Dijkstra's Algorithm

Transcription

1 Shortest Path Algorithm Dijkstra's Algorithm To find the shortest path from the origin node to the destination node No matrix calculation Floyd s Algorithm To find all the shortest paths from the nodes to every other node in a network Using matrix A typical problem in network analysis is finding the shortest path from one node to another through a network 11/8/ Dijkstra's Algorithm Let the node we are starting be called an initial node. Let a distance of a node be the distance from the initial node to it. Dijkstra's algorithm will assign some initial distance values and will try to improve them step-by-step. 1. Assign to every node a distance value. Set it to 0 for initial node and to for all other nodes. 2. Mark all nodes as unvisited. Set initial node as current. 3. For current node, consider all its unvisited neighbors and calculate their distance (from the initial node). If this distance is less than the previously recorded distance ( in the beginning, 0 for the initial node), overwrite the distance. 4. When we are done considering all neighbors of the current node, mark it as visited. A visited node will be out and not be checked ever again; its distance recorded now is final and minimal. 5. Set the unvisited node with the smallest distance (from the initial node) as the next "current node" and continue from step 3. 11/8/

2 Dijkstra's Algorithm - Procedure It starts from a source node, and in each iteration adds another vertex to the shortest-path spanning tree. This vertex is the point closest to the root which is still outside the tree. 11/8/ Dijkstra's Algorithm - Procedure 2011/11/8 60

3 Dijkstra's Algorithm - Procedure The shortest distance from node 1 to 5 is 20, and the corresponding path is /11/8 61 Dijkstra's Algorithm - Example Dijkstra's Algorithm solves the single-source shortest path problem in weighted graphs. Here we show it running on a planar graph whose edge weights are proportional to the distance between the vertices in the drawing -- thus the weight of an edge is equal to its visible length. 11/8/

4 Floyd s Algorithm The algorithm works by updating two matrices, D k and Q k, n times for a n- node network. D k, in any iteration k, gives the value of the shortest distance between all pairs of nodes (i, j) as obtained till the k th iteration. Q k gives the immediate predecessor/previous node from node i to node j on the shortest path as determined by the k th iteration. D o an Q o give the starting matrices and D n and Q n give the final matrices for an n-node system. Algorithm Steps: 1. Let k = 1 2. We calculate elements of the shortest path length matrix found after the k-th passage through algorithm D k using the following equation: 3. Elements of predecessor matrix Q k found after the k-th passage through the algorithm are calculated as follows: 4. If k = n, the algorithm is finished. If k < n, increase k by 1, i.e. K = k+1 and return to step 2. 11/8/ Floyd s Algorithm - Example A transportation network 3 4 Starting matrix D o Starting matrix Q o 11/8/

5 Floyd s Algorithm - Example 3 4 The shortest distance from node 5 to node 4 is 10, and the shortest path is from 5 to 2 to 3 to 4. 11/8/ C++ Source Code for Floyd Algorithm ( 1/11-12/lsgi521_lecture_slides/floyd.cpp) 11/8/

6 Traveling Salesman Problem Determine the best (or least cost) way to make a series of deliveries or stops Blue triangles are stops 2011/11/8 69 Assignment Problem Find the best one-to-one matching between two groups of objects Assign 8 offices that need cleaning to 8 office cleaners in order to minimize the cost Red : cleaner location Green : office location 2011/11/8 70

7 Resource Allocation Create groups of features based on proximity Create 8 compact school districts where the total number of school age children does not exceed 12, /11/8 71 Network Partition Compute the network cost between each service locations and all the links and nodes in the network Partitions the streets into three zones, one for each ambulance 2011/11/8 72

8 Spatial Interpolation Spatial interpolation: The procedure of estimating unknown values using known values at neighboring locations Usually by means of a mathematical function e.g., interpolation of elevation values Most GIS softwares offer a number of interpolation methods for use with point, line and area data Interpolation methods: Thiessen polygons Triangulated irregular network (TIN) Trend surface Inverse distance weighted (IDW) Kriging 2011/11/8 73 Control Points and Interpolation Results The number and distribution of control points can greatly influence the accuracy of spatial interpolation Two different results that could reasonably be obtained from the same set of data points INTERPOLATION RESULT A INTERPOLATION RESULT B CONTOUR MAP RESULT CORRESPONDING PERSPECTIVE VIEWS 2011/11/8 74

9 An Example: Control Points with Actual Terrain Surface 2011/11/8 75 Thiessen Polygons Thiessen polygon (Voronoi polygons) Subdividing lines joining nearest neighbor points Drawing perpendicular bisectors through these lines Using these bisectors to assemble polygon edges Every polygon contain one control point In every polygon, the distance from every point to the control point is shorter than the distance to every other control points Constructing a Thiessen polygon net 2011/11/8 76

10 Triangulated Irregular Network (TIN) Link all the control points to construct a TIN 2011/11/8 77 Trend Surface To fit a mathematically defined surface through all the control points so that the difference between the interpolated value and its original value is minimized 1 st order polynomial 2 nd order polynomial 2011/11/8 78

11 Trend Surface (cont d) 3 rd order polynomial 4 th order polynomial 2011/11/8 79 Inverse Distance Weighted (IDW) Each input point has local influence that diminishes with distance Estimates are averages of values at s known points within window R Is an exact method that enforces that the value of a point is influenced more by nearby known points than those farther away z 0 is the estimated value at point 0 z i is the z value at known point i d i is the distance between point i and point 0 s is the number of know points used K is the specified power K =1 : constant K =2 : higher rate of change near a known point 2011/11/8 80

12 Kriging Developed by South African mining engineer D.G. Krige Reasonable number of control points for interpolation The size and shape of the neighborhood points for interpolation Reasonable weights Accuracy assessment 2011/11/8 81 Characteristics Kriging Use the semivariogram, in calculating estimates of the surface at the grid nodes Can assess the quality of prediction with estimation prediction errors (stochastic) Assume spatial variation may consist of 3 components A structural component, representing a trend A spatially correlated component, representing the variation of the regionalized variable A random error term 2011/11/8 82

13 3 Components of a Spatial Variable The structural component (e.g., a linear trend) The spatially correlated component The random noise component (non-fitted) 2011/11/8 83 Semivariogram in Kriging Semivariogram Measure the spatial dependence or spatial autocorrelation of a group of points or ϒ(h) is the semi-variance between known point x i and x j, separated by the distance h; and z is the attribute value 2011/11/8 84

14 Semivariogram in Kriging 2011/11/8 85 Semivariogram in Kriging h=0 Sphere model: a h γ 1 2a γ h = c 0 + c a ( h) = c + c ( h ) 2 ( ) 1 γ ( 0 ) = 0 0 < h < a h >= a h=0 Exponent model: 3h γ ( h) = c0 + c1 exp( ) h > 0 a γ h=0 ( 0 ) = /11/8 86

15 Ordinary Kriging Assume that there is no structural component Focuses on the spatially correlated component Uses the fitted semivariogram directly for interpolation Z 0 is the estimated value, Z x is the known value at point x W x is the weight associated with point x S is the number of sample points used in estimation E.g., for a point (0) to be estimated from three known points (1, 2, 3) 2011/11/8 87 Ordinary Kriging The weight W can be determined by solving a set of simultaneous equations: C W = D The variance can be estimated by: 2011/11/8 88

16 Numeric Example of Kriging In this example, we want to estimate a value for point 0 (65E, 137N), based on the 7 surrounding sample points. The table indicates the (x,y) coordinates of the 7 sample points and their distance to point /11/8 89 Spatial Dependence Analysis γ 3h ( h) = c + c exp( ) 0 1 a Parameters: C 0 = 0, a = 10, C 1 = /11/8 90

17 Kriging Matrices λ To solve for the weights, we multiply both sides by C -1, the inverse of the left-hand side covariance matrix: 2011/11/8 91 First, the distance matrix Kriging Matrices Then, the exponent model will be used to calculate the semivariogram matrix γ 3h ( h) = c + c exp( ) 0 C 0 = 0, a = 10, C 1 = 10 1 a C(h) = 10 e 0.3 h 2011/11/8 92

18 Kriging weights: Results λ Estimated value for point 0: How can the interpolation variance be estimated? 2011/11/8 95 Analysis of Surface Slope The slope at a point is the angle measured from the horizontal to a plane tangent to the surface at that point The value of the slope will depend on the direction in which it is measured. Slope is commonly measured in the direction of the coordinate axes e.g. in the X-direction and Y-directions. The slope measured in the direction at which it is a maximum is termed the gradient Aspect The angle formed by moving clockwise from north to the direction of maximum slope ASPECT NORTH Y SLOPE 2011/11/8 96

19 Shade Slope and Aspect Image 2011/11/8 97 Hillshading Hillshading: to calculate the location of shadow and the amount of sun incident on the terrain surface when the sun is in a particular position in the sky Modelling incoming solar radiation (a f) representing morning to evening 2011/11/8 98

20 Visibility Analysis 2011/11/8 99 An Example in NASA s Mars Exploration Rover Mission Sol 1204 Sol 1210 Sol /11/8 100

21 Mapping Products at Duck Bay Distribution of the measured 3D points DTM interpolated from the 3D points 2011/11/ D Surface of Duck Bay 2011/11/8 102

22 3D Surface and Slope Maps of Duck Bay Legend Slope (degree) /11/8 103 Review Further readings Y. H. Chou, Exploring Spatial Analysis in Geographic Information Systems, Onword Press. Caroline Lafleur, 2011, MATLAB Kriging Toolbox. ( ArcGIS Network Analyst ( Summarization of the main ideas presented in this lecture: Questions? 2011/11/8 104