23 oct 2009 · Importance: Where it has been used? • Algorithm's general description • Algorithm steps in detail • Example Operations Research Methods 1
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[PDF] Lecture 18 Solving Shortest Path Problem: Dijkstras Algorithm
23 oct 2009 · Importance: Where it has been used? • Algorithm's general description • Algorithm steps in detail • Example Operations Research Methods 1
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algorithm has been obtained, which has reduced the storage space and improved the operational efficiency Keywords: Applications, Directed Graph, Dijkstra's
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Dijkstra's Shortest Path Algorithm Input: A distance matrix C for a digraph G = (V,E) with n vertices If the edge (i, j) belongs to E the c(i, j) equals the distance from i to j, otherwise c(i, j) equals ∞
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Dijkstra (1959) proposed a graph search algorithm that can be used to solve the single-source shortest path problem for any graph that has a non-negative edge
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Lecture 18
Solving Shortest Path Problem:
Dijkstra"s AlgorithmOctober 23, 2009
Lecture 18
Outline
•Focus on Dijkstra"s Algorithm •Importance: Where it has been used? •Algorithm"s general description •Algorithm steps in detail •ExampleOperations Research Methods1
Lecture 18
One-To-All Shortest Path Problem
We are given a weighted network(V,E,C)with node setV, edge setE, and the weight setCspecifying weightscijfor the edges(i,j)?E. We are also given a starting nodes?V. Theone-to-all shortest pathproblem is the problem of determining the shortest path from nodesto all the other nodes in the network.The weights on the links are also referred ascosts.Operations Research Methods2Lecture 18
Algorithms Solving the Problem
•Dijkstra"s algorithm •Solves only the problems with nonnegative costs, i.e., c ij≥0for all(i,j)?E•Bellman-Ford algorithm •Applicable to problems with arbitrary costs •Floyd-Warshall algorithm •Applicable to problems with arbitrary costs •Solves a more general all-to-all shortest path problem Floyd-Warshall and Bellman-Ford algorithm solve the problems on graphs that do not have a cycle with negative cost.Operations Research Methods3Lecture 18
Importance of Dijkstra"s algorithm
Many more problems than you might at first think can be cast as shortest path problems, making Dijkstra"s algorithm a powerful and general tool. For example:•Dijkstra"s algorithm is applied to automatically find directions between physical locations, such as driving directions on websites like Mapquest or Google Maps.•In a networking or telecommunication applications, Dijkstra"s algorithm has been used for solving the min-delay path problem (which is the shortest path problem). For example in data network routing, the goal is to find the path for data packets to go through a switching network with minimal delay.•It is also used for solving a variety of shortest path problems arising in plant and facility layout, robotics, transportation, and VLSI ?design? Very Large Scale IntegrationOperations Research Methods4Lecture 18
General Description
Suppose we want to find a shortest path from a given nodesto other nodes in a network (one-to-all shortest path problem)•Dijkstra"s algorithm solves such a problem •It finds the shortest path from a given nodesto all other nodes inthe network•Nodesis called a starting node or an initial node•How is the algorithm achieving this?
•Dijkstra"s algorithm starts by assigning some initial values for thedistances from nodesand to every other node in the network•It operates in steps, where at each step the algorithm improves the
distance values.•At each step, the shortest distance from nodesto another node is determinedOperations Research Methods5Lecture 18
Formal Description
The algorithm characterizes each node by its state The state of a node consists of two features:distance valueandstatus label •Distance value of a node is a scalar representing an estimate of the its distance from nodes.•Status label is an attribute specifying whether the distance value of anode is equal to the shortest distance to nodesor not.•The status label of a node isPermanentif its distance value is equal
to the shortest distance from nodes•Otherwise, the status label of a node isTemporary The algorithm maintains and step-by-step updates the states of the nodesAt each step one node is designated ascurrent
Operations Research Methods6
Lecture 18
Notation
In what follows:
•d?denotes the distance value of a node?.•portdenotes the status label of a node, wherepstand for permanent
andtstands for temporary•c ijis the cost of traversing link(i,j)as given by the problem The state of a node?is the ordered pair of its distance valued?and its status label.Operations Research Methods7Lecture 18
Algorithm Steps
Step 1.Initialization
•Assign the zero distance value to nodes, and label it asPermanent. [The state of nodesis(0,p).]•Assign to every node a distance value of∞and label them as Temporary. [The state of every other node is(∞,t).]•Designate the nodesas thecurrentnodeOperations Research Methods8
Lecture 18
Step 2.Distance Value Update and Current Node Designation UpdateLetibe the index of the current node.(1)Find the setJofnodes with temporary labelsthat can be reached
from the current nodeiby a link(i,j).Update the distance values of these nodes.•For eachj?J, the distance valuedjof nodejis updated as follows new d j= min{dj,di+cij}wherecijis the cost of link(i,j), as given in the network problem.(2)Determine a nodejthat has the smallest distance valuedjamong all
nodesj?J,findj?such thatminj?Jdj=dj?(3)Change the label of nodej?topermanentand designate this node asthe current node.
Operations Research Methods9
Lecture 18
Step 3.Termination Criterion
If all nodes that can be reached from nodeshave been permanently labeled, then stop - we are done. If we cannot reach any temporary labeled node from the current node, then all the temporary labels become permanent - we are done. Otherwise, go to Step 2.Operations Research Methods10Lecture 18
Dijkstra"s Algorithm: Example
We want to find the shortest path from node 1 to all other nodes using Dijkstra"s algorithm.Operations Research Methods11Lecture 18
Initialization - Step 1
•Node 1 is designated as the currentnode•The state of node 1 is(0,p)•Every other node has state(∞,t)Operations Research Methods12