Using geometric information in Euclidean graph algorithms

1
Using geometric information in Euclidean graph
algorithms

• Paul Bouman
• Seminar Graph Drawing
• 17-10-2007

2
Overview

• Euclidean graphs
• Shortest Paths
• Traveling Salesman
• A geometric approach
• Special cases of TSPs
• Conclusion
• Bibliography

3
Euclidean Graphs

• Vertices are point in a plane
• Edges have weight defined by distances in the
  plane
• Triangle Inequality
• Lower bound on path length

4
Shortest Paths

• Most important algorithms were discussed by
  Jesper
• Euclidean Distance can be used as an heuristic
• In case of A average time can became O(n)
• Important question is function h admissable
• Interesting case travel time instead of distance

5
Shortest Paths A

• Admissibility The heuristic function may never
  overestimate the true distance to the destination
• When it does, A will find an optimal solution
• With travel time, the average speed V is the
  important parameter

6
Shortest Paths

• Combining A and bidirectional search may work
  nice
• Average computational time can become O(n) in
  Euclidean Graphs
• Layering approach can reduce shortest path
  queries below linear time

7
Traveling Salesman

• In eclidean space, the quandrangle inequality
  holds
• Because of this, a minimal TSP tour cant cross
  itself

8
Traveling Salesman

• Lemma If all cities lie on the boundary of a
  convex polygon, the optimal tour is a cyclic walk
  along the boundary of the polygon (in clockwise
  or counterclockwise direction) 4

9
TSP A geometric approach

• Work with partial tours
• 1. Start with the convex hull
• 2. Sequence a unsequenced city between two
  consecutive cities on the partial tour
• 3. While unsequenced cities Repeat 2
• 4. Done

10
TSP Determining a Convex Hull

• Start with vertex h1 with lowest x coordinate
• Choose the largest angle in x
• Choose an angle vertex as h2
• Look for vertex hi with biggest angle lt
  hi-2,hi-1,hi
• Repeat until the new hi h1

11
TSP Determining a Convex Hull
12
TSP Determining a Convex Hull
13
TSP Determining a Convex Hull
14
TSP Determining a Convex Hull
15
Expanding the Partial Tour

• There are two possible techniques
• Largest Angle method
• Most eccentric ellipse method
• Methods dont guarantee optimal solutions
• Improvements possible

16
Largest Angle Method

• For each internal vertex v, look at the angle a
  lt(u v w) with u and w consequent vertices on the
  partial tour
• When we have u,v,w such that a is maximized,
  insert v between u and w on the partial tour.
• Repeat until there are no internal vertices left

17
Largest Angle Method
18
Largest Angle Method
19
Largest Angle Method
20
Largest Angle Method
21
Largest Angle Method
22
Most Eccentric Ellipse

• Look at ellipse with focal points u and w, with a
  point v on the ellipse, where u and w are
  consecutive points on a partial path and v an
  internal vertex
• Look for the most eccentric ellipse defined by
  points u,v,w and add v between u and w on the
  partial tour.

23
Most Eccentric Ellipse
24
Most Eccentric Ellipse
25
Single Point Insertion

• Test each point in the tour between each
  consecutive pair and see if the solution improves
• Start again when an improved tour is found
• Methods arent optimal tours with crossings can
  be generated

26
Performance
27
Performance
28
Special Cases of TSPs

• Pyramidally solvable TSP cases
• A tour f (1, i1, i2, , ir, n, j1, j2,
  jn-r-2) is pyramidal if i1lti2ltltir and j1gtj2gtgt
  jn-r-2
• The number of pyramidal tours is exponential in n
• The minimum cost pyramidal tour can be found in
  O(n2) time

29
Special Cases of TSPs

• Symmetric Demidenko Matrices
• ci,j cj1,l ci,j1 cj,l for all 1iltjlt
  j1ltl n
• Symetric Kalmanson Matrices
• ci,j ck,l ci,k cj,l for all 1iltjltkltln
• ci,l cj,k ci,k cj,l for all 1iltjltkltl n

30
Special Cases of TSPs
31
Special Cases of TSPs

• The k-line TSP has cities on k (almost) parallel
  lines.
• Cutler created an O(n3) time and O(n2) space
  algorithm for the k3 case
• Rote generalized this to O(nk) time
• Convex Hull and Line O(n2) time and O(n) space
• Open problem x-and-y axes TSP

32
Special Cases of TSPs
33
Conclusion

• In euclidean space, some graph problems can be
  solved more easily
• Shortest Path problems can be solved most
  efficiently by layering techniques
• TSP problems in the plane can be solved to a
  rather good extend using geometric notions
• Some special cases of TSP problems can be solved
  in polynomial time

34
Bibliography

• 1 Heuristic shortest path algorithms for
  transportation applications State of the art, L.
  Fu, D. Sun, L.R. Rilett (1995)
• 2 Hoorcollegeslides Zoekalgoritmen, Linda van
  der Gaag http//www.cs.uu.nl/docs/vakken/za/colleg
  e3.pdf
• 3 Heuristic for the Hamiltonian Path Problem in
  Euclidian Two Space, J. P. Norback R. F. Love
  (1979)
• 4 Well-solvable special cases of the traveling
  salesman problem a survey, Rainer E. Burkard,
  Vladimir G. Deineko, René van Dal, Jack A. A. van
  der Veen, Gerhard J. Woeginger (1998)
• 5 Geometric approaches to solving the traveling
  salesman problem, John P. Norback, Robert F. Love
  (1977)