Design and analysis of algorithms - Abstract View

Design and analysis of algorithms - Abstract View

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  Design and Analysisof Algorithms Waqas Nawaz, Ph.D. Innopolis University, Russia
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  Outline  Introduction  Motivation Designing an Algorithm  Analysis of Algorithms  Conclusion  References 2
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  Introduction  Algorithm  Aset of well defined rules to solve a computation problem  Specific examples: Find Minimum! and Multiplication  Design: the description of algorithm (pseudo language) and proof of its correctness  Analysis: performance evaluation (complexity analysis) 3 AlgorithmInput Output American British input output N1  N2 
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  Motivation: Algorithm Efficiencyvs. CPU Speed  Task: Sort n=106 numbers  Computer A = 109 instructions/second  Algorithm A = 2n2 instructions  Computer B = 107 instructions/second  Algorithm B = 50nlgn instructions  A =  B = 4   seconds2000 second/nsinstructio10 nsinstructio102 9 26     seconds100 second/nsinstructio10 nsinstructiolg101050 7 66   20 times better!!
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  Algorithm Design Techniques Brute Force & Exhaustive Search  follow definition / try all possibilities  Divide & Conquer  break problem into distinct subproblems  Transformation  convert problem to another one  Dynamic Programming  break problem into overlapping subproblems  Greedy  repeatedly do what is best now  Iterative Improvement  repeatedly improve current solution  Randomization  use random numbers 5
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  Analysis of Algorithms Worst case: (e.g. cards reversely ordered)  Provides an upper bound on running time  An absolute guarantee that the algorithm would not run longer, no matter what the inputs are  Best case: (e.g., cards already ordered)  Input is the one for which the algorithm runs the fastest  Average case: (general case)  Provides a prediction about the running time  Assumes that the input is random 6 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A
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  Conclusion  We studiedthe importance of designing and analysis of algorithm  Different methods and approaches are discussed to create and evaluate an algorithm  It helps us to utilize resources effectively to achieve desired results in a reasonable time  Related Study Materials  Text Books  Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein, Introduction to Algorithms, Third Edition, 2009.  Kleinberg and Tardos, Algorithm Design, 2005.  Online Lectures by Tim Roughgarden on Algorithms: Design and Analysis, Part 1, Stanford University 7
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  References 1. Adil Khan,Course slides on Data structure and algorithms, Innopolis University, 2016. 2. Monica Nicolescu, course slides on Analysis of Algorithms CS 477/677. 3. Arjun Dasgupta et. al., CSE5311, DESIGN AND ANALYSIS OF ALGORITHMS. 4. Jennifer Welch, CSCE 411, Design and Analysis of Algorithms, 2013. 5. Anany Levitin, Fundamentals of the Analysis of Algorithm Efficiency, 2nd edition, chapter 2. 8
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  Appendix: Farmer CrossesRiver Puzzle  A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage.  There is a boat that can fit himself plus either the wolf, the goat, or the cabbage.  If the wolf and the goat are alone on one shore, the wolf will eat the goat.  If the goat and the cabbage are alone on the shore, the goat will eat the cabbage.  How can the farmer bring the wolf, the goat, and the cabbage across the river? 9
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  Appendix: Our MachineModel  RAM: Generic Random Access Machine  Executes operations sequentially  Set of primitive operations:  Arithmetic. Logical, Comparisons, Function calls  Simplifying assumption: all ops cost 1 unit  Eliminates dependence on the speed of our computer, otherwise impossible to verify and to compare 10
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  Sample Algorithm FINDING LARGESTNUMBER INPUT: unsorted array ‘A[n]’of n numbers OUTPUT: largest number ---------------------------------------------------------- 1 large ← A[j] 2 for j ← 2 to length[A] 3 if large < A[j] 4 large ← A[j] 5 end
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  Space and TimeAnalysis (Largest Number Scan Algorithm) SPACE S(n): One “word” is required to run the algorithm (step 1…to store variable ‘large’) TIME T(n): n-1 comparisons are required to find the largest (every comparison takes one cycle) running time execution time for basic operation Number of times basic operation is executed input size T(n) ≈ copC(n)