Graph theory and GIS
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1 Geoinformatics FCE CTU 2009 Workshop 18th september 2009
2 Contents Graph as such Definition Alternative definition Undirected graphs Graphs as a model of reality
3 Contents Graph as such Definition Alternative definition Undirected graphs Graphs as a model of reality Geometrical and physical models States and actions Binary relations Information models
4 Contents Graph as such Definition Alternative definition Undirected graphs Graphs as a model of reality Geometrical and physical models States and actions Binary relations Information models Paths related problems Flows in networks And many many others
5 Definition Alternative definition Undirected graphs Graphs as a model of reality Definition G = (V, E, ε) V set of vertices (nodes) E set of (directed) edges (arcs) ε : E V 2 incidence edge ordered pair of vertices (initial, terminal) = (tail, head) vertices not necessarily geometrical points same for edges multigraphs are also graphs
6 Definition Alternative definition Undirected graphs Graphs as a model of reality Alternative definition G = (V, E) V set of vertices (same as above) E V 2 set of edges is a binary relation edge = ordered pair of vertices multigraph is not a graph
7 Definition Alternative definition Undirected graphs Graphs as a model of reality Undirected graphs Ignore (forget) direction of edges in directed graph Symmetric directed graph Standalone definition (and standalone theory)
8 Graphs as a model of reality Definition Alternative definition Undirected graphs Graphs as a model of reality Partial mapping : world mathematical object Model deals with only part of universum only some properties Signs of a good model Consider relevant properties of relevant part of world Findings in a model have good meaning in real world
9 Geometrical and physical models Geometrical and physical models States and actions Binary relations Information models Straightforward based on topology of a physical world Polyhedra (names of vertices and edges came from here) Systems of real world linear objects roads, railways, power lines, communication networks,... vertices represent points edges represent linear objects data from GIS Electric circuits Molecules (atoms, bonds)
10 States and actions Graph as such Geometrical and physical models States and actions Binary relations Information models Sequencing Parallel activities Project scheduling CPM, PERT activities as arrows activities as vertices Random processes (Markov chains) Games Problem solving
11 Binary relations Graph as such Geometrical and physical models States and actions Binary relations Information models Social networks Matchings: jobs workers, boys girls Dependence: logical statements packages in a Linux distribution cells in a spreadsheet
12 Geometrical and physical models States and actions Binary relations Information models Information models Data structures with pointers WWW documents and links Classes in object oriented programming Syntactic structures Flowcharts E-R diagrams etc.
13 Paths related problems Flows in networks And many many others Search Graphs are searched to find all accessible vertices or edges perform sth for all accessible vertices or edges Breadth first search produces shortest paths Depth first search easily programmed using recursive procedure basis for more complicated algorithms
14 Paths related problems Flows in networks And many many others Optimal paths Most often applied graph theoretical problem edges are assigned values (lenghts, costs, widths) value of a path computed from edges (e.g. sum) optimum over all paths shortest, longest, widest, the most probable,... Dijkstra algorithm very popular often not most efficient fails in graphs with negative lengths
15 Paths related problems Flows in networks And many many others Flows in networks Kirchhoff s law (conservation of flow in vertices) lower and upper bounds Max-flow min-cut matchings connectivity measure Minimum cost flow
16 Paths related problems Flows in networks And many many others And many many others Analysis of a structure Connected components (several types) Lengths of cycles Combinatorial problems Coloring, independent sets, covering,... Traveling Salesman Problem, Chinese Postman,... Measures of robustness
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