Geometric Networks

1
Geometric Networks
Chan-Su Shin Digital Information Engineering ????
2
Geometric Networks
3
Network Topology

• How to connect objects?

Spanning Tree
4
Network Topology

• How to connect?

Tour / Cycle
5
Measures

• What to optimize?
• minimize the total edge length of the spanning
  tree
• minimize the diameter of the spanning tree

Euclidean Minimum Spanning Tree
6
Measures

• What to optimize?
• minimize the length of the tour (TSP)
• maximize the length of the (non-crossing) tour
  (Max TSP)
• minimize the longest edge length of the tour
  (Bottleneck TSP)
• maximize the shortest edge length of the tour
  (Maximum Scatter TSP)

7
Lp Metrics
8
Objects

• What to connect?
• points
• line segments
• circles
• convex regions
• arbitrary regions
• Dimension

9
Geometric Network Problems

• Network Topology
• Measures (and Metrics)
• Objects (and Dimension)
• Problems mentioned in this talk
• MST problem and its relatives
• TST problem and its relatives

10
Euclidean MST Problem

• Definition (2 dimension)
• Given n points in the plane, find a minimum
  length spanning tree
• Applications
• Designing physical networks such as road,
  telephone line, TV cable
• Clustering using Maximum Spanning Tree
• Processing medical images
• Processing satellite images

11
Medical Image Processing

• Arrangement of nuclei in skin cell for cancer
  research

12
Medical/Satellite Image Processing

• Extraction of the key frame of vascular tree and
  road

13
MST Algorithm in graph

• Assign edges with blue or red such that MST edges
  have only blue and the others have red

14
Blue Rule

• Select a cut that no blue edge crosses. Among
  uncolored edges crossing the cut, select one of
  minimum length and color it blue

15
Blue Rule

• Select a cut that no blue edge crosses. Among
  uncolored edges crossing the cut, select one of
  minimum length and color it blue

16
Blue Rule

• Blue rule satisfies the invariant

17
Blue Rule

• Blue rule satisfies the invariant

18
Red Rule

• Select a simple cycle containing no red edges.
  Among edges on the cycle, select one of maximum
  length and color it red.

19
Red Rule

• Select a simple cycle containing no red edges.
  Among edges on the cycle, select one of maximum
  length and color it red.

20
Algorithm

• Applies Blue and Red rules repeatedly until all
  edges have a (blue or red) color.
• Implementations
• Kruskals algorithm
• Prims algorithm
• Boruvkas algorithm
• Time complexity
• m is of edges
• n is of vertices

21
Euclidean MST algorithm

• Make a complete graph for points
• The graph has quadratic edges
• Run MST algorithm with the graph
• Time complexity
• Is this satisfactory?
• No! We didnt use any geometric properties

22
Geometric property

• If an edge is in MST, then the lune has no other
  points

23
Geometric property

• If an edge is in MST, then the lune has no other
  points

24
Geometric property

• Relative Neighborhood Graph (RNG)

25
Geometric property

• Relative Neighborhood Graph (RNG)

• MST is a subgraph of the relative neighborhood
  graph
• This graph has O(n) edges only and found in O(n
  log n) time
• If apply MST algorithm to this graph, then
  Euclidean MST can be computed O(n log n) time

26
Relatives of MST problems

• Maximum spanning tree
• solvable in O(n log n) time
• Maximum non-crossing spanning tree
• no results have been known even whether or not
  its NP-hard
• k-point minimum spanning tree (k-MST)
• NP-complete, so approximation algorithms have
  developed
• -factor approximation was proposed in
  1996

27
Relatives of MST problems

• Low-degree spanning tree
• Degree of vertices of MST is lt 6 (why? kissing
  number)
• Exist deg-3, deg-4 spanning trees whose lengths
  are less than 3/2, 5/4 times that of MST,
  respectively
• minimum degree-2 spanning tree
  (Traveling Salesperson Path Problem)

28
Relatives of MST problems

• Minimum Steiner spanning tree
• Steiner points
• Steiner ratio length(MST)
• length(MSST)
• Example
• Du and Hwang proved
• Thus

29
TST Problem

• Definition
• Given n points in the plane, find a shortest tour
  that visits every point.
• Very famous NP-complete problem
• Approximation algorithms are needed
• Performance of approximation algorithms

30
2-ratio approximation algorithm

• Compute MST
• Double each edge of MST

31
2-ratio approximation algorithm

• length(MST) lt length(Optimal TST)
• length(MST2) lt 2 length(Optimal TST)

32
1.5-ratio approximation algorithm

• Find minimum matching M for odd-degree vertices
  of MST
• length(M) lt length(Optimal TST) / 2

33
1.5-ratio approximation algorithm

• Find Euler tour for union of MST and M
• (Note Every vertex has even degree)

34
1.5-ratio approximation algorithm

• Make a tour T by traversing the edges of Euler
  tour
• length(T) lt length(MST) length(M)
• lt length(Opt TST)
  length(Opt TST)/2

35
Best approximation algorithm

• -ratio approximation algorithm running
  in polynomial time Aroa1996, Mitchell1998
• length(T) lt length(Optimal TST)

36
TST problem under practical model

• Traveling sites of cities over the country on
  business
• Choose one site per city so that the length of
  yellow tour is minimized
• Constant-ratio approximation algorithm for same
  diameter cities 2000

37
TST problem under practical model

• Traveling Salesman and Buyers Tour problem
• Choose one site per city so that the length of
  yellow tour plus red access cost is minimized.
• For convex regions, there is constant-ratio
  approximation algorithm 2001

38
MST problem under practical model

• Geometric Network-base Location problem

39
Variation of TSP

• maximize the length of the (non-crossing) tour
  (Max TSP)
• For , solvable exactly in
  time, but unknown for 2D
• minimize the longest edge length of the tour
  (Bottleneck TSP)
• NP-hard
• 2-approximation, which is the best possible
  unless P NP
• maximize the shortest edge length of the tour
  (Maximum Scatter TSP)
• NP-complete in graph model, 2-approximation
  (which is best possible)
• Unknown if one can get algorithms of ratio lt 2 in
  the geometric model

40
Thanks