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Algorithms: Design and Analysis

Algorithms: Design and Analysis. 240-310 , Semester 2, 2018-2019. Objective to give some background on the course. Please ask questions. 0. Preliminaries. Who I am: Andrew Davison WiG Lab ad@fivedots.coe.psu.ac.th. 1. What is a Algorithm?.

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Algorithms: Design and Analysis

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  1. Algorithms: Design and Analysis 240-310, Semester2, 2018-2019 • Objective • to give some background on the course Please ask questions 0. Preliminaries Who I am: Andrew DavisonWiG Lab ad@fivedots.coe.psu.ac.th

  2. 1. What is a Algorithm? • An algorithm is a finite set of unambiguous instructions for solving a problem. • An algorithm is correct if on all legitimate inputs, it outputs the right answer in a finite amount of time • Can be expressed as • pseudocode • flow charts • text in a natural language (e.g. English) • computer code

  3. The Importance of Algorithms • Their impact is broad and far-reaching. • Internet. Web search, packet routing, distributed file sharing, ... • Biology. Human genome project, protein folding, ... • Computers. Circuit layout, file system, compilers, ... • Computer graphics. Movies, video games, virtual reality, ... • Security. Cell phones, e-commerce, voting machines, ... • Multimedia. MP3, JPG, DivX, HDTV, face recognition, ... • Social networks. Recommendations, news feeds, advertisements, ... • Physics. N-body simulation, particle collision simulation, ...

  4. The Top 10 Algorithms of the 20th Century • Ten algorithms having "the greatest influence on the development and practice of science and engineering in the 20th century". • Dongarra and SullivanTop Ten Algorithms of the CenturyComputing in Science and EngineeringJanuary/February 2000 • Barry CipraThe Best of the 20th Century: Editors Name Top 10 AlgorithmsSIAM NewsVolume 33, Number 4, May 2000 • http://www.siam.org/pdf/news/637.pdf

  5. What are the Top 10? • 1946: The Metropolis (Monte Carlo) Algorithm. Uses random processes to find answers to problems that are too complicated to solve exactly. • 1947: Simplex Method for Linear Programming.A fast technique for maximizing or minimizing a linear function of several variables, applicable to planning and decision-making. • 1950: Krylov Subspace Iteration Method.A technique for rapidly solving the linear equations that are common in scientific computation.

  6. 1951: The Decompositional Approach to Matrix Computations. A collection of techniques for numerical linear algebra. • 1957: The Fortran Optimizing Compiler. • 1959: QR Algorithm for Computing Eigenvalues. A crucial matrix operation made swift and practical. Application areas include computer vision, vibration analysis, data analysis. • 1962: Quicksort Algorithm. We will look at this.

  7. 1965: Fast Fourier Transform (FFT). It breaks down waveforms (like sound) into periodic components. Used in many different areas (e.g. digital signal processing , solving partial differential equations, fast multiplication of large integers.) • 1977: Integer Relation Detection. A fast method for finding simple equations that explain collections of data. • 1987: Fast Multipole Method. Deals with the complexity of n-body calculations. It is applied in problems ranging from celestial mechanics to protein folding.

  8. 2. Some Algorithm Types • Simple recursive algorithms • Backtracking • Brute force • Divide and conquer • Dynamic programming • Greedy algorithms • Randomized algorithms

  9. Simple Recursive • A simple recursivealgorithm: • Non-recursive base case • Recurs with a simpler subproblem • Examples: • Count the number of occurrence of an element in a tree • Test if a value occurs in a list • Fibonnaci number calculation • Several of the other algorithm types use recursion in more complex ways

  10. Fibonnaci numbers • The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 33, ... • Find the nth Fibonacci number: fib(int n) if (n <= 1) return 1; else return fib(n-1) + fib(n-2);

  11. lots of repeated work, that only gets worse for bigger n • Execution of fib(5):

  12. Backtracking • Backtracking algorithms use a depth-first recursive search • involves choice (non-determinism) • backtracking means "go back to where you came from" • Examples: • search a maze • color a map with no more than four colors

  13. Brute Force • A brute force algorithm tries all possibilities until a solution is found. • often this is implemented using backtracking • Brute force algorithms usually have exponential running time (very large time, so very slow) • Various heuristics and optimizations can be used • heuristic means “a rule of thumb” • e.g. look for sub-optimal solutions (not the best), which can be calculated more quickly than the best one

  14. Divide and Conquer • A divide and conquer algorithm: • Divide the problem into smaller subproblems • Combine the solutions to the subproblems into a solution to the original problem • Examples: • quicksort • merge sort • binary search

  15. Dynamic Programming • A dynamic programming algorithm remembers past results and uses them to find new results. • often use a table (2D array) to store results • memorization = remember results in a table • the problen contains repeating subproblems • since past results are stored, they can be reused instead of being calculated again

  16. Fibonacci numbers Again • Finding the nth Fibonacci number involves lots of repeated work that can be avoided using memorization:

  17. Greedy algorithms • At each step use the best solution you can get right now, and don't allow for backtracking to change the choice later • no backtracking makes the coding easier and faster • You hope that by choosing the best at each step, you will end up with the best combined final solution

  18. Example: Counting Money • Count out some money using the fewest possible notes and coins • A greedy algorithm will take the largest possible note or coin at each step • Example: count out $6.39 using: • a $5 bill • a $1 bill // to make $6 • a 25¢ coin // to make $6.25 • a 10¢ coin // to make $6.35 • four 1¢ coins // to make $6.39

  19. Greedy Algorithms Can 'Fail' • “Krons” money come in 1, 7, and 10 coins • Count out 15 krons: • A 10 kron piece • Five 1 kron pieces, for a total of 15 krons • This requires 6 coins, but a better solution is two 7 kron pieces and one 1 kron piece (3 coins) • The greedy algorithm 'fails' because its final solution is not the best

  20. Randomized Algorithms • A randomized algorithm uses a random number to make a choice during the computation • faster than calculating a choice • the choice may be just as good • Examples: • Quicksort randomly chooses a pivot • Factor a large number by choosing random numbers as possible divisors

  21. 3. Analysis of Algorithms The theoretical study of algorithm performanceand resource usage. Performance isn't the only important things for code: • modularity • user-friendliness • correctness • programmer time • maintainability • simplicity • functionality • extensibility • robustness • reliability This subject isn't about these things.

  22. Why study Algorithm Analysis? • It help us to understand algorithm scalability. • Performance often draws the line between what is feasible and what is impossible. • Algorithmic mathematics provides a precise way totalk about program behavior. • The lessons of program performance generalize to other computing resources.

  23. 4. Course Structure • Running time of programs • Backtracking • Divide-and-conquer; quicksort • Linear sorting • Hashing • Dynamic programming • Graph algorithms • search, shortest path • not minimum spanning trees, which was done in Discrete Maths • Heaps • P and NP problems

  24. Meeting Times / Locations • Monday 9:00 – 10:30 IDLWednesday 9:00 – 10:30 IDL • I want to change these times to be 3 classes/week, each of 1 hour. • Tell me your preferences.

  25. Workload • Mid-term exam: 35% (2 hours) • week 9 • Final exam: 45% (3 hours) • weeks 17-18 • Two exercises: 20%(2*10) • weeks 7-8 and weeks 15-16

  26. Non-Attendence Penalty • I may take registration at the start of a class. • If someone is not there, they lose 1%(unless they have a good excuse). • A maximum of 10% can be lost • deducted from your final mark

  27. Course Materials • All the handouts (and other materials)will be placed on-line at http://fivedots.coe.psu.ac.th/Software.coe/Algorithms/ • Print 6 slides-per-page, grayscale, and bring to class.

  28. 5. Books • Intro to Java Programming, Comprehensive, 10th ed.Y. Daniel Liang • http://www.cs.armstrong.edu/liang/intro10e/ • Topics • Generics, Collections (lists, stack, queues,sets, maps) • Sorting, searching • graph algorithms (2 chapters)

  29. Algorithms, 4th ed.Robert Sedgewick, Kevin Wayne • http://algs4.cs.princeton.edu/ • Introduction to Algorithms, 3rd ed.Thomas Cormen, Charles Leiserson, Ronald Rivest, Clifford Stein • lots of resources online; see video section

  30. การออกแบบและวิเคราะห์อัลกอริทึมการออกแบบและวิเคราะห์อัลกอริทึม ผู้แต่ง : สมชาย ประสิทธิ์จูตระกูล http://www.chulabook.com/ http://www.cp.eng.chula.ac.th/~somchai/books/ โครงสร้างข้อมูลและอัลกอริทึมData Structures and Algorithms) ผู้แต่ง : สุธี พงศาสกุลชัย&ณัฐพงษ์ วารีประเสริฐ http://ktpbook.com

  31. Fun Overviews • Algorithms UnlockedThomas H. CormenMIT Press, March 2013 • Nine Algorithms that Changed the FutureJohn MacCormickPrinceton University Press, 2011 • http://users.dickinson.edu/~jmac/9algorithms/ • search engine indexing, pagerank, public key cryptography, error-correcting codes, pattern recognition, data compression, databases, digital signatures, computablity

  32. Algorithmic PuzzlesAnany Levitin, Maria LevitinOxford University Press, , 2011 • Algorithmics: The Spirit of ComputingDavid Harel, Yishai FeldmanAddison-Wesley; 3 ed., 2004(and Springer, 2012) • http://www.wisdom.weizmann.ac.il/~harel/algorithmics.html

  33. The New Turing Omnibus: Sixty-Six Excursions in Computer ScienceA. K. DewdneyHolt, 1993 • 66 short article; e.g. detecting primes, noncomputable functions, self-replicating computers, fractals, genetic algorithms, Newton-Raphson Method, viruses

  34. 6. Videos • MIT 6.046J / 18.410J Intro. to Algorithms, Fall 2005 • http://ocw.mit.edu/6-046JF05 • original course website for Cormen book • http://videolectures.net/mit6046jf05_introduction_algorithms/ • video and slides side-by-side • http://www.catonmat.net/category/introduction-to-algorithms • notes taken while watching the videos

  35. Skiena's Algorithms Lectures • http://www3.cs.stonybrook.edu/~algorith/video-lectures/ • 1997, 2007, 2012

  36. Hi-tech TrekRoyal Institution Christmas Lectures 2008 • A hi-tech trek through the world of computer science by Professor Chris Bishop; aimed at school kids. • Lecture 1: Breaking the speed limit • Lecture 2: Chips with everything • Lecture 3: Ghost in the machine • Lecture 4: Untangling the web • Lecture 5: Digital intelligence • http://richannel.org/christmas-lectures/2008/ • I have copies on a DVD

  37. 7. Web Sites • Algorithm Tutorials • http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=alg_index • Algorithmy • http://en.algoritmy.net/ • brief explanations and code • Algorithmist • http://algorithmist.com/index.php/Main_Page • explanations and code • Wikipedia page for an algorithm • e.g. http://en.wikipedia.org/wiki/Quicksort

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