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p114_exercises_4-2
Course: STAT 601, Spring 2010School: Texas A&M
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Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
Texas A&M - STAT - 601
UCSD - MAE - 101b
NAME:MAE 101B Advanced Fluid Mechanics - Spring 2010 Midterm # 2 - May 3, 2010 50 minutes, open book, open notes, calculator allowed, no cell phones. Write your answers directly on the exam. 30 points total + 5 extra credit points (optional). The exam is
USC - CSCI - 402
Copyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California
Native American Education Services - EE - 15.082
15.082 and 6.855JSpring 2003Network OptimizationJ.B. Orlin1WELCOME!x x x x xWelcome to 15.082/6.855J Introduction to Network Optimization Instructor: James B. Orlin (jorlin@mit.edu) TA: Agustin Bompadre (abompadr@mit.edu) Website: sloanspace.mit.ed
Native American Education Services - EE - 15.082
15.082 and 6.855J February 6, 2003Data Structures1Overview of this Lecturexx xA very fast overview of some data structures that we will be using this semester lists, sets, stacks, queues, networks, trees a variation on the well known heap data struc
Native American Education Services - EE - 15.082
15.082 and 6.855J February 11, 2003Lecture 3. Graph SearchBreadth First Search Depth First Search Topological Sort1OverviewToday: Different ways of searching a graph a generic approach breadth first search depth first search data structures to suppor
Native American Education Services - EE - 15.082
15.082 and 6.855J February 13, 2003 Flow Decomposition and Transformations1Flow Decomposition and Transformationsx x x xFlow Decomposition Removing Lower Bounds Removing Upper Bounds Node splitting Arc flows: an arc flow x is a vector x satisfying: Le
Native American Education Services - EE - 15.082
15.082 and 6.855J February 20, 2003Dijkstra's Algorithm for the Shortest Path Problem1Wide Range of Shortest Path ProblemsxSources and Destinations We will consider single source problems in this lecture Properties of the costs. We will consider non-
Native American Education Services - EE - 15.082
15.082 and 6.855JShortest Paths 2: Application to lot-sizing R-Heaps1Application 19.19. Dynamic Lot Sizing (1)xK periods of demand for a product. The demand is dj in period j. Assume that dj > 0 for j = 1 to K. Cost of producing pj units in period j:
Native American Education Services - EE - 15.082
15.082J / 6.855J February 27, 2003The Label Correcting AlgorithmOverview of the Lecturexx xA generic algorithm for solving shortest path problems negative costs permitted but no negative cost cycle (at least for now) The use of reduced costs All pair
Native American Education Services - EE - 15.082
15.082J and 6.855J March 4, 2003Introduction to Maximum Flows1The Max Flow ProblemGij=(N,A)x =ijflow on arc (i,j) capacity of flow in arc (i,j) v source node sink j xij node k xki = 0 for each i s,t= v 0 xij uij for all (i,j) A.u = Maximize s
Native American Education Services - EE - 15.082
15.082 and 6.855J March 6, 2003Maximum Flows 21Network Reliabilityx xCommunication Network What is the maximum number of arc disjoint paths from s to t? How can we determine this number?Theorem. Let G = (N,A) be a directed graph. Then the maximum nu
Native American Education Services - EE - 15.082
15.082 and 6.855J March 11, 2003Max Flows 3 Preflow-Push Algorithms1Review of Augmenting PathsAt each iteration: maintain a flow x Let G(x) be the residual network At each iteration, find a path from s to t in G(x). In the shortest augmenting path alg
Native American Education Services - EE - 15.082
15.082 and 6.855J March 13, 2003 Max Flows 41Overview of today's lecturexVery quick review of Preflow Push Algorithm The Excess Scaling Algorithm O(n2 log U) non-saturating pushes O(nm + n2 log U) running time. A proof that Highest Preflow Push uses O
Native American Education Services - EE - 15.082
15.082 and 6.855J April 1, 2003The Global Min Cut problem1Global Min cutINPUT: A network G = (N, A) OUTPUT: A cut (S, N\S) such that cap(S, N\S) is minimum. Note: We do not assume that there is a source node s and destination node t. Typically, but no
Native American Education Services - EE - 15.082
15.082 and 6.855JApril 3, 2003Introduction to Minimum Cost Flows1The Minimum Cost Flow Problemuij = capacity of arc (i,j). cij = unit cost of shipping flow from node i to node j on (i,j). xij = amount shipped on arc (i,j) Minimize j xij (i,j)A cijxij
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Successive Shortest Path Algorithm and the Capacity Scaling Algorithm for the Minimum Cost Flow Problem1Pseudo-FlowsA pseudo-flow is a "flow" vector x such that 0 x u. Let e(i) denote the excess (deficit) at node i. The infeasibli
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Network Simplex Algorithm1Calculating A Spanning Tree Flow1 1 -6 2 1 3 5 3 6 -4 73A tree with supplies and demands. (Assume that all other arcs have a flow of 0) What is the flow in arc (4,3)?24See the animation.2What would
Native American Education Services - EE - 15.082
15.082 and 6.855The Minimum Cost Spanning Tree Problem1Communications SystemsConsider a communications company, such as AT&T or GTE that needs to build a communication network that connects n different users. The cost of making a link joining i and j
Native American Education Services - EE - 15.082
15.082 and 6.855JReview of Linear Programming1OverviewDescribe LP and IP min cost flow as an LP Graphical solution Basic feasible solutions. Simplex Method Basic feasible solutions in matrix form Duality Note: this will cover lots of material. We will
Native American Education Services - EE - 15.082
15.082J and 6.855JGeneralized Flows1Overview of Generalized FlowsSuppose one unit of flow is sent in (i,j). We relax the assumption that one unit arrives at node j. If 1 unit is sent from i, ij units arrive at j. ij is called the multiplier of (i,j) i
Native American Education Services - EE - 15.082
15.082 and 6.855JLagrangian Relaxation I never missed the opportunity to remove obstacles in the way of unity.-Mohandas Gandhi1On bounding in optimizationIn solving network flow problems, we not only solve the problem, but we provide a guarantee that
Native American Education Services - EE - 15.082
15.082J and 6.855JLagrangian Relaxation 2 xAlgorithms xApplication to LPs1The Constrained Shortest Path Problem2(1,10) (1,2)(1,1) (2,3) (5,7)4(1,7)1(10,1) (2,2)6(10,3)3(12,3)5Find the shortest path from node 1 to node 6 with a transit time
Native American Education Services - EE - 15.082
15.082 and 6.855The Multicommodity Flow Problem1On the Multicommodity Flow Problem O-D versionK origin-destination pairs of nodes (s1, t1), (s2, t2), ., (sK, tK) Network G = (N, A) dk = amount of flow that must be sent from sk to tk. uij = capacity on
Native American Education Services - EE - 15.082
15.082 and 6.855Multicommodity Flows 21On the Multicommodity Flow Problem O-D versionK origin-destination pairs of nodes (s1, t1), (s2, t2), ., (sK, tK) Network G = (N, A) dk = amount of flow that must be sent from sk to tk. uij = capacity on (i,j) sh
Native American Education Services - EE - 15.082