Graph_theory_Part3_students_2Pp

Graph_theory_Part3_students_2Pp - Graphs and Related Theory...

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1 2/2/2006; rev. 9/24/06 M. A. Breuer and others 1 Graphs and Related Theory and Algorithms with applications to VLSI layout Part 3 of 3 Fill-in Answer Animation Only 2/2/2006; rev. 9/24/06 M. A. Breuer and others 2 Basic graph algorithms commonly used in VLSI CAD Spanning Trees Minimum Spanning Tree (MiST) Problems Steiner Minimum Tree (StMiT) Problems Rectilinear RStMiT Problems Matching Graph Search Breadth first Depth first Shortest Path All Shortest Paths
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2 2/2/2006; rev. 9/24/06 M. A. Breuer and others 3 Making 4 nodes electrically common K 4 MST- later SMT- later A cycle Fan-out from source 2/2/2006; rev. 9/24/06 M. A. Breuer and others 4 Spanning trees A spanning tree T={V T , E T ) of a graph is a subgraph T of G, where T is a tree (no cycles) and such that V T = V and E T E.
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3 2/2/2006; rev. 9/24/06 M. A. Breuer and others 5 More on Spanning trees G 1 What is the spanning tree of this graph? G 2 How many spanning trees does G 2 have? Find all of the spanning trees of this graph G 3 answer 2/2/2006; rev. 9/24/06 M. A. Breuer and others 6 Spanning trees (Cont’d) Consider a weighted (edge) graph G, i.e., a graph where each edge e i is associated with a weight w i . Then the weight of G is w i , where the ____________ is over all edges. SF LA NY 480 2300 2000 Fill-in
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4 2/2/2006; rev. 9/24/06 M. A. Breuer and others 7 Minimum Spanning Trees (MST) Problem : Given a connected undirected weighted graph G(V, E), construct a minimum weighted tree T connecting all vertices in V. This is usually referred to as the minimal spanning tree problem . 1.2 1.3 5.2 6.1 2/2/2006; rev. 9/24/06 M. A. Breuer and others 8 Minimum Spanning Tree Algorithms Kruskal’s algorithm: O (m log n) Prim’s algorithm : O (m log n) Can be improved to O (n log n) A greedy algorithm Note: m = # of edges; n = # of vertices
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5 2/2/2006; rev. 9/24/06 M. A. Breuer and others 9 Kruskal’s Algorithm Sort the edges in ascending order of their weights Set T = φ Consider the edges in ascending order of their weights until T contains n-1 edges: Add an edge e to T iff adding e to T does not create any cycle in T What computations are required? Determining the length of each edge Sorting the edges Determining if adding an edge to T would form a loop Kruskal’s algorithm is an example of a greedy algorithm 2/2/2006; rev. 9/24/06 M. A. Breuer and others 10 Example: Kruskal’s Algorithm e
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6 2/2/2006; rev. 9/24/06 M. A. Breuer and others 11 Example: Kruskal’s Algorithm (Cont’d) e 2/2/2006; rev. Example: Kruskal’s Algorithm (Cont’d)
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Graph_theory_Part3_students_2Pp - Graphs and Related Theory...

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