L3 Network Algorithms - PDF Free Download

L3 Network Algorithms NGEN06(TEK230) Algorithms in Geographical Information Systems by: Irene Rangel, updated Nov. 2015 by Abdulghani Hasan, Nov 2017 by Per-Ola Olsson

Content 1. General issues of networks 2. Shortest path - Dijkstra s algorithm 3. Other shortest path algorithms 4. Neighbourhood graphs and clustering 5. Traveling salesman problem.

Networks in GIS By using the distances between nodes in a network we can e.g. derive the shortest path, minimum number of nodes to pass and study accessibility.

What is a network A network or graph has 2 main constitutes: Nodes Edges Edges give distances between nodes. Distances can e.g. be: - Euclidean distances - Travelling time The network can e.g. be: - street network - airplane routes

What is a network A network or graph has 2 main constitutes: Nodes Edges Many analysis in networks only consider the nodes and the edges. Geometric location of the nodes is uninteresting

General issues of networks Networks can be stored in matrixes. Not a good data storage, especially for sparse networks, since it requires a lot of memory. But a matrix gives fast access to the nodes; O(1) to access a single node. For example fast to check if two nodes are connected. In most cases relational databases or lists and trees are more suitable storage structures. A B C D A 0 2 1 - B 2 0 3 2 C 1 3 0 4 D - 2 4 0 - implies that the nodes are not connected by any edge

Graphic Representation (easy for humans To read) Adjacency matrix representation Adjacency list representation Source: Worboys and Duckham 2004

Sparse network with several zeros stored Graphic representation Adjacency matrix representation Adjacency list representation Source: Worboys and Duckham 2004

a connected to: b, distance 20 and g, distance 15 Graphic representation Adjacency matrix representation Adjacency list representation Source: Worboys and Duckham 2004

Compare to matrix with one row h connected to c, distance 10 Graphic representation Adjacency matrix representation Adjacency list representation Source: Worboys and Duckham 2004

Shortest path The path between two nodes that is shorter than all other possible paths. In the shortest path problem we are not only restricted to Euclidean distances, but all the kind of distances are of interest.

Navigation Start point target point (distance c -> a is 28) Accessibility Many points target point Shortest path

Navigation Start point target point Shortest path Accessibility Many points target point (access e from four points - c,d,f,g)

Shortest path Accessibility to parks and nature in Gothenburg (Viktor Svanerud, master thesis)

Dijkstra s algorithm - one of the most well-known shortest path algorithms. Example of finding the shortest path between Node A and Node D. We need 2 matrixes: Adjacency matrix (static) State matrix (dynamic) A B C D A 0 2 1 - B 2 0 3 2 C 1 3 0 4 D - 2 4 0 We recognise the adjacency matrix from above

Dijkstra s algorithm Example of finding the shortest path between Node A and Node D. We need 2 matrixes: Adjacency matrix (static) State matrix (dynamic) Node Distance Path Visited A 0 - Yes B 2 A No C 1 A No D - No Distance a -> b is 2 The state matrix is dynamic and changes when a node is visited.

Dijkstra s algorithm State matrix: Node is the ID of the nodes. Distance keeps track of the current shortest distance from the starting node to this node. Path stores the previous node along the shortest path (found so far) to this node. Visited states if the node has been visited in the computations of the shortest path (explained later). Node Distance Path Visited A 0 - Yes B 2 A No C 1 A No D - No Start point is A

Dijkstra s algorithm First initialize the state matrix: The start node is marked as visited. All nodes that can be reached from the start are updated in the columns distance and path based on Adjacency matrix (row-wise). Distance values are set to the distances between the start node and the respective node (derived from the adjacency matrix). Path values are set to the start node (the starting node is the current shortest path to this particular node) If there is no path to a node, the distance is set to infinity and the path value is undefined. Node Distance Path Visited A 0 - Yes B 2 A No C 1 A No D - No The state matrix is initialized for start point A

Dijkstra s algorithm After the state matrix is initialized: 1. Find the node with the shortest distance that is not yet visited. Denote this as the current node. 2. Mark the current node as visited. 3. For each node (m) that is not yet visited do: If the node m and the current node has a common edge AND if the sum of the distance from the start node to the current node and the distance from the current node to the node m (found in the adjacency matrix) is shorter than the current distance from the starting node to node m (stored in the state matrix) DO State matrix update the distance and the path for node m. 4. Proceed from (1) again until the end node is visited.

1. Find the node with the shortest distance that is not yet visited. Denote this as the current node. 2. Mark the current node as visited. 3. For each node (m) that is not yet visited do: If the node m and the current node has a common edge AND if the sum of the distance from the start node to the current node and the distance from the current node to the node m (found in the adjacency matrix) is shorter than the current distance from the starting node to node m (stored in the state matrix) DO update the distance and the path for node m. 4. Proceed from (1) again until the end node is visited. Adjacency matrix A B C D Adjacency matrix: If nodes have a common edge and for distances between current node and node m A 0 2 1 - B 2 0 3 2 C 1 3 0 4 D - 2 4 0

Result of the shortest path algorithm Distance from start to end node (Distance) Path? Node Distance Path Visited A 0 - Yes B 2 A Yes C 1 A Yes D 4 B Yes

Result of the shortest path algorithm Distance from start to end node (Distance) Path from end backwards to start. Node Distance Path Visited A 0 - Yes B 2 A Yes C 1 A Yes D 4 B Yes

Does Dijkstra s algorithm always give a correct answer? Yes, no heuristics are involved BUT all types of distances are not allowed. L3-Network Algorithms Conditions for a metric: 1. d(p,q)>=0, d(p,q)=0 p=q 2. d(p,q)=d(q,p) (symmetry) 3. d(p,q)<=d(p,r)+d(r,q) (triangle inequality) Preparation for the network algorithm exercise.

What s the computational complexity of Dijkstra s algorithm? 1. Find the node with the shortest distance and that is not yet visited. 2. Mark the current node as visited. For each node (m) that is not yet visited do: 4. Proceed from (1) again until the end node is visited.

Breadth-first traversal Q is a queu property is first-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and b to V 3. Add a (first in Q) to visited and add nodes connected to a LAST in Q. 4. Add c (first in Q) to visited and add nodes connected to c LAST in Q. 5. Add d (first in Q) to visited and add nodes connected to d LAST in Q. 6....

Breadth-first traversal Q is a queu property is first-in-first-out b is the starting point Source: Worboys and Duckham 2004 3. 4. Add c (first in Q) to visited and add nodes connected to c LAST in Q. 5. Add d (first in Q) to visited. No new nodes connected to d. 6. Add g (first in Q) to visited. No new nodes connected to g.

Depth-first traversal Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add c (first in Q) to visited and add nodes connected to c FIRST in Q. 5. Add d (first in Q) to visited and add nodes connected to d FIRST in Q. 6....

Depth-first traversal Now g is added FIRST Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add c (first in Q) to visited and add nodes connected to c FIRST in Q. 5. Add d (first in Q) to visited and add nodes connected to d FIRST in Q. 6....

Depth-first traversal Now we follow the path a -> g deeper and add e FIRST in Q Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add g (first in Q) to visited and add nodes connected to g FIRST in Q. 5. Add d (first in Q) to visited and add nodes connected to d FIRST in Q. 6....

Depth-first traversal Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add g (first in Q) to visited and add nodes connected to g FIRST in Q. 5. Add e (first in Q) to visited and add nodes connected to e FIRST in Q. 6....

Depth-first traversal Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004 1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add g (first in Q) to visited and add nodes connected to g FIRST in Q. 5. Add e (first in Q) to visited and add nodes connected to e FIRST in Q. 6. Add f (first in Q) to visited. No new nodes connected to f.

1. Start point b 2. Add nodes connected to b to Q and add b to V 3. Add a (first in Q) to visited and add nodes connected to a FIRST in Q. 4. Add g (first in Q) to visited and add nodes connected to g FIRST in Q. 5. Add e (first in Q) to visited and add nodes connected to e FIRST in Q. 6. Add f (first in Q) to visited. No new nodes connected to f. 7. Add c (first in Q) to visited and add nodes connected to c FIRST in Q. L3-Network Algorithms Depth-first traversal Q is a stack property is last-in-first-out b is the starting point Source: Worboys and Duckham 2004

Breadth-first, depth-first traversal L3-Network Algorithms

Dijkstra s algorithm Breadth-first or Depth-first algorithm?

Dijkstra s algorithm - breadth first algorithm Dijkstra s algorithm search in all directions for the shortest path. It is an breadth-first algorithm

Other shortest path algorithms Algorithms with heuristics do not always give the best answer (only a reasonably good answer is guaranteed). Heuristic 1 Direction limitation (nodes closer to the end node) A* algorithm Euclidean distances, depth-first & breadth-first More likely that the shortest path is towards the end node

Other shortest path algorithms Algorithms with heuristics do not always give the best answer (only a reasonably good answer is guaranteed). Heuristic 2 Hierarchical algorithm Minor roads and major roads Minor roads only the last and first part of the route. Major roads (highways) for the main part of the route.

Other shortest path algorithms Algorithms with heuristics do not always give the best answer (only a reasonably good answer is guaranteed). Heuristic 3 super nodes (transit nodes) Stored distances between super nodes (transit nodes)

Neighbourhood graphs minimum spanning tree (MST) Minimum Spanning Tree (MST) 1. No isolated node 2.

Kruskal s algorithm for computing MST L3-Network Algorithms Original Edges Ordered Sort the edges in ascending order after distance. While there are isolated nodes in the ordered list add the currently shortest node to the MST. Do NOT add nodes that create a closed circle Source: Sedgewick 2002

Neighbourhood graphs and Clustering Visually easy to identify clusters, but how to compute?

Neighbourhood graphs and Clustering Minimum Spanning Tree (MST) 1. No isolated node Threshold distance 2.

Traveling salesman problem (TSP) 1. Begin and end in the same node (or in different nodes) 2. Visit all nodes. 3. Minimize total length of the route.

Traveling salesman problem (TSP) How to solve the TSP problems? To compute all possible combinations and chose the shortest one is too computational demanding. It is NP-complete Use the minimum spanning tree: The maximum length of the TSP is at most 2 times the total length of the MST

Traveling salesman problem (TSP) How to solve the TSP problems? Compute the MST e.g. with Kruskal s algorithm. Do iterative improvements until no further improvements are possible. A local optimum of the TSP is found.