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Algorithms_and_Data_Structures_02

# Algorithms_and_Data_Structures_02 - Lecture 2 Computer...

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Computer Science Algorithms and Data Structures Prof. Dr. Andreas Nüchter Research I, Room 105 Jacobs University Bremen http://www.nuechti.de Lecture 2

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Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 36 Last Lecture Examples of Algorithms Summation according to C.F. Gauss Binary Search Efficient multiplication for and addition chains Sub string search – sketch of the Boyer and Moor algorithm All examples had in common that the ad-hoc solution has high runtime. Smart solutions lowered the run-time tremendously
Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 37 Questions? What’s a computability problem? What’s an algorithm? Which problems can be solved algorithmically and which cannot? Which problems can be solved in feasible time? What different algorithmic approaches are there? How depend the algorithmic solutions of a problem on each other? Can we identify “good” techniques for solving problems?

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Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 38 Questions? What is a complex problem? Is there an optimal solution to a problem? Can we verify that an algorithm solves a problem? These are the questions, we are going to discuss in this class! We need some math background for doing so.
Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 39 Set Theory – Symbols (1) for all exists and or implies from follows cardinality , is finite

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Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 40 Set Theory – Symbols (2) Cartesian product iff and n-fold Cartesian product subset empty set, i.e., Relation between and n-times
Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 41 Set Theory – Symbols (3) iff and iff and (real subset) power set union intersection

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Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 42 Set Theory – Symbols (4) set difference It holds:
Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 43 Set Theory – Symbols (5) less or equal less greater or equal greater

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Algorithms and Data Structures Dr. Andreas Nüchter September 8, 2009 44 Computability What is a computability problem? Problem Specification 1. Attempt A computability problem is specified by a triple with not empty sets and a relation is the set of possible inputs is the set of possible outputs is the set of valid inputs describes a relation between the input and the output
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Algorithms_and_Data_Structures_02 - Lecture 2 Computer...

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