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Voronoi Diagrams

Voronoi Diagrams. Computational Geometry, WS 2007/08 Lecture 11 Prof. Dr. Thomas Ottmann. Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg. Overview. Motivation Voronoi definitions Characteristics

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Voronoi Diagrams

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  1. Voronoi Diagrams Computational Geometry, WS 2007/08 Lecture 11 Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

  2. Overview • Motivation • Voronoi definitions • Characteristics • Size and storage • Construction • Divide and Conquer • Fortune’s Algorithm • Applications Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  3. The Voronoi Diagram Viewpoint 1: Locate the nearest dentist. Viewpoint 2: Find the ‘service area’ of potential customers for each dentist. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  4. The Voronoi Diagram Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  5. Voronoi Regions Eucledian distance : dist(p,q) := Let P :={ p1, p2, ...,pn } be a set of n distinct points in a plane.We define the voronoi diagram of P as the subdivision of the planeinto n cells, with the property that a point q lies in the cell correspon-ding to a site pi iff dist(q, pi ) < dist(q, pj ) for each pj P with j  i. We denote the Voronoi diagram of P by Vor(P).The cell that corresponds to a site piis denotd by V(pi ), called thevoronoi cell of pi. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  6. Example V(pi) = 1jn, jih(pi, pj) q1 q2 p q4 q3 Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  7. Computing the Voronoi Diagram Input: A set of points (sites) Output: A partitioning of the plane into regions of equal nearest neighbors Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  8. Voronoi Diagram Animations Java applet animation of the Voronoi Diagram by: Christian Icking, Rolf Klein, Peter Köllner, Lihong Ma (FernUniversität Hagen) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  9. Characteristics of the Voronoi Diagram (1) Voronoi regions (cells) are bounded by line segments. Special case : Collinear points Theorem : Let P be a set of n points (sites) in the plane.If all the sites are collinear, then Vor(P) consist of n-1 parallel lines and n cells. Otherwise, Vor(P) is a connected graph and its edges are either line segments or half-lines. e pk If pi, pjare notcollinearwithpk, thenh(pi, pj ) and h(pj, pk ) can not be parallel! pi pj h(pj,pk) h(pi,pj) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  10. Vor(P) is Connected Claim: Vor(P) is connected Proof by contradiction: If Vor(P) is not connected then there would be a Voronoicell V(Pi ) splitting the plane into two halfes. Because Voronoi cells are convex, V(Pi ) would consist of a strip bounded by two parallel full lines, but we know that edges of Voronoi diagram cannot be full lines, hence a contradiction. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  11. Other Characteristics; (2),(3) Assumption: No 4 points are on the circle. (2) Each vertex (corner) of VD(P) has degree 3 (3) The circle through the three points defining a Vertex of the Voronoi diagram does not contain any further point Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  12. Other Characteristics; (4),(5) (4) Each nearest neighbor of one point defines an edge of the Voronoi region of the point. (5) The Voronoi region of a point is unbounded iff the point lies exactly on the convex hull of the point set. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  13. Size and Storage Size of the Voronoi Diagram: V(p) can have O(n) vertices! Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  14. Size of the Voronoi Diagram Theorem: The number of vertices in the Voronoi diagram of a set of n points in the plane is at most 2n-5 and the number of edges is at most 3n-6. • Proof:1. Connect all Half-lines with fictitious point 2. Apply Euler`s formula:v – e + f = 2 • For VD(P) +  :v = number of vertices of VD(P) + 1e = number of edges of VD(P)f = number of sites of VD(P) = n • Each edge in VD(P) +  has exactly two vertices and each vertexof VD(P) + has at least a degree of 3: • sum of the degrees of all vertices ofVor(P) + = 2·( # edges of VD(P) )  3· (# vertices of VD(P) + 1) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  15. Proof (Continued) Number of vertices ofVD(P) = vp Number of edges of VD(P) = ep We can apply:(vp + 1) – ep + n = 2 2 ep  3 (vp + 1) 2 ep  3 ( 2 + ep - n) = 6 + 3ep – 3n 3n – 6  ep Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  16. Example Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  17. 4 3 2 5 3 1 4 6 1 2 5 Storage Issues Three Records (DCEL): vertex { Coordinates Incident edge };face { OuterComponent InnerComponents };halfedge { Origin Twin IncidentFace Next Prev }; e.g. : Vertex 1 = {(1,2) | 12}Face 1 = {15 | [] }Edge 54 = { 4 | 45 | 1 | 43 | 15 } Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  18. Computing the Voronoi Diagram Input: A set of points (sites) Output: A partitioning of the plane into regions of equal nearest neighbors. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  19. Divide and Conquer: Divide Step Input: A set of points (sites) Output: A partitioning of the plane into regions of equal nearest neighbors. Divide: Divide the point set into two halves Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  20. Divide and Conquer: Conquer Step Conquer: Recursively compute the Voronoi diagrams for the smaller point sets. Abort condition: Voronoi diagram of a single point is theentire plane. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  21. Divide and Conquer: Merge Merge the diagrams by a (monotone) sequence of edges) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  22. The Result The finished Voronoi Diagram Running time: With n given points is O(n log n) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  23. P2 P1 T DAC: Construction of the Voronoi Diagram Divide:Divide P by a vertical dividing line T into 2 equal size subsets say P1 and P2. If |P|= 1completed. Conquer:Compute VD(P1 ) and VD(P2 ) recursively. Merge: Compute the edge sequence K separating P1 and P2Cut VD(P1) and VD(P2 ) by means of K starting from VD(P1 ) and VD(P2 ) and K Theorem:If K can be computed in time O(n), then the running time of the D&C-algorithm is T(n) = O(n logn) Proof :T(n) = 2 T(n/2) + O(n),T(1) =O(1) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  24. First edge in K P1 P2 Last edge in K Computation of K 4 tangentialpointsP1 P2Observation:K is y - monotonous Incremental (sweep line) construction(p1 in P1 and p2 in P2perpendicular with m, Sweep l) Determines intersections1 of m with Vor(p1) below l Determines intersection s2 of m with Vor(p2) belowl Extend K by line segment l si Set l = si Compute new K defining pair p1,p2 Theorem: Running time O(n) Proof:Vor(pi) are convex, therefore each one‘sforward - edge are only visitedonce. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  25. Example Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  26. Observations: Intersection of the parabolas define edges New "telephones" ( ) define new parabolas Parabola intersection disappear, if C(P, q) has 3 points Fortune’s Algorithm Beach - line Sweep - line Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  27. Utilizing the Voronoi Diagram Search for nearest neighbour • Input: A fixed (static) set P of n points in the plane, a query point p • Output: A nearest neighbour of p in P Solution • Construct the Voronoi diagram for P in time O(n log n) • Solve the point location problem. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  28. Enclosing Triangle Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  29. a A b c Use (Dynamic Object Set) Search for next neighbour : Idea: Hierarchical subdivision of VD(P) Step 1 :Triangulation of final Voronoi regions Step 2 : Summary of triangles and structure of a search tree Rule of Kirkpatrick: Remove in each case points withdegree < 12, its neighbor is already far. A a c b Theorem:Using the rule of Kirkpatrick a search tree of logarithmic depth develops. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  30. Utilizing the Voronoi Diagram Search for nearest neighbour • Input: A fixed (static) set P of n points in the plane, a query point p • Output: A nearest neighbour of p in P Solution • Construct the Voronoi diagram for P in time O(n log n) • Solve the point location problem. Theorem: For a given set of n points we can construct a structure in time O(n log n) such that a nearest neighbour query can be answered in time O(log n). Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  31. Use (Static Point Set) Closest pair of points: Go through edge list for VD(P) and determine minimum All next neighbors : Go through edge list for VD(P) for all points and get next neighbors in each case Minimum Spanning tree (after Kruskal) Theorem: The MST can be computed in time O(n log n) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  32. MST (after Kruskal): Informal Algorithm Minimum Spanning Tree (after Kruskal): Construct for a graph G = (V, E) a minimum spanning tree in time O(|E| log |E|). • Each point p from P defines1-node tree; start with the forest of these 1-node trees. • If there are more than one tree T in the current forest, • 2.1) find p, p´ with p in T and p´ not in Twith d(p, p´)minimal. • 2.2) connect T containing p and T´ containing p´ (union of T and T´) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  33. Kruskal’s Algorithm Minimum Spanning Tree (after Kruskal): Manipulates a forest of trees. Find(v) returns the tree to which node v belongs. Union(v, w) merges two trees with names v and w to a new one with the name w. Make-set(v) returns the tree which has v as its only node. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  34. Formal Algorithm Minimum Spanning Tree (after Kruskal): G = (V, E). • Initialise E´ as empty set of edges. • Sort the edges in E according to increasing lengths. For allvVdoMake-set(v); For all (v, w) E in increasing length order doif Find(v) Find(w) then {Select edge (v, w):}{Union(Find(v), Find(w));E´ = E + {(v, w)}} Naïve Implementation: E is the set of all n(n-1)/2 edges (v, w) with points v and w in P. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  35. Summary Minimum Spanning Tree (after Kruskal) Crucial Observation: It is sufficient to consider the O(n) distances between (nearest) neighbours in order to find a pair of points with minimal distance! It suffices to apply Kruskal’s algorithm to the dual graph of the Voronoi diagram with only O(n) edges! Theorem: If the Voronoi diagram for a set of n points is known, the minimum spaning tree can be constructed in time O(n log n). Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

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