l3-network algorithms l3 – network algorithms ngen06(tek230) – algorithms in geographical...

L3-Network Algorithms

L3 – Network Algorithms

NGEN06(TEK230) –

Algorithms in Geographical Information Systems

by: Irene Rangel, updated Nov. 2015 by Abdulghani Hasan


Background

By using the distances between nodes in a

network we can derive the shortest path,

Minimum number of nodes to pass,

and other interesting quantities in GIS.

Shortest path -> Dijkstra’s algorithm


Content

1. General issues of networks

2. Dijkstra’s algorithm

3. Other shortest path algorithms

4. Neighbourhood graphs and clustering

5. Traveling salesman problem.


Aim

Understand how an algorithm can use geographic information stored in a network.

Get knowledge about the shortest path problem and Dijkstra’s algorithm.


General issues of networks

Many analysis in networks only consider the nodes and the edges.

• A network or graph has 2 main constitutes:– Nodes– Edges

street network

airplane routes

Geometric location of the nodes is uninteresting

Edges = (distances between nodes)


General issues of networks

A B C D

A 0 2 1 -

B 2 0 3 2

C 1 3 0 4

D - 2 4 0

Networks can be stored in matrixes.

”-” implies that the nodes are not connected by any edge

In most cases relational databases or lists and trees are more suitable storage structures.

Not a good data storage since it requires a lot of memory.


Graphicrepresentation

Adjacency matrixrepresentation

Adjacency listrepresentation

Source: Worboys and Duckham 2004


Shortest path

• The path between two nodes that is shorter than all the 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.

• Dijkstra’s algorithm is one of the most well-known shortest path algorithms.


Shortest path

Navigation

Start point target point

Accessibility

Many points target point


Dijkstra’s algorithm

• Example of finding the shortest path between nodes A and D.

• We need 2 matrixes:– Adjacency matrix

– 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

Node Distance Path Visited

A 0 A Yes

B 2 A No

C 1 A No

D - No



State matrix– Distance keeps track of the current shortes distance from the

starting node to this node.– Path stores the previous node along the shortes path (found

so far) to this node.– Visited describes is the node has been visited in the

computations of the shortest path.

Start point is A


A

B

C

D


A 0 A Yes

B 2 A No

C 1 A No

D - No


A 0

B 2

C 1

D


A 0 A

B 2 A

C 1 A

D -



Initialized state matrix

Initialization of the state matrix:– The start node is marked as visited.– All the nodes that can be reached from the start node have been updated in the

columns distance and path.• Distance values are set to the distances between the start node and the

respective node (use 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 known path to some nodes, their distance is set to infinity and their

path value are undefined.

The state matrix is initialized for start point A


A 0 A Yes

B 2 A No

C 1 A No

D - No


A 0 A Yes

B 2 A No

C 1 A No

D - No



0. Initialize the state matrix.1. Find the node which has the shortest distance and is not yet

visited. Denote this as the current node.2. Mark the current node as visited.

Stop Criteria: End node visited?3. For each node (m) that is not yet visited do:

• If the node m and the current node has a common edge ANDthe sum of the distances for the start node to current node and from current node to 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)DOupdate the distance and the path for node m.

4. Proceed from (1) again until the end node is visited.


Result of the shortest path algorithm

• Distance from start to end node• Path


A 0 - Yes

B 2 A Yes

C 1 A Yes

D 4 B Yes



This algorithm does not search along a possible path direction

depth-first

BUT

rather in all directions

breadth-first.


Does Dijkstra’s algorithm always give a correct answer?

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)


Other shortest path algorithms

Heuristic 1 – Direction limitation

A* algorithm

Euclidean distances, depth-first & breadth-first

Heuristic 2 – Hierarchical algorithm

Minor roads and major roads

Heuristic 2 – super nodes (transit nodes)

Algorithms with heuristics are not always giving the best answer (only a reasonably good answer is guaranteed).


Breadth-first traversal

Q is a queu – property is first-in-first-out

b is the starting point



Depth-first traversal

S is a stack – property is last-in-first-out

b is the starting point



Neighbourhood graphs – minimum spanning tree (MST)

Minimum Spanning Tree (MST)

1. No isolated node

2.


Source: Sedgewick 2002

Kruskal’s algorithm for computing MST

Original Ordered

Edges


Neighbourhood graphs and Clustering

Minimum Spanning Tree (MST)

1. No isolated node

2.

Threshold distance


Traveling salesman problem (TSP)



1. Begin and end in the same node.

2. Visit all nodes.

3. Minimize total length of the route.

NP-complete



A possible way is to compute all possible combinations and chose the shortest one. This would, of course, take exponential time.

How to solve the TSP problems?

A more sensible approach, is to use heuristics. One approach is to start to compute the MST (by e.g. Kruskal’s algorithm).

The maximum length of the TSP is at most 2 times the total length of the MST



This solution can greatly been improve by making short cuts.

Start by the MST

Make iterative improvements when they found.

until no further improvements are possible.

A local optimum of the TSP is found.

How to solve the TSP problems?

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