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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 33 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 25, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 1 / 34 What Weve Learned Part One 1 Summation Formulae, Induction and Bounding 2 How to compare functions: o,,O, , 3 How to count the running time of algorithms 4 How to solve recurrences that occur when we do (3) 5 Data Structures Hash Binary Search Trees Heaps Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 2 / 34 What Weve Learned Part Deux Dynamic Programming (15.[1,3]) Greedy Algorithms (16.[1,2]) Graphs and Search (22.*) Spanning Trees (23.*) (Single Source) Shortest Paths (24.[1,2,3]) (All Pairs) Shortest Paths (25.[1,2]) Max Flow (26.[1,2,3]) Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 3 / 34 Stuff To Know: EVERYTHING! DP and Greedy Develop (and potentially solve small) problems via DP Activity Selection (or related problems): Greedy Works Graphs BFS, DFS, and Analysis. Classifying edges in directed and undirected graphs Topological Sorting Finding Strongly Connected Components Spanning Trees Kruskals Algorithm (and analysis) Prims Algorithm (and analysis) Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 4 / 34 More Stuff To Know... Single Source Shortest Paths Distance Labels and Relax Path Relaxation Property BellmanFord Algorithm How to do it When (Why?) it works Analysis SSSP Dag How to do it When (Why?) it works Analysis Dijkstras Algorithm How to do it When (Why?) it works Analysis Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 5 / 34 Even More Stuff To Know... All Pairs Shortest Paths Analogue to Matrix Multiplication FloydWarshall How to do it? When (Why?) it works? Analysis Flows What is a flow? What is a cut? What is MFMC Theorem? How to create residual graph G f ? How to do Augmenting Paths algorithm (Ford Fulkerson/Edmonds Karp) Analysis Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 6 / 34 What Weve Learned, Part Trois Matrix Review. Linear (in)dependence, positive definiteness, singularity, range, nullspace, etc. Matrix manipulation: Matrix Multiplication Solving Triangular Systems Cholesky Factorization (Least Squares) Gaussian Elimination Relationship to LUfactorization PA = LU Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 7 / 34 O, , definitions ( g ) = { f : c 1 ,c 2 ,n > 0 such that c 1 g ( n ) f ( n ) c 2 g ( n ) n n } ( g ) = { f  constants c,n > s.t. cg ( n ) f ( n ) n n } O ( g ) = { f  constants c,n > 0 s.t. f ( n ) cg ( n ) n n } Jeff Linderoth (Lehigh University) IE170:Lecture 33 Lecture Notes 8 / 34 o, Notation f o ( g ) lim n f ( n ) g ( n ) = 0 f ( g ) g o ( f ) lim n f ( n ) g ( n ) = f ( g ) lim n f ( n ) g ( n ) = c...
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This note was uploaded on 08/06/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .
 Spring '07
 Ralphs
 Systems Engineering

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