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Unformatted text preview: Chapter 9 Shortest Paths and Discrete Dynamic Programming Example 9.1 Littleville Suppose that you are the city traffic engineer for the town of Littleville. Figure 9.1(a) depicts the arrangement of one and twoway streets in a proposed improvement plan for Littleville’s downtown, including the estimated average time in seconds that a car will require to transit each block. From survey and other data we can estimate how many driver trips originate at various origin points in the town, and the destination for which each trip is bound. But such survey data cannot indicate what streets will be selected by motorists to move from origin to destination in a street network that does not yet exist. Example 9.1 Littleville One of the tasks of a traffic engineer is to project the route that drivers will elect, so that city leaders can evaluate whether flows will concentrate where they hope. A good starting point is to assume that drivers will do the most rational thingfollow the shortest time path from their origin to their destination. We need to compute all such shortest paths. Example 9.1 Littleville 20 18 12 18 13 2 8 38 32 30 18 28 36 2 5 4 9 2 1 4 Example 9.1 Littleville 1 2 3 4 1 9 7 6 5 8 20 12 18 13 2 5 2 1 2 8 4 9 18 28 36 4 18 38 32 30 9.1 Shortest Path Models • Urban traffic, Satellite communications, or surface of a microchip. • Mathematical graphs model travel, flow, and adjacency patterns in a network . [9.1] • The nodes or vertices of a graph represent entities, intersections, and transfer points of the network. [9.2] • The arcs of a graph model available directed (oneway) links between nodes. Edges represent undirected (two way) links. [9.3] 9.1 Shortest Path Models • A path is a sequence of arcs or edges connecting two specified nodes in a graph. Each arc or edge must have exactly one node in common with its predecessors in the sequence, any arcs must passed in the forward direction, and no node may be visited more than once. [9.4] Shortest Path Problems • Shortest path problems seek minimum total length paths between specified pairs of nodes in a graph. [9.5] Example 9.1 Littleville 1 2 3 4 1 9 7 6 5 8 20 12 18 13 2 5 2 1 2 8 4 9 18 28 36 4 18 38 • Name: Littleville • Graph: arcs and edges • Costs: nonnegative • Output: shortest paths • Pairs: all nodes to all others Example 9.2 Texas Transfer Texas Transfer, a major trucker in the southwest, ships goods from its hub warehouse in Ft. Worth to all the cities shown. Trucks leave the hub and proceed directly to their destination city, with no intermediate dropoffs or pickups. Texas Transfer drivers are allowed to choose their own route from Ft. Worth to their destination. However, management’s proposal in current labor negotiations calls for drivers to be paid on the basis of shortest standard mileage to the location. That is, they will be paid according to the length of the shortest path from Ft. Worth to their destination city in the network of Figure 9.3. To see the impact of this proposal, we need to compute such shortest path distances for all cities. Example 9.2 Texas TransferExample 9....
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This note was uploaded on 12/03/2011 for the course ESI 6316 taught by Professor Staff during the Summer '11 term at FIU.
 Summer '11
 STAFF

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