381notes - Discrete Mathematical Modeling Math 381 Course...

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Unformatted text preview: Discrete Mathematical Modeling Math 381 Course Notes University of Washington Prof. Sara Billey Fall Quarter, 2008 c R. J. LeVeque 2 Acknowledgments : These notes are based on many quarters of Math 381 at the University of Washington. The original notes were compiled by Randy LeVeque in Applied Math at UW. Tim Chartier and Anne Greenbaum contributed significantly to the current version in 2001-2002. Jim Burke contributed a portion of the chapter on linear programming. Sara Billey contributed a portion of the chapter on Stochastic Processes. We thank them for their efforts. Since this document is still evolving, we appreciate the input of our readers in improving the text and its content. Math 381 Notes University of Washington Fall 2008 Contents 1 Introduction to Modeling 1 1.1 Lifeboats and life vests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The goals of 381 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Graph Theory and Network Flows 5 2.1 Graphs and networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The traveling salesman problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Brute force search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Counting the possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Enumerating all possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.4 Branch and bound algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Shortest path problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1 Word ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2 Enumerating all paths and a branch and bound algorithm . . . . . . . . . . . . . 10 2.4.3 Dijkstras algorithm for shortest paths . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Minimum spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 Kruskals algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.2 Prims algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5.3 Minimum spanning trees and the Euclidean TSP . . . . . . . . . . . . . . . . . . 12 2.6 Eulerian closed chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 P , NP , and NP-complete problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Stochastic Processes 17 3.1 Basic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Probability Distributions and Random Variables . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Expected Value ....
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This note was uploaded on 10/17/2008 for the course STAT 390 taught by Professor Marzban during the Spring '08 term at Washington State University .

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381notes - Discrete Mathematical Modeling Math 381 Course...

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