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+# SYLLABUS
+
+## UNIT- I
+### Introduction:
+- Introduction
+- Steps for algorithmic problem solving
+- Important Problem Types
+- Asymptotic Notations and Efficiency Classes
+- Mathematical Analysis for recursive Algorithms and Non- recursive Algorithms
+- Empirical Analysis
+- Algorithm Visualization
+
+## UNIT-II:
+### Brute Force:
+- Selection and Bubble sort
+- Sequential Search
+- Closest- Pair
+- Convex Hull Problem
+
+### Exhaustive Search:
+- Travelling Salesman problem
+- Knapsack problem
+- Assignment Problem
+
+### Divide-and-Conquer:
+- General Method
+- Binary Search
+- Merge sort
+- Quick sort
+- Stassen’s Matrix Multiplication
+- Multiplication of large integers
+
+## UNIT-III:
+### Transform and conquer:
+- Pre-sorting
+- Gauss Elimination
+- Balanced Trees – 2-3 Trees
+- Heap sort
+- Horner’s rule and binary exponentiation
+
+### Problem reduction
+- Least Common Multiple
+- Counting paths in a graph
+
+### Dynamic Programming:
+- All Pair Shortest Path Problem
+- Optimal Binary Search Trees
+- 0/1 Knapsack Problem and Memory Functions
+
+## UNIT-IV:
+### Greedy Technique:
+- Minimum Spanning Tree Algorithm
+- Single Source Shortest Path Problem
+- Huffman Trees
+
+### Space and Time Trade-offs:
+- Sorting by computing
+- Input Enhancement in String Matching- Horspool’s Algorithm
+- Boyer-Moore Algorithm
+
+### Backtracking:
+- N-queen problem
+- Sum of subsets problem
+- Graph Coloring
+- Hamiltonian Cycle Problem
+
+## UNIT-V:
+### Branch and Bound:
+- General method
+- Applications:
+    - 0/1 knapsack proble
+    - Travelling salesman problem
+    - Assignment Problem
+
+### NP-HARD & NP-COMPLETE PROBLEMS:
+- Lower Bound Arguments
+- P, NP, NP - Hard and NP Complete classes
+- Challenges in numerical algorithms
+
+
+# Notes
+
+
+
+## Unit-V
+### Branch and Boumd:
+1. General Method:
+Branch and Bound is a general algorithm design paradigm that is used to solve
+optimization problems. It works by exploring the solution space in a systematic
+way, dividing it into smaller subproblems and bounding the solutions to prune
+the search space efficiently.
+
+Key Steps in Branch and Bound:
+- Initialization: Initialize the algorithm with the initial problem instance.
+- Branching: Divide the problem into smaller subproblems by branching on a
+  variable or decision.
+- Bounding: Calculate an upper or lower bound on the solution to prune the
+  search space.
+- Backtracking: Explore the subproblems recursively, backtracking when a
+  solution is infeasible or suboptimal.
+- Termination: Stop when all subproblems have been explored or the optimal
+  solution is found.
+
+Applications of Branch and Bound:
+- Combinatorial Optimization: Traveling Salesman Problem, Knapsack Problem
+- Scheduling Problems: Job Shop Scheduling, Task Assignment
+- Integer Programming: Mixed Integer Linear Programming
+- Graph Problems: Shortest Path, Minimum Spanning Tree
+- Machine Learning: Decision Trees, Feature Selection
+
+2. 0/1 Knapsack Problem:
+The 0/1 Knapsack Problem is a classic optimization problem that falls under the
+NP-Hard category. It involves selecting a subset of items with given weights
+and values to maximize the total value while keeping the total weight within a
+specified limit.
+
+Key Concepts:
+- Decision Variables: Binary variables indicating whether an item is selected
+  or not
+- Objective Function: Maximize the total value of selected items
+- Constraints: Total weight of selected items should not exceed the knapsack
+  capacity
+
+Solving the 0/1 Knapsack Problem using Branch and Bound:
+- Branching Strategy: Branch on the decision variables (item selection)
+- Bounding Strategy: Calculate the upper bound based on the remaining items
+- Backtracking: Explore the search space efficiently to find the optimal solution
+- Termination: Stop when all subproblems have been explored
+
+
+### NP-HARD, NP-COMPLETE, & Numerical Problems:
+1. Lower Bound Arguments
+Lower bound arguments are fundamental in computational complexity theory,
+helping us understand the inherent difficulty of solving computational
+problems. A lower bound is the minimum time or space complexity required to
+solve a problem, regardless of the algorithm used.
+
+Key aspects of lower bound arguments include:
+- Proving Minimum Computational Complexity: Lower bounds demonstrate the
+    theoretical minimum resources (time or space) needed to solve a problem.
+- Proof Techniques:
+  - Comparison Model: Analyzing algorithms that can only compare elements
+  - Decision Tree Arguments: Examining the minimum number of decisions required to solve a problem
+  - Information-Theoretic Arguments: Using information theory to establish computational limits
+
+Classic Examples:
+- Sorting Lower Bound: Comparison-based sorting algorithms have a Ω(n log n) lower bound
+- Searching in Unsorted Arrays: Require Ω(n) time complexity
+- Minimum number of comparisons to find the maximum in an array: Ω(n-1)
+
+2. P, NP, NP-Hard, and NP-Complete Classes
+These complexity classes are crucial in understanding computational complexity
+and problem difficulty:
+P (Polynomial Time):
+- Problems solvable in polynomial time by a deterministic Turing machine
+- Efficient, typically with complexity O(n^k) where k is a constant
+- Examples: Sorting, shortest path, linear programming
+
+NP (Nondeterministic Polynomial Time):
+- Problems verifiable in polynomial time
+- Can be solved by a nondeterministic Turing machine in polynomial time
+- Verification is easy, but finding the solution might be hard
+- Examples: Traveling Salesman Problem, Graph Coloring
+
+NP-Hard:
+- At least as hard as the hardest problems in NP
+- No known polynomial-time algorithm
+- Not necessarily in NP
+- Example: Halting Problem
+
+NP-Complete:
+- Subset of NP-Hard problems that are also in NP
+- If a polynomial-time solution is found for any NP-Complete problem, all NP
+  problems can be solved in polynomial time
+- Famous NP-Complete Problems:
+  - Boolean Satisfiability Problem (SAT)
+  - Traveling Salesman Problem
+  - Graph Coloring
+  - Subset Sum Problem
+
+The P vs NP Problem:
+- One of the most important open problems in computer science
+- Asks whether every problem whose solution can be quickly verified can also be solved quickly
+- Unsolved as of now, with a $1 million Millennium Prize for its solution
+
+3. Challenges in Numerical Algorithms
+Numerical algorithms face unique challenges due to computational limitations
+and numerical precision:
+Numerical Stability:
+- Small changes in input can lead to large changes in output
+- Rounding errors can accumulate and cause significant inaccuracies
+- Examples: Matrix inversion, solving linear systems
+
+Floating-Point Arithmetic Limitations:
+- Finite precision representation
+- Cannot exactly represent all real numbers
+- Leads to approximation errors
+- Challenges in comparing floating-point numbers
+
+Algorithmic Challenges:
+- Ill-Conditioned Problems: Small input changes cause large output variations
+- Convergence Issues in Iterative Methods
+- Numerical Integration Accuracy
+- Eigenvalue and Eigenvector Computation
+- Linear Algebra Transformations
+
+Mitigation Strategies:
+- Use of extended precision
+- Error analysis techniques
+- Regularization methods
+- Preconditioning in linear algebra
+- Adaptive algorithms
+
+Specific Numerical Algorithm Challenges:
+- Linear System Solving (Gaussian Elimination)
+- Eigenvalue Computation
+- Numerical Integration
+- Ordinary Differential Equation Solving
+- Machine Learning Optimization Algorithms
+
+Computational Complexity Considerations:
+- Time Complexity
+- Space Complexity
+- Numerical Stability
+- Convergence Rate
+- Approximation Error
+
+These challenges require sophisticated techniques like:
+- Iterative Refinement
+- Preconditioning
+- Regularization
+- Advanced Numerical Linear Algebra Methods

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+# SYLLABUS
+## Unit-I
+Introduction to DBMS: Overview of DBMS, File system versus a DBMS, Advantages of a
+DBMS, Three Schema architecture of DBMS, Data Models, Database Languages,
+Transaction Management, Structure of a DBMS, Client/Server Architecture, Database
+Administrator and Users.
+Entity-Relationship Model: Design Issues, ER Modeling concepts, Cardinality constraints,
+Weak-entity types, Subclasses and inheritance, Specialization and Generalization,
+Conceptual Database Design with the ER Model.
+Learning Outcomes: At the end of this unit the students will be able to:
+1. To design and build a simple database system and demonstrate competence with the
+fundamental tasks involved with modeling, designing, and implementing a DBMS.
+2. Describe the fundamental elements of relational database management systems and
+design ER-models to represent simple database application scenarios
+
+## Unit-II
+Relational Model: Structure of Relational Databases, Basics of Relational Model, Integrity
+Constraints, Logical Database Design, Introduction to Views, Destroying/ Altering
+Tables and Views, Relational Algebra, Relational Calculus.
+Learning Outcomes: At the end of this unit the students will be able to
+1. Design entity relationship and convert entity relationship diagrams into RDBMS and
+formulate SQL queries on the respect data into RDBMS and formulate SQL queries
+on the data.
+2. Recall Relational Algebra concepts, and use it to translate queries to Relational
+Algebra statements and vice versa.
+
+## Unit-III
+SQL: Concept of DDL, DML, DCL, set operations, Nested queries, Aggregate Functions,
+NullValues, Referential Integrity Constraints, assertions, views, Embedded SQL, Cursors
+Stored
+procedures and triggers, ODBC and JDBC, Triggers and Active Database, designing active
+databases.
+Learning Outcomes: At the end of this unit the students will be able to
+1. Perform PL/SQL programming using concept of Cursor Management,
+Error Handling, Package and Triggers.
+2. Use the basics of SQL and construct queries using SQL in database
+creation andinteraction
+
+
+## Unit-IV
+Database Design: Schema Refinement, Functional Dependencies, Reasoning about
+Functional
+Dependencies, Normalization using functional dependencies, Decomposition, Boyce-Codd
+Normal Form, 3NF, Normalization using multi-valued dependencies, 4NF, 5NF
+Security: Access Control, Discretionary Access Control - Grant and Revoke on Views
+and IntegrityConstraints, Mandatory Access Control.
+Learning Outcomes: At the end of this unit the students will be able to
+1. Apply various Normalization techniques, functional Dependency and Functional
+Decomposition to improve the database design.
+2. Implement typical security techniques in real time applications.
+
+## Unit-V
+Transaction Management: The ACID Properties, Transactions & Schedules, Concurrent
+Execution of Transactions, Lock- Based Concurrency Control.
+Concurrency Control: 2PL, Serializability and Recoverability, Introduction to Lock
+Management, Lock Conversions, Dealing with Deadlocks, Specialized Locking Techniques,
+Concurrency Control without Locking.
+Crash Recovery: Introduction to ARIES, The Log, Other Recovery-Related Structures,
+TheWrite-Ahead Log Protocol, Check pointing, recovering from a System Crash, Media
+Recovery.