Using geometric information in Euclidean graph algorithms Paul Bouman Seminar Graph Drawing

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 h 1 with lowest x coordinate 2. Choose the largest angle in x 3. Choose an angle vertex as h 2 4. Look for vertex hi with biggest angle < h i-2,h i-1,h i 5. Repeat until the new h i = h 1

TSP: Determining a Convex Hull

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

Largest Angle Method

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.

Most Eccentric Ellipse

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

Special Cases of TSPs Pyramidally solvable TSP cases A tour φ = (1, i 1, i 2, …, i r, n, j 1, j 2, … j n-r-2 ) is pyramidal if i 1 j 2 >…> j n-r-2 The number of pyramidal tours is exponential in n The minimum cost pyramidal tour can be found in O(n 2 ) time

Special Cases of TSPs Symmetric Demidenko Matrices c i,j + c j+1,l ≤ ci,j+1 + c j,l for all 1≤i<j< j+1<l ≤ n Symetric Kalmanson Matrices c i,j + c k,l ≤ c i,k + c j,l for all 1≤i<j<k<l≤n c i,l + c j,k ≤ c i,k + c j,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(n 3 ) time and O(n 2 ) space algorithm for the k=3 case Rote generalized this to O(n k ) time Convex Hull and Line: O(n 2 ) time and O(n) space Open problem: x-and-y axes TSP

Special Cases of TSPs

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 [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)