FUZZY DIJKSTRA ALGORITHM FOR SHORTEST PATH PROBLEM IN GEOGRAPHIC INFORMATION SYSTEMS

Transcription

1 ENGLISH SEMINAR FUZZY DIJKSTRA ALGORITHM FOR SHORTEST PATH PROBLEM IN GEOGRAPHIC INFORMATION SYSTEMS Presenter: Huy Nguyen Phuong

2 Contents 1. Introduction to GIS 2. Shortest Path Problem in GIS 3. Dijkstra's algorithm 4. Fuzzy logic in GIS 5. Fuzzy Dijkstra's algorithm in GIS 6. Conclusion 7. Reference 10/19/2015 2

3 Introduction to GIS What is GIS? GIS: a simplified view of the real world Discrete features Continuous features 10/19/2015 Points Lines Areas Networks A series of interconnecting lines Road network River network Sewage network Surfaces Elevation surface Temperature surface 3

4 Introduction to GIS What is GIS? A computer system for - collecting, - storing, - manipulating, - analyzing, - displaying, and - querying Geographically related information. 10/19/2015 4

5 Introduction to GIS Data in GIS Geospatial data tells you where it is Attribute data tells you what it is. Metadata describes both geospatial and attribute data. In GIS, we call geographic data as GIS data or spatial data 10/19/2015 5

6 Introduction to GIS Two basic data models B G B G G B B B B B B G B G B G B G G G BK Raster Representation Real World Y-AXIS River X-AXIS Trees 500 Trees House 600 Vector Representation 10/19/2015 6

7 Introduction to GIS Two basic data models RASTER VECTOR Real World 10/19/2015 7

8 Introduction to GIS Two basic data models Advantages Good representation of reality Compact data structure Topology can be described in a network Accurate graphics Disadvantages Complex data structures Simulation may be difficult Some spatial analysis is difficult or impossible to perform 10/19/2015 Vector Representation Raster Representation Advantages Simple data structure Easy overlay Various kinds of spatial analysis Uniform size and shape Cheaper technology Disadvantages Large amount of data Less pretty Projection transformation is difficult Different scales between layers can be a nightmare May lose information due to generalization 8

9 Shortest Path Problem in GIS Networks and GIS What is a network? Road network Computer network Drainage network Water supply network Power supply network Social network Economic network 10/19/2015 9

10 Shortest Path Problem in GIS Networks and GIS Land Use Transportation networks in GIS Flows Transportation Network Layers A 10/19/2015 road network what can be considered as an impedance? 10

11 Shortest Path Problem in GIS Networks and GIS Link Impedance (also Resistance, Friction) IMPEDANCE Node impedance 10/19/

12 Shortest Path Problem in GIS Shortest Path Problem Navigation is a main application in location based systems. Shortest path analyses - commonly based on vector maps. Used to calculate the shortest path between two points in a network or route. Currently used in many application such as Google Earth, Yahoo Map etc... 10/19/

13 Shortest Path Problem in GIS Shortest Path Algorithms In graph theory, It s the problem of finding a path between two vertices (or nodes) such that the sum of the weights of its constituent edges is minimized. Ex: On a road map; vertices locations & edges roads. The problem is also sometimes called the single-pair shortest path problem, to distinguish it from the following generalizations The single-source shortest path problem (a source vertex v to all other vertices in the graph). The single-destination shortest path (can be reduced to the singlesource shortest path problem by reversing the edges in the graph). The all-pairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph. 10/19/

14 Shortest Path Problem in GIS Single source shortest path problem Dijkstra's algorithm Solves the single-source shortest path problem, with non negative edge path costs, common in routing. Bellman-Ford algorithm Label correcting algorithm, computes single-source shortest paths in a weighted digraph A* search algorithm, Floyd Warshall algorithm, J C Green Algo, Johnson's algorithm and Pertubation Theory 10/19/

15 Dijkstra's algorithm Introduction Explanation of Dijkstra's Algorithm Edsger Wybe Dijkstra, May 11,1930 August 6, /19/

16 Programming Dijkstra's algorithm 10/19/

17 Dijkstra's algorithm Example 10/19/

18 Dijkstra's algorithm Applications z 11? 2 10! 4 X 1 (6) X 2 (9) 6 3 v 10/19/

19 Fuzzy Logic Fuzzy logic in GIS It deals with uncertainties by simulating the process of human reasoning. These systems allow computers to behave less precisely and logically than conventional computers do. The idea behind this approach is that decision making is not always a matter of true or false, black and white. It often involves gray areas where the terms approximately, possible, and similar are more appropriate. 10/19/

20 Fuzzy logic in GIS How the Fuzzy models work Crisp data Inputs converted to degrees of membership of fuzzy sets. Fuzzy rules applied to get new sets of members. Fuzzifier Member 90% hot, 10% cold Fuzzy rules IF 90% hot THEN 80% open IF 10% cold THEN 20% closed These sets are then converted back to real numbers. Fuzzy output set 80% open, 20% closed Defuzzifier 10/19/2015 Crisp data 20

21 Fuzzy logic in GIS Geographic Data Uncertainty There are thousands of ways to measure the position, shape, orientation and size of phenomena or objects Data uncertainty varies spatially and over time Ambiguity of concepts (semantics, geometry) Poorly known resolution and precision Lack of up-to-dateness and timeliness Incompleteness Low level of «processability» Everytime data are reused or transformed, additional uncertainty is introduced Uncertainty can be reduced Better observation technologies and methods (Fuzzy) Standards Training 10/19/

22 Fuzzifier GIS data Fuzzy logic in GIS 10/19/2015 Map of Area 22

23 Fuzzifier GIS data The degree of Fuzzy sets is shown as follows: Layers Settlements Fuzzy logic in GIS Membership function MF = 0, if x < 500 MF = 1, if x > 1500 MF = ((x-500)/1000), if 500 x /19/2015 Visual interpretation of Membership Functionwith respective graphs 23

24 Fuzzifier GIS data Fuzzy logic in GIS Rivers Roads Slope Settlement Aspect Wells Aquifer Legend /19/

25 Fuzzy logic in GIS Fuzzifier GIS data Layers Membership function Membership Functions Slope MF = 1, if 0 <x<5 MF = ((x-5)/10), if 5 x 15 MF = 0, if x>15 Aspect MF = 0, MF = ((x-135)/90) MF = ((x-315)/90) MF = 1, MF = 1, if 225 x 315 if 135<x<225 if 315<x<45 (315<x<405) if 45 x 135 if x=361(flat areas) Wells River MF = 0, if x<200 MF = ((x-400)/200), if 400 x 600 MF = 1, if x>600 MF = 0, if x<250 MF = ((x-250)/500), if 250 x 750 MF = 1, if x>750 Road MF = 0, if x>500 MF = ((500-x)/500), if 0<x 500 settlements Geology 10/19/2015 MF = 0, if x<250 MF = ((x-500)/1000), if 500 x 1500 MF =1, if x>1500 MF = 0, if x > +125 MF = ((125-x)/250), if -125 x +125 MF = 1, if x <

26 Fuzzy numbers Fuzzy Dijkstra's algorithm in GIS Triangular fuzzy number The graded mean integration representation of triangular fuzzy number 10/19/

27 Fuzzy numbers Fuzzy Dijkstra's algorithm in GIS Trapezoidal fuzzy number The graded mean integration representation of trapezoidal fuzzy number 10/19/

28 Example Fuzzy Dijkstra's algorithm in GIS A transportation network. 10/19/

29 Example Fuzzy Dijkstra's algorithm in GIS Arc 1 (17,23) = Arc(17, 20) Arc(20, 23)= (7, 10, 11, 12) (13, 14,16, 17) =1/6( )+1/6( )= 151/6 10/19/

30 Example Fuzzy Dijkstra's algorithm in GIS 10/19/

31 Example Fuzzy Dijkstra's algorithm in GIS 10/19/

32 Example Fuzzy Dijkstra's algorithm in GIS 10/19/

33 Fuzzy Dijkstra's algorithm in GIS Fuzzy and Boolean Comparisions Boolean - sharp distinction with YES and NO areas Fuzzy - gradual delineation for selected landfill Flexibility to decide on threshold for fuzzy logic No need for repeated analysis No need for change in criteria and rules Saves time and reduces effort Decisions on threshold can be supplimented by field work 10/19/

34 Conclusions GIS is an important information system in modern life Dijkstra's algorithm is a very good sollution for Shortest Path Problem in path finding application of GIS Fuzzy Dijkstra's algorithm is more suitable due to Geographic Data Uncertainty We can replace fuzzy by Hedge Algebras 10/19/

35 Reference 1. Ananthanarayanan M., V. Pusparaj (2014), A Study on Comparison Between Fuzzy Shortest Path With Conventional Method, Indian Journal of Applied Research, Volume 4, Issue Elizabeth S., L. Sujatha (2011), Fuzzy Shortest Path Problem Based on Index Ranking, Journal of Mathematics Research. 3. Jingxiong Zhang, Michael F. Goodchild (2003), Uncertainty in Geographical Information, The Taylor & Francis e-library. 4. Rajveer Kaur (2012), A Study on Fuzzy Shortest Path Problems, Thesis of Masters of Science in Mathematics and Computing. 5. Tzung-Nan Chuang, Jung-Yuan Kung (2005), The fuzzy shortest path length and the corresponding shortest path in a network, Computers & Operations Research. 6. Tzung-Nan Chuang, Jung-Yuan Kung (2006), A new algorithm for the discrete fuzzy shortest path problem in a network, Applied Mathematics and Computation, Elsevier. 7. Yong Deng, Yuxin Chen, Yajuan Zhang, Sankaran Mahadevan (2012), Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment, Applied Soft Computing, Elsevier. 8. Wolfgang Kainz (2010), The Mathematics of GIS, University of Vienna, Austria. 10/19/

36 THANK YOU!