT-3: LOOPS & FUNCTIONS

ITERATION AND LOOPS: USE OF WHILE, DO WHILE AND FOR LOOPS, MULTIPLE

LOOP VARIABLES, USE OF BREAK AND CONTINUE STATEMENTS.

FUNCTIONS: INTRODUCTION, TYPES OF FUNCTIONS, FUNCTIONS WITH ARRAY,

PASSING PARAMETERS TO FUNCTIONS, CALL BY VALUE, CALL BY REFERENCE,

RECURSIVE FUNCTIONS.

UNIT-4 : ARRAYS & BASIC ALGORITHM

ARRAYS: ARRAY NOTATION AND REPRESENTATION, MANIPULATING ARRAY

ELEMENTS, USING MULTI DIMENSIONAL ARRAYS. CHARACTER ARRAYS AND STRINGS,

STRUCTURE, UNION, ENUMERATED DATA TYPES, ARRAY OF STRUCTURES, PASSING

ARRAYS TO FUNCTIONS. BASIC ALGORITHMS: SEARCHING & BASIC SORTING

ALGORITHMS (BUBBLE, INSERTION AND SELECTION), FINDING ROOTS OF

EQUATIONS, NOTION OF ORDER OF COMPLEXITY.

UNIT-5: POINTERS AND FILE HANDLING

POINTERS: INTRODUCTION, DECLARATION, APPLICATIONS, INTRODUCTION TO

DYNAMIC MEMORY ALLOCATION (MALLOC, CALLOC, REALLOC, FREE), USE OF POINTERS

IN SELF-REFERENTIAL STRUCTURES, NOTION OF LINKED LIST (NO IMPLEMENTATION).

FIE HANDLING: FILE 1/0 FUNCTIONS, STANDARD C PREPROCESSORS, DEFINING

AND CALLING MACROS, COMMAND-LINE ARGUMENTS.

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Quantum Series AKTU | AKTU Quantum PDF BTech First Year

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Programming is not merely about writing lines of code; it’s a creative and strategic process aimed at solving real-world problems. In the realm of problem-solving, programming serves as a dynamic tool, empowering individuals to tackle challenges across diverse domains.

At its core, programming instills a structured approach to problem-solving. It encourages breaking down complex issues into smaller, more manageable tasks. Through the logical flow of code, programmers devise solutions, leveraging the power of algorithms and data structures.

One of the key strengths of programming in problem-solving lies in its versatility. From automating repetitive tasks to optimizing intricate algorithms, code becomes a problem-solver’s best friend. Whether in scientific research, business analytics, or creative endeavors, programming provides a means to transform abstract ideas into functional solutions.

The iterative nature of programming promotes a mindset of continuous improvement. As developers encounter obstacles, they refine and enhance their code, learning from each challenge. This adaptive process not only resolves immediate issues but also hones problem-solving skills for future endeavors.

Programming languages serve as the languages of problem-solving, each with its unique strengths. From the simplicity of Python to the efficiency of C++ or the web development prowess of JavaScript, choosing the right language is akin to selecting the perfect tool for the job at hand.

Moreover, programming fosters collaboration and knowledge-sharing within a global community. Online forums, open-source projects, and collaborative platforms enable programmers to learn from one another, contributing to a collective pool of problem-solving expertise.

In essence, programming for problem-solving is a dynamic journey where logic meets creativity. It empowers individuals to transform challenges into opportunities, providing a skill set that transcends industries and fuels innovation. As technology continues to evolve, the ability to code for problem-solving remains a valuable asset, opening doors to a world of endless possibilities.

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Design and Analysis of Algorithm Aktu Question Paper Quantum Notes Pdf 22-23

Design and Analysis of Algorithm Important Questions 2022–2023 in Quantum Book Pdf with Solved Exam Paper, Repetition of Most Important Questions, Syllabus, and Aktu Notes

Section A: Design and Analysis of Algorithm Aktu Notes Short Answers

a. How analyze the performance of an algorithm in different cases ?

Ans. Analysis/complexity of an algorithm: 

The complexity of an algorithm is a function g(n) that gives the upper bound of the number of operation (or running time) performed by an algorithm when the input size is n.

Types of complexity: 

Space complexity: Space complexity of an algorithm is the amount of memory it needs to run to complection, including the space for input values for execution.

Time complexity: Time complexity of an algorithm is the number of times a statement executes. It is not the actual time required to execute a particular code. 

Cases of complexity:

• 1. Worst case: It is the upper bound on the running time of an algorithm. It is the case that causes a maximum number of operation to be executed.
• 2. Average case complexity: The running time for any given size input will be the average number of operations over all problem instances for a given size.
• 3. Best case complexity: The best case complexity of the algorithm is the function defined by the minimum number of steps taken on any instance of size n.  

b. Derive the time complexity of merge sort.

c. Explain left rotation in RB tree. 

• 1. In rotation operation, the positions of the nodes of a subtree are interchanged. 
• 2. Rotation operation is used for maintaining the properties of a red black tree when they are violated by other operations such as insertion and deletion. 
• 3. In left-rotation, the arrangement of the nodes on the right is transformed into the arrangements on the left node.  

d. Write down the properties of Fibonacci Heap. 

Ans. Fibonacci Heap is a collection of trees with min-heap or max-heap property. In Fibonacci Heap, trees can have any shape even all trees can be single nodes. Fibonacci Heap maintains a pointer to minimum value (which is root of a tree). 

e. Explain greedy programming in brief.  

Ans. Greedy is an algorithmic paradigm that builds up a solution piece by piece, always choosing the next piece that offers the most obvious and immediate benefit. So the problems where choosing locally optimal also leads to global solution are best fit for Greedy. 

A Greedy algorithm makes greedy choices at each step to ensure that the objective function is optimized. 

f. What do you mean by convex hull ?

Ans. There exists a set of points on a plane which is said to be convex if for any two pointsA and B in the set, the entire line segment with the end points at A and B belongs to the set.

g. Write down the Floyd Warshal algorithm. 

h. Explain branch and bound method in brief. 

Ans. Branch and bound is one of the techniques used for problem solving. It is similar to the backtracking since it also uses the state space tree. It is used for solving the optimization problems and minimization problems. If we have given a maximization problem then we can convert it using the Branch and bound technique.  

i. Explain randomized algorithm in brief. 

• 1. A randomized algorithm is defined as an algorithm that is allowed to access a source of independent, unbiased bits and it is then allowed to use these random bits to influence its computation. 
• 2. An algorithm is randomized if its output is determined by the input as well as the values produced by a random number generator.
•  3. A randomized algorithm makes use of a randomizer such as a random number generator.
• 4. The execution time of a randomized algorithm could also vary from run to run for the same input. 
• 5. The algorithm typically uses the random bits as an auxiliary input to guide its behaviour in the hope of achieving good performance in the “average case”.
• 6. Randomized algorithms are particularly useful when it faces a malicious attacker who deliberately tries to feed a bad input to the algorithm. 

j. Explain NP-complete and NP-Hard. 

Ans. NP-complete : The group of problems which are both in NP and NP-hard are known as NP-complete problem. 

• 1. We say that a decision problem P i is NP-hard if every problem in NP is polynomial time reducible to P i .
• 2. In symbols,

• 3. This does not require P i to be in NP.
• 4. Highly informally, it means that P i is ‘as hard as’ all the problem in NP.  
• 5. If P i can be solved in polynomial time, then all problems in NP.
• 6. Existence of a polynomial time algorithm for an NP-hard problem implies the existence of polynomial solution for every problem in NP.  

Section B: Aktu Long Questions of Design and Analysis of Algorithm

a. Solve the recurrence  

i. T (n) = 3T (n/4) + cn 2 using recursion tree method. 

ii. T (n) =n + 2T (n/2) using iteration method. (Given T (1) = 1) 

Repeat upto k times  

b. What is Binomial Heap ? Write down the algorithm for decrease key operation in Binomial Heap also write its time complexity. 

Ans. Binomial heap: 

1. Binomial heap is a type of data structure which keeps data sorted and allows insertion and deletion in amortized time. 

2. A binomial heap is implemented as a collection of binomial tree.  

Decrease key operation: BINOMIAL-HEAP-DECREASE-KEY (H, x, k) 

Deleting a key: The operation BINOMIAL-HEAP-DELETE (H,x) is used to delete a node x’s key from the given binomial heap H. The following implementation assumes that no node currently in the binomial heap has a key of – ∞. 

BINOMIAL-HEAP-DELETE (H, x) 

1. BINOMIAL-HEAP-DECREASE-KEY (H, x, ∞) 

2. BINOMIAL-HEAP-EXTRACT-MINED 

Time complexity: The time complexity of finding the minimum key in binomial heap is O(logn).

c. Write and explain the Kruskal algorithm to find the Minimum Spanning Tree of a graph with suitable example. 

Ans. Kruskal algorithm: 

i. In this algorithm, we choose an edge of G which has smallest weight among the edges of G which are not loops. 

ii. This algorithm gives an acyclic subgraph T of G and the theorem given below proves that Tis minimal spanning tree of G. Following steps are required:

Step1: Choose e 1 , an edge of G, such that weight of e 1 w(e 1 ) is as small as possible and e 1 is not a loop. 

Step 2: If edges e 1 ,e 2 ,……..,e i have been selected then choose an edge e i+1 not already chosen such that

Step 3: If G has n vertices, stop after n – 1 edges have been chosen. Otherwise repeat step 2.

If G be a weighted connected graph in which the weight of the edges areall non-negative numbers, let T be a subgraph of G obtained by Kruskal’s algorithm then, T is minimal spanning tree.

Step 1: Arrange the edge of graph according to weight in ascending order. 

Step 2: Now draw the vertices as given in graph, 

Now draw the edge according to the ascending order of weight. If any edge forms cycle, leave that edge. 

Step 3: Select edge 13 

All the remaining edges, such as 34, 41, 12, 35, 56 are rejected because they form cycle. 

All the vertices are covered in this tree. So, the final tree with minimum cost of given graph is 

Minimum cost = 2+3+ 4+ 5 +6 20 

d. What is N queens problem ? Draw a state space tree for 4 queens problem using backtracking. 

Ans. N-Queens problem:

• 1. In N-Queens problem, the idea is to place queens one by one in different columns, starting from the leftmost column.
• 2. When we place a queen in a column, we check for clashes with already placed queens.
• 3. In the current column, if we find a row for which there is no clash, we mark this row and column as part of the solution.
• 4. If we do not find such a row due to clashes then we backtrack and return false. 

4-Queens problem:

• 1. Suppose we have 4 x 4 chessboard with 4-queens each to be placed in non-attacking position.  

• 2. Now, we will place each queen on a different row such that no two queens attack each other.
• 3. We place the queen q 1 in the very first accept position (1, 1).
• 4. Now if we place queen q 2 in column 1 and 2 then the dead end is encountered.  
• 5. Thus, the first acceptable position for queen q 2 is column 3 ie., (2, 3) but then no position is left for placing queen q 3 safely. So, we backtrack one step and place the queen q 2 in (2, 4). 
• 6. Now, we obtain the position for placing queen q 3 which is (3, 2). But later this position lead to dead end and no place is found where queen q 2 can be placed safely.  

• 7. Then we have to backtrack till queen q 1 and place it to (1, 2) and then all the other queens are placed safely by moving queen q 2 to (2, 4), queen q 3 to (3, 1) and queen q 4 to (4, 3) i.e., we get the solution < 2, 4, 1, 3>. This is one possible solution for 4-queens problem. 

• 8. For other possible solution the whole method is repeated for all partial solutions. The other solution for 4-queens problem is <3, 1, 4, 2> i.e., 

• 9. Now, the implicit tree for 4-queen for solution <2, 4, 1,3> is as follows: 
• 10. Fig. shows the complete state space for 4-queens problem. But we can use backtracking method to generate the necessary node and stop if next node violates the rule i.e., if two queens are attacking.  

Section 3: Design and Analysis of Algorithm Quantum Pdf

a. Write merge sort algorithm and sort the following sequence {23, 11, 5, 15, 68, 31,4, 17} using merge sort.

Ans. Merge sort algorithm:

• 1. Merge sort is a sorting algorithm that uses the idea of divide and conquer.
• 2. This algorithm divides the array into two halves, sorts them separately and then merges them.
• 3. This procedure is recursive, with the base criteria that the number of elements in the array is not more than 1. 

Algorithm: 

MERGE_SORT (a,p, r) 

MERGE (A,p, q,r) 

Numerical: 

Given array: {23, 11, 5, 15, 68, 31, 4, 17}

1. Divide into two halves:

2. Consider first part: Again divide into two sub arrays  

3. Consider the second half: Again divide into two sub arrays  

4. Merge these two sorted sub arrays 

b. What do you understand by stable and unstable sorting? Sort the following sequence {25, 57, 48, 36, 12, 91, 86, 32} using heap sort. 

Ans. Stable sorting: 

• 1. A sorting algorithm is said to be stable if two objects with equal keys appear in the same order in sorted output as they appear in the input sorted array.
• 2. A stable sort is one where the initial order of equal items is preserved.
• 3. Some sorting algorithms are stable by nature, such as bubble sort, insertion sort, merge sort, counting sort etc.
• 4. Let A be an array, and let < be a strict weak ordering on the elements of A. Sorting algorithm is stable if:

• 5. Stability means that equivalent elements retain their relative positions, after sorting.  

For example:  

Unstable sorting: In an unstable sorting algorithm the ordering of the same values is not necessarily preserved after sorting. 

Unstable sorting 

Sorting algorithms which are not considered stable are selection sort, shellsort, Quicksort, Heapsort. 

Originally the given array: A[25, 57, 48, 36, 12, 91, 86, 32]

First we call BUILD-MAX HEAP

heap size [A] = 8

So, final tree after BUILD-MAX heap is 

Section 4: Aktu Quantum Pdf of Design and Analysis of Algorithm

a. Discuss the various cases for insertion of key in red-black tree for given sequence of key in an empty red-black tree {15, 13, 12, 16, 19, 23, 5, 8}.

Ans. Insertion: 

Delete 15: No tree 

b. What is skip list ? Explain the search operation in skip list with suitable example also write its algorithm. 

Ans. Skip list:

• 1. A skip list is built in layers. 
• 2. The bottom layer is an ordinary ordered linked list.
• 3. Each higher layer acts as an “express lane”, where an element in layer i appears in layer (i+ 1) with some fixed probability p (two commonly used values for p are ½  and ¼ .).
• 4. On average, each element appears in 1/(1-p) lists, and the tallest element (usually a special head element at the front of the skip list) in all the lists.
• 5. The skip list contains log 1/p n (i.e., logarithm base 1/p of n). 

Searching operations algorithm:

• 1. Let’stake an example to understand the working of the skip list. In this example, we have 14 nodes, such that these nodes are divided into two layers, as shown in the diagram. 
• 2. The lower is a common line that links all nodes, and the top layer is an express line that links only the main nodes, as you can see in the diagram. 
• 3. Suppose you want to find 47 in this example. You will start the search from the first node of the express line and continue running on the express line until you find a node that is equal a 47 or more than 47. 
• 4. You can see in the example that 47 does not exist in the express line, so you search for a node of less than 47, which is 40. Now, you go to the normal line with the help of 40, and search the 47, as shown in the diagram. 

Section 5: Design and Analysis of Algorithm Aktu Solved Question Paper

a. What is Knapsack problem ? Solve Fractional Knapsack problem using greedy programming for the following four items with their weights w = {3,5, 9, 5} and values P = {45, 30, 45, 10} with Knapsack capacity is 16. 

Ans. Knapsack problem:

• 1. The knapsack problem is a problem in combinatorial optimization.
• 2. Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.  

Approach use to solve the problem: 

• 1. In knapsack problem, we have to fill the knapsack of capacity W, with a given set of items I 1 , I 2 …..I n having weight w 1 , w 2 …..w n in such a manner that the total weight of items cannot exceed the capacity of knapsack and maximum possible value (v) can be obtained. 
• 2. Using branch and bound approach, we have a bound that none of the items can have total sum more than the capacity of knapsack and must give maximum possible value. 
• 3. The implicit tree for this problem is a binary tree in which left branch implies inclusion and right implies exclusion. 
• 4. Upper bound of node can be calculated as:

To fulfill capacity W= 16 we will have 

1. Add item of weight 3, W = 16 – 3 = 13 

2. Add item of weight 5, W = 13 – 5 = 8 

3. Skip item of weight 9 

4. Add item of weight 5, W= 8 – 5 = 3 

Total maximum value = 45 + 30 + 10 = 85

b. Write Clown the Bellman Ford algorithm to solve the single source shortest path problem also write its time complexity. 

Ans. Bellman Ford algorithm:  

• 1. Bellman-Ford algorithm finds all shortest path length from a source s 𝜖 V to all u 𝜖 V or determines that a negative-weight cycle exists. 
• 2. Bellman-Ford algorithm solves the single source shortest path problem in the general case in which edges of a given digraph G can have negative weight as long as G contains no negative cycles. 
• 3. This algorithm, uses the notation of edge relaxation but does not use with greedy method. 
• 4. The algorithm returns boolean TRUE if the given digraph contains no negative cycles that are reachable from source vertex otherwise it returns boolean FALSE. 

Bellman-Ford (G, w, s): 

RELAX (u, v, w): 

If Bellman-Ford returns true, then G forms a shortest path tree, else there exists a negative weight cycle. 

Time complexity: Time complexity of Bellman-Ford algorithm is OVE), where V is vertex and E is edge of the graph.  

Section 6: Aktu Btech Notes Pdf Design and Analysis of Algorithm

a. What is travelling Salesman Problem (TSP) ? Find the solution of following TSP using branch and bound method. 

Ans. Travelling Salesman Problem (TSP): Travelling salesman problem is the problem to find the shortest possible route for a given set of cities and distance between the pair of cities that visits every city exactly once and returns to the starting point.

1. Reduce each column and row by reducing the minimum value from each element in row and column. 

2. So, total expected cost is: 10 + 2 + 2 + 3 + 4 + 1 + 3 = 25. 

3. We have discovered the root node V 1 so the next node to be expanded will be V 2 , V 3 , V 4 , V 5 Obtain cost of expanding using cost matrix for node 2.

4. Change all the elements in 1 st row and 2 nd column.

5. Now, reducing M 2 in rows and columns, we get:   

Find the solution of following TSP using branch and bound method. 

12. Now, the promising node is V 4 = 25. Now, we can expand V 2 , V 3 , and V 5 . Now, the input matrix will be M 4 . 

19. Now, promising node is V 2 = 28. Now, we can expand V 1 and V 5. Now, the input matrix will be M 7 .

20. Change all the elements in 2 nd row and 3 rd columnn. 

24. Here V 5 is the most promising node so next we are going to expand this node further. Now, we are left with only one node not yet 5. traversed which is V 5 .

So, total eost of traversing the graph is:

10 + 6  +2 + 7 + 3 = 28 

b. Explain the method of finding Hamiltonian cycles in a graph using backtracking method with suitable example. 

• 1. Given a graph G = (V, E), we have to find the Hamiltonian circuit using backtracking approach.
• 2. We start our search from any arbitrary vertex, say ‘a’. This vertex ‘a’ becomes the root of our implicit tree. 
• 3. The first element ofour partial solution is the first intermediate vertex of the Hamiltonian cycle that is to be constructed. 
• 4. The next adjacent vertex is selected on the basis of alphabetical (or numerical) order.  
• 5. If at any stage any arbitrary vertex makes a cycle with any vertex other than vertex ‘a’ then we say that dead end is reached.
• 6. In this case we backtrack one step, and again search begins by selecting another vertex and backtrack the element from the partial solution must be removed. 
• 7. The search using backtracking is successful if a Hamiltonian cycle is obtained. 

Section 7: Design and Analysis of Algorithm Pdf notes of Aktu Quantum

a. Write and explain the algorithm to solve vertex cover problem using approximation algorithm.  

Ans. A vertex cover of an undirected graph G = (V, E) is a subset of V’ ⊆ V such that if edge (u, v) 𝜖 G then u 𝜖 V or v 𝜖 V (or both).  

Problem: Find a vertex cover of maximum size in a given undirected graph. This optimal vertex cover is the optimization version of an NP-Complete problem but it is not too hard to find a vertex cover that is near optimal.  

Approx-vertex-cover (G: Graph)

Analysis: It is easy to see that the running time of this algorithm is O(V+ E), using adjacency list to represent E’.

b. Explain and write the Knuth-Morris-Pratt algorithm for pattern matching also write its time complexity.  

Ans. Knuth-Morris-Pratt algorithm:

COMPUTE-PREFIX-FUNCTION (P)  

KMP-MATCHER calls the auxiliary procedure COMPUTE-PREFIX-FUNCTION to compute 𝜋.

KMP-MATCHER (T, p)  

Time complexity: The time complexity of KMP algorithm is O(n) in the worst case.  

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AKTU Question Paper

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Download Design and Analysis of Algorithm Quantum pdf For Aktu B-tech 3rd Year:

Introduction.

In the world of computer science and engineering, algorithms play a crucial role in solving complex problems efficiently. The design and analysis of algorithms is a fundamental subject for students pursuing a Bachelor of Technology (B-Tech) degree in AKTU (Dr. A.P.J. Abdul Kalam Technical University). This article aims to provide a comprehensive overview of the key topics covered in the Design and Analysis of Algorithm Quantum PDF for AKTU B-Tech 3rd year students.

How to Download Design and Analysis of Algorithm Quantum PDF for free?

Simply click at the given link to download Design and Analysis of Algorithm Quantum PDF.

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Table of Contents

• Introduction to Algorithm Design
• Asymptotic Notations
• Divide and Conquer Algorithms
• Greedy Algorithms
• Dynamic Programming
• Backtracking Algorithms
• Branch and Bound
• Graph Algorithms
• Network Flow
• NP-Completeness
• Approximation Algorithms
• Randomized Algorithms
• String Matching Algorithms
• Sorting and Searching Algorithms
• Case Studies and Practical Applications

Key topics covered in Design and Analysis of Algorithm Quantum pdf

1. introduction to algorithm design.

In this section, students will learn about the fundamental concepts of algorithm design, including problem-solving techniques, algorithmic thinking, and the importance of efficiency in algorithm design.

2. Asymptotic Notations

Asymptotic notations, such as Big O, Omega, and Theta, are vital tools for analyzing the efficiency of algorithms. This topic covers the mathematical representation of algorithmic complexity and the classification of algorithms based on their growth rates.

3. Divide and Conquer Algorithms

Divide and Conquer is a popular algorithmic paradigm that involves breaking down a problem into smaller subproblems, solving them independently, and combining the solutions to obtain the final result. Students will explore various divides and conquer algorithms, such as merge sort, quick sort, and the closest pair problem.

4. Greedy Algorithms

Greedy algorithms make locally optimal choices at each step in the hope of finding a global optimum. This section covers greedy algorithms’ characteristics, applications, and examples, such as the Knapsack problem and minimum spanning trees.

5. Dynamic Programming

Dynamic programming is a powerful algorithmic technique that breaks down a problem into overlapping subproblems and solves them in a bottom-up manner. Students will learn about dynamic programming principles, memorization, and examples like the 0/1 Knapsack problem and the Fibonacci sequence.

6. Backtracking Algorithms

Backtracking algorithms are used to explore all possible solutions to a problem by incrementally building candidates and backtracking when a solution is found or deemed impossible. This topic delves into backtracking algorithms, such as the N-Queens problem and the Sudoku solver.

7. Branch and Bound

Branch and Bound is an algorithmic paradigm for solving optimization problems by systematically exploring the solution space. This section covers the principles of branch and bound, branch and bound tree, and applications like the Traveling Salesman Problem.

8. Graph Algorithms

Graph algorithms play a crucial role in various domains, including network analysis, social networks, and recommendation systems. This topic encompasses graph traversal algorithms (DFS and BFS), shortest path algorithms (Dijkstra’s algorithm), and minimum spanning tree algorithms (Prim’s and Kruskal’s algorithms).

9. Network Flow

Network flow algorithms are used to optimize the flow of resources in a network. Students will learn about maximum flow, minimum cut, and Ford-Fulkerson algorithm, along with their applications in transportation networks and network planning.

10. NP-Completeness

NP-Completeness is a classification of computational problems that are believed to have no efficient solution. This section introduces students to the concept of NP-Completeness, complexity classes, and the famous P vs. NP problem.

11. Approximation Algorithms

Approximation algorithms provide near-optimal solutions for NP-Hard problems. This topic covers approximation ratios, greedy approximation algorithms, and examples like the traveling salesman problem and the knapsack problem.

12. Randomized Algorithms

Randomized algorithms utilize randomization to solve problems more efficiently or with better approximations. Students will explore concepts such as Monte Carlo algorithms and Las Vegas algorithms, along with their applications.

13. String Matching Algorithms

String matching algorithms are used to find occurrences of a pattern within a larger text. This section covers various string matching algorithms, including the brute-force approach, Knuth-Morris-Pratt algorithm, and Rabin-Karp algorithm.

14. Sorting and Searching Algorithms

Sorting and searching algorithms are fundamental to data processing and retrieval. This topic includes popular sorting algorithms like bubble sort, selection sort, insertion sort, and efficient searching algorithms such as binary search and interpolation search.

15. Case Studies and Practical Applications

In this final section, students will explore real-world case studies and practical applications of algorithm design and analysis. Examples may include data compression algorithms, image recognition algorithms, and optimization algorithms used in logistics and scheduling.

The Design and Analysis of Algorithm Quantum PDF for AKTU B-Tech 3rd year provides a comprehensive foundation in algorithmic problem-solving and analysis. By mastering the key topics mentioned above, students can develop efficient algorithms and tackle complex computational problems with confidence.

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