This document outlines a course on design and analysis of algorithms. It covers topics like algorithm complexity analysis using growth functions, classic algorithm problems like the traveling salesperson problem, and algorithm design techniques like divide-and-conquer, greedy algorithms, and dynamic programming. Example algorithms and problems are provided for each topic. Reference books on algorithms are also listed.
Solution of N Queens Problem genetic algorithm MohammedAlKazal
Artificial Intelligence
Supervised by
Dr. Fawzia R.
Mosul university
Computer Sciences Department
Masters Students 2018 - 2019
Mohammed Al-Gazal
and
Aalaa Alrashidy
Muna Mohammah Saeed
Mustafa Ameen
Mayada Al-Obaidy
Solution of N Queens Problem genetic algorithm MohammedAlKazal
Artificial Intelligence
Supervised by
Dr. Fawzia R.
Mosul university
Computer Sciences Department
Masters Students 2018 - 2019
Mohammed Al-Gazal
and
Aalaa Alrashidy
Muna Mohammah Saeed
Mustafa Ameen
Mayada Al-Obaidy
The Engineer of Industrial Universtiy of Santander, Elkin Santafe, give us a little summary about direct methods for the solution of systems of equations
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Mastering Greedy Algorithms: Optimizing Solutions for Efficiency"22bcs058
Greedy algorithms are fundamental techniques used in computer science and optimization problems. They belong to a class of algorithms that make decisions based on the current best option without considering the overall future consequences. Despite their simplicity and intuitive appeal, greedy algorithms can provide efficient solutions to a wide range of problems across various domains.
At the core of greedy algorithms lies a simple principle: at each step, choose the locally optimal solution that seems best at the moment, with the hope that it will lead to a globally optimal solution. This principle makes greedy algorithms easy to understand and implement, as they typically involve iterating through a set of choices and making decisions based on some criteria.
One of the key characteristics of greedy algorithms is their greedy choice property, which states that at each step, the locally optimal choice leads to an optimal solution overall. This property allows greedy algorithms to make decisions without needing to backtrack or reconsider previous choices, resulting in efficient solutions for many problems.
Greedy algorithms are commonly used in problems involving optimization, scheduling, and combinatorial optimization. Examples include finding the minimum spanning tree in a graph (Prim's and Kruskal's algorithms), finding the shortest path in a weighted graph (Dijkstra's algorithm), and scheduling tasks to minimize completion time (interval scheduling).
Despite their effectiveness in many situations, greedy algorithms may not always produce the optimal solution for a given problem. In some cases, a greedy approach can lead to suboptimal solutions that are not globally optimal. This occurs when the greedy choice property does not guarantee an optimal solution at each step, or when there are conflicting objectives that cannot be resolved by a greedy strategy alone.
To mitigate these limitations, it is essential to carefully analyze the problem at hand and determine whether a greedy approach is appropriate. In some cases, greedy algorithms can be augmented with additional techniques or heuristics to improve their performance or guarantee optimality. Alternatively, other algorithmic paradigms such as dynamic programming or divide and conquer may be better suited for certain problems.
Overall, greedy algorithms offer a powerful and versatile tool for solving optimization problems efficiently. By understanding their principles and characteristics, programmers and researchers can leverage greedy algorithms to tackle a wide range of computational challenges and design elegant solutions that balance simplicity and effectiveness.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
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2. Course Outline
1. Introduction to Algorithms
2. Mathematical Preliminaries
3. Growth Functions
4. Recurrences
5. Overview of Data Structures
6. Sorting and Searching Algorithms
7. Tree Algorithms
8. Graph Algorithms
9. Strings and Pattern Matching
10. Dynamic Programming
11. Theory of NP-Completeness
3. Reference Books
1. T. Corman, E. Leiserson, L. Rivest, C. Stein, Introduction to
Algorithms, 3rd Edition, Prentice Hall
2. R. Sedgewik, P. Flajolet, An Introduction to the Analysis of
Algorithms, 2nd Edition, Addison-Wesley
3. A. Levitin, Introduction to Design and Analysis of Algorithms, 3rd
Edition, Pearson Education Inc
5. Algorithms - History
• Procedures for solving geometric and arithmetic problems were formulated by ancient
Greeks.
• Some two thousands years ago, the celebrated procedure for finding greatest
common divisor (g c d) was discovered by Euclid.
• In tenth century, Persian mathematician Muhammad- inbne- Musa Al lkowarzimi
stated procedures for solving algebraic equations. The procedure are described in his
famous book Kitab al Jabr wa’l muqabla (‘Rules of restoring and equating’).
The word algorithm is coined from Alkhowarizm
• In the mid- twentieth century D.E. Knuth undertook in depth study and analysis of
algorithms. His work is embodied in the comprehensive book ‘Art of Computer
Programming’, which serves as foundation for modern study of algorithms.
6. Algorithms – Formal Definition
An algorithm is an orderly step-by-step procedure, which has the
characteristics:
1) It accepts one or more input value
2) It returns at least one output value
3) It terminates after finite steps
8. Algorithms – Classic Problems
• Hamiltonian Circuit Problem
• Traveling Salesperson Problem
• Graph Coloring Problem
• The Sum-Set Problem
• The n-Queen Problem
9. Hamiltonian Circuit Problem
Problem: Determine if a given graph has a Hamiltonian circuit.
• A Hamiltonian circuit , also called tour, is path that starts and ends at
the same vertex, and passes through all other vertices exactly once.
• A graph can have more than one Hamiltonian circuit, or none at all.
10. Traveling Salesperson Problem
Problem(a): A salesperson has to travel to n cities, such he starts at one city
and ends up at the same city, visits each city exactly once , and covers
minimum distance.
Problem(b): Determine minimum tour for a graph with n vertices.
11. Graph Coloring Problem
• Problem :. For given graph, find the minimum number of colors that can be
used to color the vertices so that no two adjacent vertices have the same
color.
12. Graph Coloring Problem
• Problem :. For given graph, find the minimum number of colors that can be
used to color the vertices so that no two adjacent vertices have the same
color.
13. Sum of Subset Problem
Problem : Given a set of set S = {s1, s2, s3, … sn} of n positive integers and an
integer w, find all subsets of S such that sum of integers, in each subset, equals
w.
Suppose S = {s1, s2, s3, s4, s5}
Let s1=5, s2 = 6, s3 = 10, s4 = 11 ,s5 = 16, and w = 21
• s3 + s4 = 10 + 11 = 21
In general, for a set of n integers, there are 2n subsets need to be examined.
For example, for n=20, the number of subsets is 220 = 1,048,576
14. Sum of Subset Problem
Problem : Given a set of set S = {s1, s2, s3, … sn} of n positive integers and an
integer w, find all subsets of S such that sum of integers, in each subset, equals
w.
Suppose S = {s1, s2, s3, s4, s5}
Let s1=5, s2 = 6, s3 = 10, s4 = 11 ,s5 = 16, and w = 21
• s1 + s2 + s3 = 5 + 6 + 10 =21
• s1 + s5 = 5 + 16 = 21
• s3 + s4 = 10 + 11 = 21
In general, for a set of n integers, there are 2n subsets need to be examined.
For example, for n=20, the number of subsets is 220 = 1,048,576
15. n-Queen Problem
Problem : n queens are to be placed on an n by n chessboard so that no two
queens threat each other. A queen threats another queen if is on the same row,
or same column, or same diagonal of the chess board.
17. Divide-and-Conquer Algorithm
The divide and conquer algorithm has the following approach:
1) The problem is split (divided) into two distinct independent sub problems
2) The sub problems are solved (conquered) separately
3) The solutions to the sub problems are merged together to obtain solution to
main problem.
19. Divide-and-Conquer Algorithm
• Quick Sort
• Merge Sort
• Binary Search
• Multiplication of Large Integers
• Matrix Multiplication
• Fast Fourier Transform
20. Divide-and-Conquer Algorithm
Divide and conquer algorithm provides efficient sorting and searching methods.
It, however, has following limitations:
• The divide-and-conquer algorithm is implemented using recursive function
calls, which is not supported by some of the programming languages.
• Recursive calls need large stacks to store intermediate results, utilize
computer’s special memory called heap which may not be available on some
systems and decrease data access speed
• The divide-and-conquer algorithm is in-efficient when sub problems are not
of nearly equal size.
21. Greedy Algorithm
Greedy Algorithm is designed to solve optimization problems. The greedy algorithm
has the follows approach:
1) The problem is solved in steps. At each step, several choices are available for
partial solutions. A greedy choice is made by selecting the option with maximum
( or minimum) cost. This is referred to as locally optimum solution, as opposed
to globally optimum solution, which is expected to be the overall optimum
solution.
2) It is checked if the selected option satisfies problems constraint ( for example, the
cost does not exceed the prescribed limit ).This is referred to as feasible solution.
3) If current solution is not feasible, the next best option is selected.
4) Once a choice is made, it is not changed in subsequent steps. This condition is
referred to as irrevocability
23. Greedy Algorithm
Minimum Cabling in network
• Prim’s Algorithm
• Kruskal’s Algorithm
Shortest route in a network
• Dijkstara Algorithm
Huffman coding for text compression
Optimum Job Scheduling
24. Greedy Algorithm
The greedy algorithm provides is a simple and elegant solution to optimization
problems. It has, however, the following limitations:
• The greedy algorithm does not always produce a globally optimum solution ,
(for example, in case of traveling salesperson problem).
• In order to check whether or not the resulting solution is optimal, further
analysis is required.
• Often the analysis for globally optimum solution is quite complex.
• Analysis may requires additional resources.