Algorithms_and_Data_Structures_08a

Algorithms_and_Data_Structures_08a - Lecture 8 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 andreas@nuechti.de Lecture 8
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 232 Announcements Midterm October 15 9.45-11.00 in class Remember grading policy: Assignments: 20% of the final grade Midterm: 30% of the final grade Final exam: 50% of the final grade Midterm has 3 parts: 1. Applying discussed algorithms to problems 2. Solving problems similar to homework 3. Solving new problems October 8 review and discussion session
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 233 Last Lecture – Quicksort • Idea: divide divide divide
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 234 Last Lecture – Quicksort • Idea: • Main function: // pivot element // elements less than the pivot // elements greater than the pivot
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 235 Last Lecture – Quicksort One implementation of the Partition function Iteration over the array from left to right using two indices •N o t e : This is based on the Cormen at al. book! // pivot element
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 236 Last Lecture – Quicksort An example, more on assignment #6.
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 237 Lower Bounds for Sorting Precondition : For sorting only key comparisons are allowed. Next, we aim at proving the following theorem. Theorem : The average number of comparisons for sorting keys is
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 238 Lower Bounds for Sorting Example : Consider the following sorting problem with 3 keys. yes yes yes yes yes no no no no no
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 239 Lower Bounds for Sorting Average runtime = average path length Given a binary decision tree , i.e., a tree with root and all nodes have less or equal 2 out edges. Semantic : At the nodes comparisons are per- formed.
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 240 Lower Bounds for Sorting An optimal decision tree has two child nodes for a comparison node, thus we extend the definition: To the extended binary decision tree is that is defined as leaves of where inner nodes of and every inner node has two child nodes. In other words: To every node of with degree < 2 child nodes are attached, such that has out-degree 2.
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 241 Lower Bounds for Sorting Semantic : At the leaves of the tree there are the permutations that correspond to the sortings.
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Algorithms and Data Structures Dr. Andreas Nüchter September 29, 2009 242 Lower Bounds for Sorting Lemma : In other words: In there exists exactly one leave more than inner nodes.
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This note was uploaded on 05/04/2010 for the course CS 320251 taught by Professor Nuechter during the Fall '09 term at Jacobs University Bremen.

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Algorithms_and_Data_Structures_08a - Lecture 8 Computer...

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