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Review for Exam 1 Chinese Postman Problem Traveling Salesman Problem Network Problem Critical Path Analysis Spring 2015 Mathematics in Management Science Exam 1 – Wed Feb 11 Chapters 1 & 2. Please bring pencils and a valid UC ID . Please know your Section # and M # . No phones allowed. Seats will be assigned – come early. Exam 1 – Wed Feb 11 • Please sit in every other seat starting in the front rows. • Please be certain to include your: M # , Version Letter • Please have your ID out. • No phones out. Example What are critical paths? What is ECT? Example Spp task 6 time reduced by 4. What is new ECT? 3 Example Spp task 7 time increased by 4. What is new ECT? 12 Example What are critical paths and ECT for pictured job? Chinese Postman Problem Given connected graph: Find a minimal length circuit traversing entire graph. If the graph has an Euler circuit, it is such a minimal length circuit. If not: Eulerize the graph Find an EC on new graph Transfer back to original graph Solving CPP – easy case If the graph has an Euler circuit (or path), it is a minimal length circuit (or path) traversing the entire graph. Euler’s Circuit/Path Theorems tells us exactly when a graph has EC/EP. How do we actually find an Euler circuit or Euler path? (Fleury’s Algorithm) Euler’s Theorem A graph has an Euler circuit if and only if the graph is connected, and, all its vertices have even valence. Fleury’s Algorithm Check that the graph is connected and either has no odd vertices (for circuit), or has two odd vertices (for path). Choose a starting vertex. For a circuit, can be any vertex; for path, it must be one of the two odd vertices. At each step, if you have a choice, don’t choose a bridge of the yet-tobe-traveled part of the graph. Solving CPP – harder case What if there are no ECs or EPs? No EC means: some odd vertex! No EP means….(?) No EC/EP: gotta retrace edge(s). • Eulerize the graph. • Find an EC (or EP) on new graph. • Transfer back to original graph. Eulerizing Graphs – to get EC If all vtxs have even, done (bcuz have an EC). Otherwise, have odd vtxs: mark them. (Must have an even number of odd vtxs!) Pick an odd vtx. Duplicate an existing edge having this vtx as an endpoint. Change degrees of two vtxs just connected. All degrees even – done; otherwise, repeat the above step. Finding Good Eulerizations Locate odd vertices. Try to duplicate any connecting edges between pairs of odd vertices. Sometimes vertices are more than one edge apart; in this case, gotta duplicate edges between vertices. Look for the fewest edges to duplicate to make all vertices even. Solving a TSP TSP Given complete weighted graph, find a minimum cost Hamiltonian circuit. Brute Force Algorithm Nearest Neighbor Algorithm Repetitive Nearest Neighbor Algorithm Sorted Edges Algorithm Nearest Neighbor Algorithm From a given vertex, go to the nearest vertex not already visited. Repetitive Nearest Neighbor Do NNA with each vertex of the graph as the starting vertex. Of all the circuits obtained, keep the best one. If there is a designated starting vertex, rewrite this best circuit using that vertex as the reference point. Sorted Edges Algorithm Pick the cheapest link (edge with smallest weight) available. Continue picking next cheapest link available that does not close a circuit which is not an HC, nor create three edges coming out of a single vertex. Done when get a Hamilton circuit. CPP versus TSP CPP Given a connected graph, find a minimal length circuit (or path) traversing all edges of the graph. TSP Given a complete weighted graph, find a minimum weight Hamilton circuit (visiting all vtxs just once). Network Problem Find a minimal cost spanning tree in a given weighted graph. Of all possible spanning trees, the one whose sum of the associated weights is as small as possible is called a minimum-cost spanning tree. Kruskal’s Algorithm Use the edges, in order of increasing cost, so that –no circuits get formed Done when –all vertices on some edge –connected Critical Paths Look at all paths; each has an associated completion time (the sum of all processing times for tasks along the path). A path with the longest processing time is called a critical path. Can find these via backflow algorithm; we use eyeball approach. Earliest Completion Time The shortest possible time it takes to complete the overall project. You cannot do better than this. A major factor in the overall cost. Just the length of a critical path: ECT=length of critical path