1. www.spatialanalysisonline.com Chapter 7 Part A: Network analysis

2. 3 rd editionwww.spatialanalysisonline.com2 Network analysis Networks – basic components: Collections of interconnected linear forms: Lines (or Edges, E) Intersections (or Vertices, V) Regions (or Cells, C) - created by the partitioning of space by the lines Planar - e.g. streets, all on same level, vertices at every intersection of edges Non-planar - e.g. airline routes, highways with bridges/flyovers, electronic circuits

3. 3 rd editionwww.spatialanalysisonline.com3 Network analysis Sample problems : Shortest (least time/cost) between vertices (SPA) Shortest path (tree) connecting all vertices (MST) Shortest route visiting all locations once and returning to start point (Travelling Salesman Problem, or TSP) Minimum cost of constructing a network between a set of vertices Identification of zones within specified travel time/cost Designing a network with minimum cost of USE Designing a network with minimum travel time to specified vertices Including constraints, e.g. edge capacity, maximum distances/times permitted, vehicle capacity

4. 3 rd editionwww.spatialanalysisonline.com4 Network analysis Networks – basic components: Directed (with predefined directions or flows) Non-directed (symmetric access/flows) Common level or hierarchical Abstracted as graphs and/or tables Connected or collection of sub-graphs Principal forms: Paths, trees, circuits, cells

5. 3 rd editionwww.spatialanalysisonline.com5 Network analysis Networks – basic components: Degree (of a vertex) Path Connected graph Cycles/circuits Tree

6. 3 rd editionwww.spatialanalysisonline.com6 Network analysis Networks – basic components: Paths - edges and verticesTree - no circuitsCircuitCells: V-E+C=2 Intermediate data coding points - not vertices - can be ignored/eliminated

7. 3 rd editionwww.spatialanalysisonline.com7 Network analysis Networks – compare topologies:

8. 3 rd editionwww.spatialanalysisonline.com8 Network Analysis Vertex connectivity or adjacency matrix: Symmetric, binary, 0=non-connected or self- connected, sparse; positive valued To vertex From vertex Networks: Binary connectivity matrix

9. 3 rd editionwww.spatialanalysisonline.com9 Network analysis Networks – basic components: Direction tree networks may have a consistent direction e.g. river flows, broadcast data communications circuit networks may have mixture of directions closed circuits may exist in directed networks Magnitude (edge length, time, cost…) Volume (flow from vertex to vertex) Weights/demand at vertices

10. 3 rd editionwww.spatialanalysisonline.com10 Network analysis Networks: Source data Network construction Set of points in the plane Existing network to be augmented Network analysis Existing set of vertices, edges and associated attribute data A pre-defined or imposed topology Data representation issues

11. 3 rd editionwww.spatialanalysisonline.com11 Network analysis Networks – sample attributes: turn attributes at intersections: permitted/not- permitted, turn penalties, U-turn permissions definition of weights/impedances, by direction definition of one-way edges and their direction specification of any permanent or temporary barriers demand and capacity constraint levels (edge and/or vertex based)

12. 3 rd editionwww.spatialanalysisonline.com12 Network analysis Computational complexity Optimisation problems and decision-making Provably optimal Provably within defined bounds of optimality Good in practice Big O notation and complexity Polynomial (P): e.g. O(n 2 ) and O(nlogn) O(logn) 3 Non-polynomial (NP): e.g. O(n!) and O(2 n ) O(2n) 3 NP-complete problems Heuristics

13. 3 rd editionwww.spatialanalysisonline.com13 Network analysis Key problems - 1: Hamiltonian circuit (HC) – NP-complete Eulerian circuit (EC) Shortest path (SP) – P (linear--) Spanning tree (ST) Minimal spanning tree (MST) – P (linear--) Steiner MST – NP-complete Steiner tree – NP-complete Travelling salesman problem (TSP) – NP- complete

14. 3 rd editionwww.spatialanalysisonline.com14 Network analysis Key problems – 2: NP-hard or NP-complete: Vehicle routing problems (VRP) Transportation problems Trans-shipment problems Arc routing problems (ARP) Facility location on a network: p median/p centre/coverage

15. 3 rd editionwww.spatialanalysisonline.com15 Network analysis Typical problem parameters: Objective function (e.g. length, cost, time…) Constraints on the path (e.g. direct or via specified nodes) Input geometry (e.g. obstacles/barriers) Dimension of the problem (2D, planar?) Type of moving object (simple, constraints, friction) Single shot vs. repetitive mode queries (e.g. 1 st, 2 nd..) Static vs. dynamic environments Exact vs. approximate algorithms Known vs. unknown map

16. 3 rd editionwww.spatialanalysisonline.com16 Network analysis Example logistics software facilities: vehicle routing taking one-way streets into account trip routing taking restricted junctions into account varying speeds by road type and time of day trip routing of vehicles to avoid toll roads and toll bridges delivery routing taking account of customer access constraints by time of day night time/weekend truck routing controls weight and height restrictions (e.g. for truck routing) vehicle routing costs per mile/km and/or per hour weight/climb related vehicle routing costs

17. 3 rd editionwww.spatialanalysisonline.com17 Network analysis Minimum spanning tree connect every point to its nearest neighbour typically this will result in a collection of unconnected sub-networks connect each sub-network to its nearest neighbour sub-network iterate step 2 until every sub-network is inter- connected

18. 3 rd editionwww.spatialanalysisonline.com18 Network analysis Minimum spanning tree (Euclidean) A. Point set (nodes or vertices)B. MST

19. 3 rd editionwww.spatialanalysisonline.com19 Network analysis Gabriel network A. Gabriel network constructionB. Gabriel network included excluded

20. 3 rd editionwww.spatialanalysisonline.com20 Network analysis Steiner tree (unweighted, Euclidean) A. MSTB. MST with 1 Steiner point

21. 3 rd editionwww.spatialanalysisonline.com21 Network analysis Shortest paths Input: existing network, source vertex (s) and target vertex (t) or vertices Output: shortest path – length, d(t), and vertex list; set of shortest paths (1 st, 2 nd,… shortest); source to all vertices (shortest path tree) Solve by systematic search algorithm (single paths in near linear time) Large problems solve by A* heuristics

22. 3 rd editionwww.spatialanalysisonline.com22 Network Analysis Dantzig Shortest path algorithm (SPA) s=1, t=3: Step 1: identify the shortest (least distance/cost/time) link from vertex 1 - this is to vertex 2 (cost = 4). Add vertex 2 and link from 1 to 2 to the tagged set Step 2: identify the shortest (least cost/time) link from vertex 1 or from vertex 2 plus link 1 distance - this is to vertex 4 from 2 (cost=6). Add vertex 4 and link 2 to 4 to the tagged set Step 3: identify the shortest (least cost/time) link from the tagged set - this is from vertex 1 to 2 to 4 to 3 (cost=7) Stop - all vertices reached; repeat from vertex 2, 3 and 4

23. 3 rd editionwww.spatialanalysisonline.com23 Network analysis SPA: Dijkstra algorithm 1: initialise all vertices such that d(t)= and d(s)=0 2: For each edge leading from s, add the edge length from s to the current value of the path length at s. If this new distance is less than the current value for d(t) replace this with the lower value 3: choose the lowest value in the set d(t) and move the current (active) vertex to this location 4: iterate steps 2 and 3 until the target vertex is reached or all vertices have been scanned Optionally augment with preceding SP vertex list

24. 3 rd editionwww.spatialanalysisonline.com24 Network analysis SPA – sample problem – specified tour A. Delivery locations and road barriersB. Solution (sequential tour from 1-4) S T Obstacles

25. 3 rd editionwww.spatialanalysisonline.com25 Network analysis Travelling salesman problems (TSPs) Basic problem: given N locations in the plane, what is the shortest complete circuit Very difficult to solve for N large (NP-complete) Modest sized problems can be solved exactly, e.g. by systematic tree-based search, LP + cutting planes Larger problems can be solved using heuristic methods, e.g. Genetic Algorithms, Cross-entropy methods, Simulated annealing Applications: salesmen visiting customers; rubbish trucks servicing business premises; delivery trucks servicing retail outlets; security staff patrolling premises; VLSI design; analysis of DNA sequences…

26. 3 rd editionwww.spatialanalysisonline.com26 Network analysis Sample TSP problem and exact solution A. Source vertices (130) – from TSPLibB. Exact TSP solution (Concorde)

27. 3 rd editionwww.spatialanalysisonline.com27 Network analysis Sample TSP problem and heuristic solution (L-K) A. Source vertices (130) – from TSPLibB. Heuristic TSP solution

28. 3 rd editionwww.spatialanalysisonline.com28 Network analysis Travelling salesman problems - extensions Multiple tours (e.g. divided point set) Should the tours start at the same point (e.g. warehouse, bus depot…?) What if demand varies across the target points? Capacity constraints – e.g. service vehicles may have limited capacity and vary in type what mix would be optimal? do tours/deliveries have to be made in certain time windows?

29. 3 rd editionwww.spatialanalysisonline.com29 Network analysis Drive time zones Network-derived zones of equal time from sample location Created as polygon layer(s) Can use map algebra techniques to compute estimated demand May include differential speeds for route type and off-road Can be slow to generate