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Design & Analysis of Algorithms - Vol 3
This course will cover basic concepts in the design and analysis of algorithms. Asymptotic complexity, O() notation Sorting and search Algorithms on graphs: exploration, connectivity, shortest paths, directed acyclic graphs, spanning trees Design techniques: divide and conquer, greedy, dynamic programming Data structures: heaps, union of disjoint sets, search trees Intractability
Course Chapters
Grid Paths
The discussion centers around the problem of counting grid paths in a rectangular grid, focusing on movements from the bottom left corner to the top right corner with specific rules. It explores combinatorial methods to calculate the number of unique paths, including scenarios where certain intersections are blocked. The concepts of inclusion and exclusion are introduced to account for multiple blocked intersections and their effect on path counting.
Inductive Formulation of the Grid Path
The chapter focuses on dynamic programming and its application in solving grid path problems. It explains how paths from the origin to various points can be calculated using recursive relations, while also addressing the concepts of memoization and the significance of handling obstacles or holes in the paths. The distinction between recursive calculations and dynamic programming is highlighted alongside practical examples.
Common Subwords and Subsequences
The chapter focuses on the concepts of common subwords and subsequences, specifically the longest common subword problem. It introduces various methods for determining the length of the longest common subword between two sequences, including brute force and dynamic programming approaches, while also demonstrating how to recover the actual subword from these computations.
Longest Common Subsequence
This chapter explores the concept of the longest common subsequence (LCS) and its significance in fields like bioinformatics and text comparison. It details the algorithmic approach to finding LCS, comparing it to the longest common subword problem, and discusses the computational efficiency using dynamic programming. The application of LCS in real-world scenarios, such as genetic sequencing and text file comparison, highlights its relevance.
Edit Distance
The chapter discusses the concept of Edit Distance, primarily focusing on how it measures the similarity between two documents through the minimum number of operations required for transformation. It elaborates on specific operations such as insertion, deletion, and substitution of characters, demonstrating practical applications in spell checking and genetics through the Levenshtein distance. Additionally, the chapter explores algorithm design around computing edit distance efficiently using dynamic programming techniques.
Matrix Multiplication
The chapter discusses the efficient multiplication of matrices using dynamic programming techniques. It highlights the importance of the order of multiplication in determining the computational cost, with specific examples illustrating how different orders yield varying levels of efficiency. The chapter concludes with a discussion on formulating the problem inductively and filling up a cost matrix in a structured manner.
Linear Programming
Linear programming is a mathematical optimization technique that deals with maximizing or minimizing a linear function subject to linear constraints. The chapter covers the formulation of linear programming problems through practical examples, particularly in the context of maximizing profit from product sales with various constraints. It also explains the geometric interpretation of feasible regions and solutions through vertices.
LP Modeling: Production Planning
The chapter discusses the application of Linear Programming (LP) for production planning in a carpet manufacturing company. It outlines the intricacies of managing workforce, overtime production, hiring, firing, and storage costs linked to fluctuating demand. The chapter emphasizes how to formulate these aspects into a linear programming model to optimize costs while maintaining production efficiency.
Introduction to the Problem
The chapter discusses a linear programming problem related to bandwidth allocation in a communication network involving three users, A, B, and C. It explores how to allocate bandwidth while ensuring sufficient connectivity between users and maximizing revenue based on different rates for each connection type. The process includes identifying variables, constraints, and objective functions to model the optimal allocation of bandwidth within given limits on connections.
Network Flows
The chapter delves into the concept of network flows, specifically in the context of linear programming and the Ford-Fulkerson algorithm. It explains the representation of network flows using directed graphs containing source and sink vertices, and highlights the significance of flow conservation and optimization. Additionally, it discusses the relationship between maximum flow and minimum cut, demonstrating how these principles are crucial for efficiently managing resources in a network.
Reductions
The concept of reductions in problem-solving is explored, specifically in relation to bipartite matching and network flows. A course allocation problem is used as an example, demonstrating how to match teachers with courses based on their preferences. It emphasizes the importance of using existing efficient algorithms for related problems to indirectly solve more complex issues, showcasing the process of translating problems into forms suitable for established algorithms.
Intractability: Checking Algorithms
The chapter delves into the concept of intractability in algorithms, emphasizing the distinction between generating and checking solutions. It highlights important problems such as Boolean satisfiability and the traveling salesman problem, noting that while finding efficient solutions may be difficult or impossible, checking their validity often is not. The chapter concludes by illustrating the relationship between various computational problems and their checking algorithms.