algorithms: an introduction

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Algorithms: An Introduction. ‘Algorithm’ is a distortion of Al-Khawarizmi, a Persian mathematician. Section 3.1 of Rosen Fall 2010 CSCE 235 Introduction to Discrete Structures Course web-page: cse.unl.edu/~cse235 Questions : cse235@cse.unl.edu. Outline. Introduction & definition - PowerPoint PPT Presentation

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  • Algorithms: An Introduction

    Section 3.1 of RosenFall 2010CSCE 235 Introduction to Discrete StructuresCourse web-page: cse.unl.edu/~cse235Questions: cse235@cse.unl.edu Algorithm is a distortion of Al-Khawarizmi, a Persian mathematician

    Algorithms: An IntroductionCSCE 235*

    OutlineIntroduction & definitionAlgorithms categories & typesPseudo-codeDesigning an algorithmExample: MaxGreedy AlgorithmsChange

    Algorithms: An IntroductionCSCE 235*

    Computer Science is About Problem SolvingA Problem is specified byThe givens (a formulation)A set of objectsRelations between themThe queryThe information one wants to extract from the formulation, the question to answer

    An algorithm is a method or procedure that solves instances of a problem

    Real WorldComputing WorldObjectsrepresented bydata Structures, ADTs, ClassesRelationsimplemented withrelations & functions (e.g., predicates)ActionsImplemented withalgorithms: a sequence of instructions

    Algorithms: An IntroductionCSCE 235*

    Algorithms: Formal DefinitionDefinition: An algorithm is a sequence of unambiguous instructions for solving a problem.Properties of an algorithmFinite: the algorithm must eventually terminateComplete: Always give a solution when one existsCorrect (sound): Always give a correct solutionFor an algorithm to be an acceptable solution to a problem, it must also be effective. That is, it must give a solution in a reasonable amount of timeEfficient= runs in polynomial time. Thus, effective efficientThere can be many algorithms to solve the same problem

    Algorithms: An IntroductionCSCE 235*

    OutlineIntroduction & definitionAlgorithms categories & typesPseudo-codeDesigning an algorithmExample: MaxGreedy AlgorithmsChange

    Algorithms: An IntroductionCSCE 235*

    Algorithms: General TechniquesThere are many broad categories of algorithmsDeterministic versus Randomized (e.g., Monte-Carlo)Exact versus ApproximationSequential/serial versus Parallel, etc.Some general styles of algorithms includeBrute force (enumerative techniques, exhaustive search)Divide & ConquerTransform & Conquer (reformulation)Greedy Techniques

    Algorithms: An IntroductionCSCE 235*

    OutlineIntroduction & definitionAlgorithms categories & typesPseudo-codeDesigning an algorithmExample: MaxGreedy AlgorithmsChange

    Algorithms: An IntroductionCSCE 235*

    Good Pseudo-Code: ExampleIntersectionInput: Two finite sets A, BOutput: A finite set C such that C = A BC If |A|>|B| Then Swap(A,B) End For every x A Do If x B Then C C {x} Union(C,{x}) End End Return C

    Algorithms: An IntroductionCSCE 235*

    Algorithms: Pseudo-CodeAlgorithms are usually presented using pseudo-codeBad pseudo-code gives too many details or is too implementation specific (i.e., actual C++ or Java code or giving every step of a sub-process such as set union)Good pseudo-code Is a balance between clarity and detailAbstracts the algorithmMakes good use of mathematical notationIs easy to read andFacilitates implementation (reproducible, does not hide away important information)

    Algorithms: An IntroductionCSCE 235*

    Writing Pseudo-Code: AdviceInput/output must properly definedAll your variables must be properly initialized, introduced Variables are instantiated, assigned using All `commands' (while, if, repeat, begin, end) bold face \bfFor i 1 to n DoAll functions in small caps Union(s,t) \sc All constants in courier: pi 3.14 \ttAll variables in italic: temperature 78 (\it, \em)LaTeX: Several algorithm formatting packages exist on WWW

    Algorithms: An IntroductionCSCE 235*

    OutlineIntroduction & definitionAlgorithms categories & typesPseudo-codeDesigning an algorithmExample: MaxGreedy AlgorithmsChange

    Algorithms: An IntroductionCSCE 235*

    Designing an AlgorithmA general approach to designing algorithms is as followsUnderstanding the problem, assess its difficultyChoose an approach (e.g., exact/approximate, deterministic/ probabilistic)(Choose appropriate data structures)Choose a strategyProve TerminationCompletenessCorrectness/soundnessEvaluate complexityImplement and test itCompare to other known approach and algorithms

    Algorithms: An IntroductionCSCE 235*

    Algorithm Example: MaxWhen designing an algorithm, we usually give a formal statement about the problem to solveProblemGiven: a set A={a1,a2,,an} of integersQuestion: find the index i of the maximum integer aiA straightforward idea is Simply store an initial maximum, say a1Compare the stored maximum to every other integer in AUpdate the stored maximum if a new maximum is ever encountered

    Algorithms: An IntroductionCSCE 235*

    Pseudo-code of MaxMaxInput: A finite set A={a1,a2,,an} of integersOutput: The largest element in the set temp a1 For i =2 to n Do If ai > temp Then temp ai End End Return temp

    Algorithms: An IntroductionCSCE 235*

    Algorithms: Other ExamplesCheck Bubble Sort and Insertion Sort in your textbooks which you should have seen ad nauseum in CSE 155 and CSE 156 And which you will see again in CSE 310Let us know if you have any questions

    Algorithms: An IntroductionCSCE 235*

    OutlineIntroduction & definitionAlgorithms categories & typesPseudo-codeDesigning an algorithmExample: MaxGreedy AlgorithmsChange

    Algorithms: An IntroductionCSCE 235*

    Greedy AlgorithmsIn many problems, we wish to not only find a solution, but to find the best or optimal solutionA simple technique that works for some optimization problems is called the greedy techniqueAs the name suggests, we solve a problem by being greedy: Choose what appears now to be the best choiceChoose the most immediate best solution (i.e., think locally)Greedy algorithmsWork well on some (simple) algorithmsUsually they are not guaranteed to produce the best globally optimal solution

    Algorithms: An IntroductionCSCE 235*

    Change-Making ProblemWe want to give change to a customer but we want to minimize the number of total coins we give themProblemGiven: An integer n an a set of coin denominations (c1,c2,,cr) with c1>c2>>crQuery: Find a set of coins d1,d2,,dk such thati=1k di = n and k is minimized

    Algorithms: An IntroductionCSCE 235*

    Greedy Algorithm: ChangeChangeInput: An integer n and a set of coin denominations {c1,c2,,cr} with c1 > c2> >crOutput: A set of coins d1,d2,,dk such that i=1k di = n and k is minimizedC For i =1 to n Do While n ci Do C C {ci} End Return C

    Algorithms: An IntroductionCSCE 235*

    Change: Analysis (1)Will the algorithm always produce an optimal answer?ExampleConsider a coinage system where c1=20, c2=15, c3=7, c4=1We want to give 22 cents in change

    What is the output of the algorithm?Is it optimal?

    It is not optimal because it would give us two c4 and one c1 (3 coins). The optimal change is one c2 and one c3 (2 coins)

    Algorithms: An IntroductionCSCE 235*

    Change: Analysis (2)What about the US currency system: is the algorithm correct in this case?Yes, in fact it is. We can prove it by contradiction.For simplicity, let us considerc1=25, c2=10, c3=5, c4=1

    Algorithms: An IntroductionCSCE 235*

    Optimality of Change (1)Let C={d1,d2,,dk} be the solution given by the greedy algorithm for some integer n.By way of contradiction, assume there is a better solution C={d1,d2,,dl} with lq, the greedy algorithm would have used q by construction. Thus, it is impossible that the greedy uses q

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