18_Matchings - Wednesday 17/11/10 Dr. Daniel Hughes

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CSC 30155 Wednesday 17/11/10 Dr. Daniel Hughes daniel.hughes@xjtlu.edu.cn
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Today’s Lecture will be Short l FYP Specification Marks (10 mins) l Matching Theory (30 mins) l Matchings in Trees (30 mins) l Matchings in Bipartite Graphs (20 mins) l Feedback Forms (10 mins)
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CSC 30155 Feedback on FYP Specification Marks Dr. Daniel Hughes daniel.hughes@xjtlu.edu.cn
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FYP Specification Feedback l Remember: l Assignment marks are private . Do not look at anyone else’s score or script unless they offer. l At the end of this session, the projects will be collected and returned to your supervisor for safekeeping. l You may now collect your assignment from the desk at the front.
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50 55 60 65 70 75 80 85 90 95 100 1 2 3 4 5 6 7 8 9 10 11 12 13
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Average Overall Marks l 67% was the average mark for this assignment, which is a little high. The marking may be a little harder for the next assignment. l No marks were docked this time for reports that were too long or not handing in two copies. l Next time: 2% will be docked for each extra page (10 pages max). 5% will be docked if you don’t hand in two copies of your report.
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54 56 58 60 62 64 66 68 70 Quality References Plan Conduct Deliverables Solution Aims Understanding
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Distribution of Marks l There are no major issues here. l There was a very small difference between the marks for each section. l You may take five minutes to review your assignment.
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Any Questions?
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CSC 30155 Matching Theory Dr. Daniel Hughes daniel.hughes@xjtlu.edu.cn
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Graph Terminology l A graph G = (V,E) consists of a set V of vertices and a set E of edges. Each edge is an unordered pair of vertices. l V = {1, 2, 3, 4, 5, 6}. l E = { (1,2), (2,3), (3,4), (4, 5), (5, 6), (3, 5), (3, 6) }.
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Graph Terminology l We will assume that E does not contain a loop, which is an edge from a vertex to itself – this causes problems in many instances: remember the shortest path problem? l If G is a graph, we will refer to the vertex set of G as V(G) and the edge set of G as E(G) .
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Graph Terminology l We say that two edges e and f are adjacent (written e ~ f ) if they have a vertex in common. (2,3) ~ (3,5) but (2,3) is not adjacent to (4, 5) . l The degree of a vertex v , written deg(v) is the number of edges adjacent to it, so deg(5) = 3 .
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Some Quick Questions l Is edge (5,6) adjacent to (3,4) ? l What is the value of deg(3) ?
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Matchings l A matching in G is a subset M of E(G) such that no two edges in M are adjacent. So this is a matching.
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Matchings l And this is a matching.
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Matchings l But this is not a matching.
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Maximal Matchings l A matching M is maximal if no edges can be added . That is, for every e E \ M there is an f M such that e ~ f . This matching is maximal.
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Maximal Matchings l But this matching is not maximal:
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Maximum Matchings l The size of a matching M is just the number of edges in M . So this matching has size 2 . l A matching is said to be a maximum matching if it is the largest possible matching for the graph.
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