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Geometric Networks

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minimize the longest edge length of the tour (Bottleneck TSP) ... Choose one site per city so that the length of yellow tour is minimized ... – PowerPoint PPT presentation

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Title: 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
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