4-ISYE3400_Ch 9 and 10_MST&SPP-Spring2018.pptx

# 4-ISYE3400_Ch 9 and 10_MST&SPP-Spring2018.pptx -...

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1 Minimum Spanning Tree & Shortest Path Problem

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Learning Objectives 1. Know definitions of network terminology 2. Define the Minimum Spanning Tree problem and solve with greedy algorithm 3. Identify Shortest Path Problems and solve with Dijkstra’s Algorithm and LP Solver
Minimum Spanning Tree This problem involves choosing for the network the links that have the shortest total length while providing a path between each pair of nodes. Remember a path is a sequence of distinct arcs connecting two nodes of a network (i.e. direct or indirect connection between nodes) These links must be chosen so that the resulting network forms a tree that spans all the given nodes of the network, i.e., a spanning tree with minimum total length of the links. 3

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Minimum Spanning Tree Suppose that each arc (i,j) in a network has a length associated with it and that arc (i,j) represents a way of connecting node i to node j. In many applications it must be determined that the set of arcs in a network that connects all nodes such that the sum of the length of the arcs is minimized. Clearly, such a group of arcs contain no loop . For a network with n nodes, a spanning tree is a group of n-1 arcs that connects all nodes of the network and contains no loops . A spanning tree of minimum length in a network is a minimum spanning tree (MST) . 4
MST Example 5 1 2 3 4 5 6 1 9 7 4 5 6 3 1 0 8 3

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Applications of MST Design of telecommunications networks (fiber-optic networks, computer networks, leased-line telephone networks, cable television networks, etc.) Design of highly used transportation network to minimize the total cost of providing the links (rail lines, roads, etc.) Design of high voltage electrical power transmission lines Design of network of wiring on electrical equipment to minimize total length of wire Design of a network of pipelines (oil, sewer) to connect many locations 6
Minimum Spanning Tree Problem MST solution methods Formulate and solve as LP ( difficult due to no-looping constraint) Use greedy MST algorithm A “greedy” algorithm does whatever is best at each step. Usually greedy algorithms do not produce optimal solutions. However, greedy MST algorithm does produce optimal solution! 7

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MST Algorithm 1. Start with any node and join it to its closest node in the network. The resulting two nodes now form a connected set, and the remaining nodes comprise the unconnected set. 2. Choose a node from the unconnected set that is closest to any node in the connected set and add it to the connected set. 3. Redefine the connected and unconnected sets accordingly. Repeat the process until the connected set includes all the nodes in the network. 4. Ties may be broken arbitrarily; however, ties may indicate the existence of alternative minimal spanning trees.
• Spring '10
• CHEN
• Graph Theory, Gary, Shortest path problem, Fort Wayne, Terre Haute South Bend

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