Computational Geometry Exercise session #7: Voronoi diagrams - - PDF document

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Computational Geometry

Exercise session #7: Voronoi diagrams

• Fields of applications of Voronoi Diagrams
• Applications:
• 1. The largest empty circle problem
• 2. Euclidean minimum spanning trees
• 3. Euclidean traveling salesman problem
• 4. Polyhedral terrain reconstruction
• Graphs related to the Delaunay graph
• Homework 3 solution

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Fields of application

• Partial list of fields:

Anthropology Archaeology Astronomy Biology Cartography Chemistry Crystallography Ecology Forestry Geography Geology Linguistics Marketing Metallography Meteorology Operations research Physics Physiology Statistics Remote sensing Urban planning

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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The largest empty circle problem

• Problem definition: find a largest radius circle

whose center is in the (closed) convex hull of a set of n sites S, and contains no sites within its interior.

• Motivation: placement of a nuclear reactor as

far away from point-cities (maximize minimal distance while keeping maximum small).

• An optimization problem with seemingly

infinite choice of options (location of center).

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Centers inside the convex hull

• Imagine inflating a circle from a point p inside the

hull until it hits a site (radius = f(p)).

• f(p) cannot be a maximum if only one site is hit (p

inside Voronoi cell).

• f(p) also cannot be a maximum if only two sites are

hit (p on edge of Voronoi diagram).

• f(p) is possibly at local maximum if three sites are hit

at the same time (Voronoi vertex), but not always (sites must encompass more than semicircle).

• Conclusion: look at circles at Voronoi vertices.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Centers on convex hull boundary

• Previous arguments do not apply when point is
• n boundary of hull.
• f(p) cannot be a maximum on a hull vertex.
• If circle hits one site, f(p) can increase by

moving to one of the two sides of the hull edge.

• f(p) can be at local maxima if two sites are hit,

since outside of hull is illegal.

• Conclusion: maxima of f(p) can occur at

intersection of Voronoi edges and hull boundary.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Largest empty circle algorithm

1. Compute Voronoi Diagram of sites VD(S). 2. Compute convex hull CH(S). 3. For each Voronoi vertex v do:

• If v is inside CH(S) then
• Compute radius of circle centered on v, update

max 4. For each Voronoi edge e do:

• Compute p = e ∩ ∂CH(S)
• Compute radius of circle centered on p, update

max

• Return max

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Algorithm complexity

• Naïve implementation: O(n2) time.
• Efficient implementation:

Step 3 – preprocess CH(S) for point inclusion queries costing O(logn) time each. Step 4 – find intersection by binary search, discrimination based on vertex-edge sidededness tests: O(logn) time each. Total O(nlogn) time for entire algorithm.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Polyhedral terrain reconstruction

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Fat triangulations

• There are many ways to triangulate a point set.
• Good heuristic: maximize minimal angle.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Lexicographical order of triangulations

• Let t be the number of triangles in

triangulation T (constant for fixed point set S).

• Let α1,α2,…,α3t be the angles of the triangles,

sorted by increasing order.

• Definition: Triangulation T > T’ if there exists

an index i ≤ 3t such that αj=αj’ for all j < i and αi>αi’.

• Theorem: the Delaunay triangulation DT is

maximal, that is DT ≥ T, for any T of S.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Sketch of proof

• The edge flip operation on an ‘illegal edge’ of

the triangulation results in T’ > T.

• An edge is illegal iff the enclosing circle

contains one of the quadrilateral points.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Sketch of proof, continued

• From definition of Delaunay graph as the dual
• f the Voronoi diagram:

there is an edge between p and q iff there exists a closed disc with p and q on its boundary and no

• ther site in its interior.

Three points p,q,r are the vertices of the same face

• f the Delaunay graph iff the circle through them

contains no other point.

• The delaunay triangulation is obtained from

repeated edge flipping until no more improvements are possible.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Euclidean Minimum Spanning Tree

• The Minimum Spanning Tree (MST) of a

graph G=(V,E) is a tree that covers V with minimal total weight of the edges.

• Consider the complete graph on a set of points

S, with edge weights equal to the Euclidean

• distance. The MST of this graph is called the

EMST of S.

• Motivation: minimal cost of a (telephone,

cable, …) network that reaches every target.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Kruskal’s algorithm

• The greedy approach to growing the tree:

Sort the edges by their weight: e1,e2,… Initialize tree T to be empty, i = 1. While T is not spanning do: If T + ei is acyclic then T = T + ei i = i + 1

• The algorithm can be implemented in

O(|E|log|E|) time.

• In our case, |E| = n(n-1)/2.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Relation between EMST and DG

• Theorem: EMST(S) is a subset of the Delaunay

graph DG(S).

• Proof:

Assume edge e=(p,q) in EMST(S) but not in DG(S). No circle with p, q on boundary is empty. Circle with e as diameter is not empty (r inside). Remove e from EMST(S), put either (p,r) or (q,r) instead to obtain shorter tree contradiction.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Efficient EMST algorithm

• Compute the Delaunay graph of S.
• Compute the EMST using Kruskal’s algorithm
• n the edges of DG.
• Complexity: O(nlogn).

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Euclidean Traveling Salesman Problem

• TSP problem on graphs: visit every vertex and

return to origin with minimum total cost.

• ETSP problem: similar, except no graph edges

specified – every shortcut is possible.

• Problem is NP-complete. Known 2-

approximation: use the MST as the path, running on all edges twice.

• Improvement – make detours where vertex

already visited.

• The Christofides heuristc: 1.5 approximation.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Graphs related to Delaunay graph

• The boundary of the convex hull is a subgraph
• f the Delaunay graph.
• Definition: the Gabriel Graph (GG) has an

edge (p,q) iff the circle with (p,q) as diameter is empty of points in S.

• Definition: the Relative Neighborhood Graph

(RNG) has an edge (p,q) iff d(p,q) ≤ minr≠p,q{d(p,r),d(q,r)}

i.e. lune defined by intersection two discs is empty.

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Relation between graphs

• Definition: the Nearest Neighbor Graph

(NNG) has a directed edge pq iff q is the nearest neighbor of p.

DG(S) ∂CH(S) GG(S) RNG(S) EMST(S)