Fundamentals of Network Analysis: Components, Problems, and Computational Complexity

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

2. 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 www.spatialanalysisonline.com

3. 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 www.spatialanalysisonline.com

4. 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 www.spatialanalysisonline.com

5. Network analysis • Networks – basic components: • Degree (of a vertex) • Path • Connected graph • Cycles/circuits • Tree www.spatialanalysisonline.com

6. Network analysis • Networks – basic components: Intermediate data coding points - not vertices - can be ignored/eliminated Paths - edges and vertices Tree - no circuits Circuit Cells: V-E+C=2 www.spatialanalysisonline.com

7. Network analysis • Networks – compare topologies: www.spatialanalysisonline.com

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

9. 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 www.spatialanalysisonline.com

10. 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 www.spatialanalysisonline.com

11. 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) www.spatialanalysisonline.com

12. 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(n2) and O(nlogn) • O(logn)<O(n)<O(nlogn)<O(n2), n>3 • Non-polynomial (NP): e.g. O(n!) andO(2n) • O(2n)<<O(n!), n>3 • NP-complete problems • Heuristics www.spatialanalysisonline.com

13. 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 www.spatialanalysisonline.com

14. 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 www.spatialanalysisonline.com

15. 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. 1st, 2nd..) • Static vs. dynamic environments • Exact vs. approximate algorithms • Known vs. unknown map www.spatialanalysisonline.com

16. 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 www.spatialanalysisonline.com

17. 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 www.spatialanalysisonline.com

18. Network analysis • Minimum spanning tree (Euclidean) www.spatialanalysisonline.com

19. Network analysis • Gabriel network included excluded www.spatialanalysisonline.com

20. Network analysis • Steiner tree (unweighted, Euclidean) www.spatialanalysisonline.com

21. 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 (1st, 2nd,… 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 www.spatialanalysisonline.com

22. Network AnalysisDantzig 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 www.spatialanalysisonline.com

23. 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 www.spatialanalysisonline.com

24. Network analysis • SPA – sample problem – specified tour T S Obstacles www.spatialanalysisonline.com

25. 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… www.spatialanalysisonline.com

26. Network analysis • Sample TSP problem and exact solution www.spatialanalysisonline.com

27. Network analysis • Sample TSP problem and heuristic solution (L-K) www.spatialanalysisonline.com

28. 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? www.spatialanalysisonline.com

29. 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 www.spatialanalysisonline.com