using geometric information in euclidean graph algorithms paul bouman seminar graph drawing...
Post on 20-Dec-2015
214 views
TRANSCRIPT
Using geometric information in Euclidean
graph algorithms
Paul Bouman
Seminar Graph Drawing
17-10-2007
Overview
Euclidean graphs Shortest Paths Traveling Salesman
A geometric approach Special cases of TSPs
Conclusion Bibliography
Euclidean Graphs
Vertices are point in a plane Edges have weight defined by distances in
the plane
Triangle Inequality Lower bound on path length
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
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
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
Traveling Salesman
In eclidean space, the quandrangle inequality holds
Because of this, a minimal TSP tour can’t cross itself
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]
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
TSP: Determining a Convex Hull
1. Start with vertex h1 with lowest x coordinate
2. Choose the largest angle in x
3. Choose an angle vertex as h2
4. Look for vertex hi with biggest angle < hi-2,hi-1,hi
5. Repeat until the new hi = h1
Expanding the Partial Tour
There are two possible techniques Largest Angle method Most eccentric ellipse method
Methods don’t guarantee optimal solutions
Improvements possible
Largest Angle Method
For each internal vertex v, look at the angle α = <(u v w) with u and w consequent vertices on the partial tour
When we have u,v,w such that α is maximized, insert v between u and w on the partial tour.
Repeat until there are no internal vertices left
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.
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 aren’t optimal: tours with crossings
can be generated
Special Cases of TSPs
Pyramidally solvable TSP cases A tour φ = (1, i1, i2, …, ir, n, j1, j2, … jn-r-2) is
pyramidal if i1<i2<…<ir and j1>j2>…> jn-r-2
The number of pyramidal tours is exponential in n The minimum cost pyramidal tour can be found in
O(n2) time
Special Cases of TSPs
Symmetric Demidenko Matrices ci,j + cj+1,l ≤ ci,j+1 + cj,l for all 1≤i<j< j+1<l ≤ n
Symetric Kalmanson Matrices ci,j + ck,l ≤ ci,k + cj,l for all 1≤i<j<k<l≤n
ci,l + cj,k ≤ ci,k + cj,l for all 1≤i<j<k<l ≤n
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 k=3 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
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
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/college3.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)