L3-Network Algorithms
L3 – Network Algorithms
NGEN06(TEK230) –
Algorithms in Geographical Information Systems
by: Irene Rangel, updated Nov. 2015 by Abdulghani Hasan,
Nov 2016 by Per-Ola Olsson
L3-Network Algorithms
Content
1.
2.
3.
4.
5.
General issues of networks
Shortest path - Dijkstra’s algorithm
Other shortest path algorithms
Neighbourhood graphs and clustering
Traveling salesman problem.
L3-Network Algorithms
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.
L3-Network Algorithms
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
L3-Network Algorithms
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
L3-Network Algorithms
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
L3-Network Algorithms
Graphic
Representation
(easy for humans
To read)
Adjacency matrix
representation
Source: Worboys and Duckham 2004
Adjacency list
representation
L3-Network Algorithms
Sparse network with
several zeros stored
Graphic
representation
Adjacency matrix
representation
(as previous slide)
Adjacency list
representation
Source: Worboys and Duckham 2004
L3-Network Algorithms
a connected to:
b, distance 20 and
g, distance 15
h connected to c,
distance 10
Graphic
representation
Adjacency matrix
representation
Adjacency list
representation
Source: Worboys and Duckham 2004
L3-Network Algorithms
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.
L3-Network Algorithms
Shortest path
Navigation
Start point  target point
(distance c -> a is 28)
Accessibility
Many points target point
L3-Network Algorithms
Shortest path
Navigation
Start point  target point
Accessibility
Many points target point
(access e from four points - c,d,f,g)
L3-Network Algorithms
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
L3-Network Algorithms
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”.
L3-Network Algorithms
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
L3-Network Algorithms
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
L3-Network Algorithms
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
update the distance and the path for node m.
4.
Proceed from (1) again until the end node is visited.
L3-Network Algorithms
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.
A
B
C
D
A
0
2
1
-
B
2
0
3
2
C
1
3
0
4
D
-
2
4
0
L3-Network Algorithms
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
L3-Network Algorithms
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
L3-Network Algorithms
Does Dijkstra’s algorithm always give a
correct answer?
Yes, no heuristics are involved BUT all types of
distances are not allowed.
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.
L3-Network Algorithms
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.
3.
For each node (m) that is not yet visited do:
……………………………
4.
Proceed from (1) again until the end node is visited.
L3-Network Algorithms
Breadth-first traversal
Q is a queu – property is first-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a LAST in Q.
Add c (first in Q) to visited and add nodes connected to c LAST in Q.
Add d (first in Q) to visited and add nodes connected to d LAST in Q.
...
L3-Network Algorithms
Breadth-first traversal
Q is a queu – property is first-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a LAST in Q.
Add c (first in Q) to visited and add nodes connected to c LAST in Q.
Add d (first in Q) to visited and add nodes connected to d LAST in Q.
...
L3-Network Algorithms
Breadth-first traversal
Q is a queu – property is first-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a LAST in Q.
Add c (first in Q) to visited and add nodes connected to c LAST in Q.
Add d (first in Q) to visited and add nodes connected to d LAST in Q.
...
L3-Network Algorithms
Breadth-first traversal
Q is a queu – property is first-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a LAST in Q.
Add c (first in Q) to visited and add nodes connected to c LAST in Q.
Add d (first in Q) to visited and add nodes connected to d LAST in Q.
...
L3-Network Algorithms
Depth-first traversal
Same as
breadth-first
Q is a stack – property is last-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a FIRST in Q.
Add c (first in Q) to visited and add nodes connected to c FIRST in Q.
Add d (first in Q) to visited and add nodes connected to d FIRST in Q.
...
L3-Network Algorithms
Depth-first traversal
Now g is
added FIRST
Q is a stack – property is last-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a FIRST in Q.
Add c (first in Q) to visited and add nodes connected to c FIRST in Q.
Add d (first in Q) to visited and add nodes connected to d FIRST in Q.
...
L3-Network Algorithms
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
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a FIRST in Q.
Add g (first in Q) to visited and add nodes connected to g FIRST in Q.
Add d (first in Q) to visited and add nodes connected to d FIRST in Q.
...
L3-Network Algorithms
Depth-first traversal
Now we follow the
path a -> g “deeper”
Q is a stack – property is last-in-first-out
b is the
starting point
1.
2.
3.
4.
5.
6.
Source: Worboys and Duckham 2004
Start point b
Add nodes connected to b to Q and b to V
Add a (first in Q) to visited and add nodes connected to a FIRST in Q.
Add g (first in Q) to visited and add nodes connected to g FIRST in Q.
Add e (first in Q) to visited and add nodes connected to e FIRST in Q.
...
L3-Network Algorithms
Breadth-first, depth-first traversal
L3-Network Algorithms
Dijkstra’s algorithm
Breadth-first or
Depth-first algorithm?
L3-Network Algorithms
Dijkstra’s algorithm - breadth first algorithm
Dijkstra’s algorithm search in all
directions for the shortest path. It is an
breadth-first algorithm
L3-Network Algorithms
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
L3-Network Algorithms
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.
L3-Network Algorithms
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)
L3-Network Algorithms
Neighbourhood graphs – minimum spanning tree (MST)
Minimum Spanning Tree (MST)
1. No isolated node
2.
L3-Network Algorithms
Kruskal’s algorithm for computing MST
Edges
Original
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
L3-Network Algorithms
Neighbourhood graphs and Clustering
Visually easy to identify clusters,
but how to compute?
L3-Network Algorithms
Neighbourhood graphs and Clustering
Minimum Spanning Tree
(MST)
1. No isolated node
2.
Threshold distance
L3-Network Algorithms
Neighbourhood graphs and Clustering
Minimum Spanning Tree
(MST)
1. No isolated node
2.
Threshold distance
L3-Network Algorithms
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.
L3-Network Algorithms
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
L3-Network Algorithms
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.