Using geometric information in Euclidean graph algorithms Paul Bouman Seminar Graph Drawing 17-10-2007

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

Performance

Performance

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


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



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)

Documents

Using geometric information in Euclidean graph algorithms Paul Bouman Seminar Graph Drawing 17-10-2007