# Proximity problems. Voronoi diagrams. Properties - UPC

Proximity problems. Voronoi diagrams. Properties - UPC

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCLOSEST PAIRFind the pair of points closest to each other

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCLOSEST PAIRFind the pair of points closest to each other• Brute force solution:O(n 2 ) timecheck every pair in• This procedure does not exploit the metricproperties• In dimension 1, it can be solved in O(n log n)time by sorting the input first and using thefact that the solution must be a pair of consecutivepointsAPPLICATIONAirplanes in danger of collision

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCAll NEAREST NEIGHBORSBuild the directed graph where the existance ofan (oriented) edge −−→ p i p j means that p j is thepoint of P closest to p i

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCAll NEAREST NEIGHBORSBuild the directed graph where the existance ofan (oriented) edge −−→ p i p j means that p j is thepoint of P closest to p i• Brute force solution:O(n 2 ) timecheck every pair in• This process does not exploit the metric properties• In dimension 2, at most 6 points of P havep i as their closest point• In dimension 1 it can be solved in O(n log n)time (same as before)APPLICATIONIn Ecology, to study the territoriality of species

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCEUCLIDEAN MINIMUM SPANNING TREEBuild the tree connecting all points of P andminimizing the sum of the lenghts of its edges

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCEUCLIDEAN MINIMUM SPANNING TREEBuild the tree connecting all points of P andminimizing the sum of the lenghts of its edges• Applying Prim’s or Kruskal’s algorithms tothe complete graph of P , with euclideanweights: O(e log e) = O(n 2 log n) time• Applying certain algorithms to the completeweighted graph: O(e) = O(n 2 ) time• These procedures do not exploit the metricproperties• In dimension 1 it can be done in O(n log n)time (same as before)APPLICATIONConnections in general, telephone prices in the US

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMIN-MAX FACILITY LOCATION:MINIMUM SPANNING CIRCLEFind the point x on the plane achievingmin max d(x, p i)x∈R 2 p i ∈PGeometrically: find the center of the circleof minimum radius enclosing P

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMIN-MAX FACILITY LOCATION:MINIMUM SPANNING CIRCLEFind the point x on the plane achievingmin max d(x, p i)x∈R 2 p i ∈PGeometrically: find the center of the circleof minimum radius enclosing P• Brute force solution: consider everyset of three points, in O(n 4 ) time• In dimension 1 it can be solved inO(n) time: the solution is the midpointof min(P ) and max(P )APPLICATIONLocation of emergency services (hospitals, fire departments,...), of radio and tv repeaters, antennasfor mobile phones,...

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMAX-MIN FACILITY LOCATION:MAXIMUM EMPTY CIRCLEFind the point x in the region A achievingmax min d(x, p i)x∈A p i ∈PGeometrically: find in A the center of circle ofmaximum radius not enclosing any point of P

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMAX-MIN FACILITY LOCATION:MAXIMUM EMPTY CIRCLEFind the point x in the region A achievingmax min d(x, p i)x∈A p i ∈PGeometrically: find in A the center of circle ofmaximum radius not enclosing any point of P

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMAX-MIN FACILITY LOCATION:MAXIMUM EMPTY CIRCLEFind the point x in the region A achievingmax min d(x, p i)x∈A p i ∈PGeometrically: find in A the center of circle ofmaximum radius not enclosing any point of P• Brute force solution: O(n 4 ) time (as before)• In dimension 1 it can be solved in O(n log n)time by sorting the input first and using thefact the solution is the mid-point of the maximumgap of PAPPLICATIONLocation of undesired services (polluting industry,landfills, ...) or new shopping malls in competencewith preexistent centers

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCNEAREST NEIGHBOR QUERYGiven a new point q, quickly find its closestsite p i

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCNEAREST NEIGHBOR QUERYGiven a new point q, quickly find its closestsite p i

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCNEAREST NEIGHBOR QUERYGiven a new point q, quickly find its closestsite p i• Brute force: compare q with every site p i ,in O(n) time• Preprocessing P in order to speed up thequeries is worth the effort when the queryis to be repeatedly called• In dimension 1, first sort the points of P inO(n log n) time and then each query will beaswered in O(log n) time by binary searchAPPLICATIONThe post office problem, pattern (voice) recognitionand, in general, all classifications

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCSOLUTIONVORONOI DIAGRAM• Structure capturing the proximity information• Decomposes the plane into regions,each one is associated by proximityto one of the points of P• Given two points, the boundary separatingthe portion of the plane closerto one point than to the other isthe perpendicular bisector of the segmentconnecting the two points

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCSOLUTIONVORONOI DIAGRAM• Structure capturing the proximity information• Decomposes the plane into regions,each one is associated by proximityto one of the points of P• Given two points, the boundary separatingthe portion of the plane closerto one point than to the other isthe perpendicular bisector of the segmentconnecting the two points

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCSOLUTIONVORONOI DIAGRAM• Structure capturing the proximity information• Decomposes the plane into regions,each one is associated by proximityto one of the points of P• Given two points, the boundary separatingthe portion of the plane closerto one point than to the other isthe perpendicular bisector of the segmentconnecting the two points

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCSOLUTIONVORONOI DIAGRAM• Structure capturing the proximity information• Decomposes the plane into regions,each one is associated by proximityto one of the points of P• Given two points, the boundary separatingthe portion of the plane closerto one point than to the other isthe perpendicular bisector of the segmentconnecting the two points

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCHistory of the rediscoveries of the Voronoi diagram• Descartes: distribution of matter in the solar system and surroundings (1644).• Dirichlet en dim 2 i 3, i Voronoi en dim d. Estudi de formes quadràtiques definides positives.Tesel . lació de Dirichlet (1850), Diagrama de Voronoi (1908).• Cristall . llografia: distribució regular dels punts Wirkung Sbereich, domini d’acció, àrea d’activitat,àrea d’influència (finals del S XIX).• Meteorologia: estimació de promitjos de pluges a regions, a partir de dades puntuals,poĺıgons de Thiessen (1911).• Geologia: estimació de la presència de mineral d’or en un sediment, Copy from a partir the de original catesbook puntuals, ofpoĺıgons d’àrea d’influencia (1909).René Descartes Le Monde, ouTraité de la Lumière (1644), reproducedregionsin the edition de Wigner-Seitz of M. S.• Fisica i química: estudi de propietats quimiques de certs elements,(1933); modelatge d’estructures d’aleacions, domini d’un àtom Mahoney, (1958). 1979.• Ecologia: supervivència d’organismes en competència per aliments o llum, com arbres alsboscos, àrea potencialment disponible (1965); plantes, poĺıgonsQuoteddeinplantes A. Okabe,(1966).B. Boots, K.Sugihara, S. N. Chiu, Spatial Tessellations,2nd ed., J. Wiley & Sons,2000.• Medicina: regions dels capilars que irriguen el teixit muscular, dominis de capilaritat (1985).

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCHistory of the rediscoveries of the Voronoi diagram• Descartes: distribution of matter in the solar system and surroundings (1644).• Dirichlet in dim 2 and 3, and Voronoi in dim d. Study of positive definite quadratic forms.Dirichlet tessellation (1850), Voronoi Diagram (1908).• Crystallography: regular distribution of the points Wirkung Sbereich, action domain, activityarea, influence area (end of the XIX century).• Meteorology: rainfall average estimation based on local data, Thiessen’s polygons (1911).• Geology: estimation of the presence of gold in a sediment, based on trial pits, influence areapolygons (1909).• Physics and Chemistry: study of the chemical properties of some elements, Wigner-Seitz’sregions(1933); alloy structure modelling, atom domain (1958).• Ecology: survival of organisms in competition for nourishment or light, like trees in thewoods, area potentially available (1965), plants polygons (1966)• Medicine: region of muscle tissue supplied by a capillary, capillarity domains (1985)

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCHistory of the rediscoveries of the Voronoi diagram (continued)• Economics: market areas of centers in competition, under various conditions of market pricesand transportation costs (since the mid XIX century)• Archaeology: study of the diffusion of the use of tools in order to analyze the influence ofthe civilization centers (sixties-seventies)modelling territorial human organization under several aspects (sixties-• Anthropology:seventies)

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMore recent applications of the Voronoi diagramIreland counties. On the left, theoretical partitioncreated with the Voronoi diagram ofthe counties capitals. On the right, the realcounties.K. R. Cox, J. A. Agnew, The spatial correspondenceof theoretical partitions, Discussion paper, SyracuseUniversity,Dept. of Geography, 1976. Quoted inA. Okabe, B. Boots, K. Sugihara, S. N. Chiu, SpatialTessellations, 2nd ed., J. Wiley & Sons, 2000.

PROXIMITYMore recent applications of the Voronoi diagramThe school districts at Tsukuba. Left, realdistricts and the location of the schools (circles).Right, relocation of the schools (squares)so to minimize the area of the studentsthat do not attend their closest school.Computational Geometry, Facultat d’Informàtica de Barcelona, UPCA. Suzuki, M. Iri, Aproximation on a tessellation ofthe plane by a Voronoi diagram,journal of the OperationResearch Society of Japan, 29, 1986. Quoted inA. Okabe, B. Boots, K. Sugihara, S. N. Chiu, SpatialTessellations, 2nd ed., J. Wiley & Sons, 2000.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMore recent applications of the Voronoi diagramInterpolation of discrete sample valuesA. Okabe, B. Boots, K. Sugihara, S. N. Chiu,Spatial Tessellations, 2nd ed., J. Wiley & Sons,2000.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMore recent applications of the Voronoi diagramInterpolation of discrete sample valuesA. Okabe, B. Boots, K. Sugihara, S. N. Chiu,Spatial Tessellations, 2nd ed., J. Wiley & Sons,2000.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCMore recent applications of the Voronoi diagramInterpolation of discrete sample valuesA. Okabe, B. Boots, K. Sugihara, S. N. Chiu,Spatial Tessellations, 2nd ed., J. Wiley & Sons,2000.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCConstructing the Voronoi diagramSixties• Application concepts clear• Lack of tools to build the diagramsSeventies• Start of the development of algorithm to compute the diagrams,thanks to computer science• Discovery of new propierties associated to the diagrams

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCDEFINITIONLet P = {p 1 , . . . , p n } be a finiteset of points in the plane.The Voronoi diagram of P isthe decomposition of the planeV (P ) = {V (p 1 ), . . . , V (p n )}such thatV (p i ) = {x | d(x, p i ) ≤ d(x, p j ) ∀ j ≠ i}.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCDEFINITIONLet P = {p 1 , . . . , p n } be a finiteset of points in the plane.The Voronoi diagram of P isthe decomposition of the planeV (P ) = {V (p 1 ), . . . , V (p n )}such thatV (p i ) = {x | d(x, p i ) ≤ d(x, p j ) ∀ j ≠ i}.CHARACTERIZATIONIf b ij is the perpendicular bisectorof the segment p i p j , andH ij is the halfplane defined byb ij enclosing p i , thenV (p i ) = ⋂ j≠iH ij .

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCDEFINITIONLet P = {p 1 , . . . , p n } be a finiteset of points in the plane.The Voronoi diagram of P isthe decomposition of the planeV (P ) = {V (p 1 ), . . . , V (p n )}such thatV (p i ) = {x | d(x, p i ) ≤ d(x, p j ) ∀ j ≠ i}.Voronoi edgeVoronoi vertexCHARACTERIZATIONIf b ij is the perpendicular bisectorof the segment p i p j , andH ij is the halfplane defined byb ij enclosing p i , thenV (p i ) = ⋂ j≠iH ij .Voronoi region

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROPERTIES1. V (p i ) is a convex polygonal region.2. V (p i ) is bounded if and only if p i lies inthe interior of ch(P ).3. Voronoi edges can be:• lines, if the points of P are allaligned;• half-lines, if the two points determiningthe edge are consecutivevertices of ch(P );• segments, if at least one ofthe points lies in the interiorof ch(P ).

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROPERTIES (continued)4. The planarity of the Voronoi diagramallows applying Euler’s formula to thegraph adding a vertex at infinity:v + n = e + 1.Since each vertex is incident to at least3 edges, and each edge has exactly 2endpoints,from where we get:2e ≥ 3(v + 1),v ≤ 2n − 5, e ≤ 3n − 6.Thus, the complexity of V (P ) is O(n).5. Although each V (p i ) can have n − 1edges, on average they have 6.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHE DUAL GRAPHProximity between points: two points of Pare neighbors if they share an edge in theVoronoi diagram of P .A proximity graph is obtained by connectingeach point to its Voronoi neighbors. Thisgraph is called Delaunay graph.The Delaunay graph is the rectiliniear dualof the Voronoi diagram.In general, it is a triangulation of P , althoughsometimes it is not:• when the points are aligned,• when 4 or more points are concyclic.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.TheoremA point q is a Voronoi vertex iff C m (q)has three or more points of P in its boundary.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.TheoremA point q is a Voronoi vertex iff C m (q)has three or more points of P in its boundary.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.TheoremA point q is a Voronoi vertex iff C m (q)has three or more points of P in its boundary.The bisector b ij of two points p i and p jdefines a Voronoi edge iff there exists apoint q of the plane such that C m (q) hasp i and p j and no other point of P in itsboundary. In such case, the Voronoi edgeis the set of all these points q.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.TheoremA point q is a Voronoi vertex iff C m (q)has three or more points of P in its boundary.The bisector b ij of two points p i and p jdefines a Voronoi edge iff there exists apoint q of the plane such that C m (q) hasp i and p j and no other point of P in itsboundary. In such case, the Voronoi edgeis the set of all these points q.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCPROXIMITY AND EMPTY CIRCLESFor any point q in the plane, let C m (q) bethe largest cirgle, centered at q, which doesnot contain any point of P in its interior.TheoremA point q is a Voronoi vertex iff C m (q)has three or more points of P in its boundary.The bisector b ij of two points p i and p jdefines a Voronoi edge iff there exists apoint q of the plane such that C m (q) hasp i and p j and no other point of P in itsboundary. In such case, the Voronoi edgeis the set of all these points q.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 1Closest neighbors digraphFor each point p i ∈ P , the closest point of Pis a Voronoi neighbor. Therefore, the closestneighbors digraph is a subgraph of the Delaunaygraph.Application 2The closest pairI particular, the shortest edge p i p j belongs tothe Delaunay graph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.Prim’s algorithm to compute the minimumspanning tree in a weighted graph builds thetree T by adding, at each iteration, the edgeof minimum weight incident to T which doesnot close a cycle when added to T .

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.Prim’s algorithm to compute the minimumspanning tree in a weighted graph builds thetree T by adding, at each iteration, the edgeof minimum weight incident to T which doesnot close a cycle when added to T .

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.Prim’s algorithm to compute the minimumspanning tree in a weighted graph builds thetree T by adding, at each iteration, the edgeof minimum weight incident to T which doesnot close a cycle when added to T .

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCCorollary (blue and red)Given any partition of P into two classes (bluepoints and red points), the shortest red-bluesegment is a Delaunay edge, because the circlehaving it as diameter is empty.Application 3Euclidean minimum spanning treeThe euclidean minimum spanning tree is a subgraphof the Delaunay graph.Prim’s algorithm to compute the minimumspanning tree in a weighted graph builds thetree T by adding, at each iteration, the edgeof minimum weight incident to T which doesnot close a cycle when added to T .At each step, if we consider the vertices of Tto be red, and the remainig ones to be blue,the added edge belongs to the Delaunay graph.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplication 4Finding the closest siteGiven any point q in the plane, findingits closest site p i ∈ P means detectingthe Voronoi region V (p i ) containing q.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplication 4Finding the closest siteGiven any point q in the plane, findingits closest site p i ∈ P means detectingthe Voronoi region V (p i ) containing q.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplicationMax-min facility locationThe center of the larger empty circle, restrictedto a given region A, can be locatedat:• a Voronoi vertex,• the intersection of a Voronoi edgewith the boundary of A.• a vertex of A.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplication 6Min-max facility locationThe center of the minimum spanningcircle does not seem tobe related with the Voronoi diagram!How could it be?

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCApplication 6Min-max facility locationThe center of the minimum spanningcircle does not seem tobe related with the Voronoi diagram!How strange!

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.segmentsdisks.barriers

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.segmentsdisks.barriers

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.segmentsdisks.barriersAPPLICATIONS- How do car emissions affect the treesalong the highways.- Rivers and mountains influence in determiningschool districts.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension d

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension d

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension d

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension d

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension d

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.spherecylinder.dimension dAPPLICATIONS- Sphere: study of global Earth problems- Cilindre: study of problems with cyclicperiodicity in time

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetrics

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetrics

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetricsAPPLICATIONMassive storage system whitha reading/writing head basedon the Manhattan or the maximummetrics

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetricsNumbers indicate multiplicative weights

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetricsNumbers indicateadditive weights

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.othermetricsAPPLICATIONS- Population of each city- Amount of emissions of some polluting product- Size of each atom in a crystalline structure- Combination of transportation prices and costsNumbers indicateadditive weights

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.214568k37

Computational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.kPROXIMITY(1, 2)21(1, 3)(2, 4)(4, 5)(2, 5)(1, 5)(3, 5)345(4, 6)(5, 6)6(5, 7)(6, 7)7(4, 8)(6, 8)8(7, 8)(3, 7)

Computational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.kPROXIMITY(1, 2)APPLICATIONS- Finding the k-th closest emergency service- Spatial interpolation in statistics estimations21(1, 3)(2, 4)(4, 5)(2, 5)(1, 5)(3, 5)345(4, 6)(5, 6)6(5, 7)(3, 7)(6, 7)7(4, 8)(6, 8)8(7, 8)

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.kThe Voronoi diagram of order k = n − 1is the Furthest Point Voronoi diagram.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCGENERALIZATIONS OFTHE VORONOI DIAGRAMThe Voronoi diagram ofa finite set P of points inthe plane is a partition ofthe plane such that eachregion is the locus of thepoints that lie closer toone element of P thanto the remaining ones.kThe Voronoi diagram of order k = n − 1is the Furthest Point Voronoi diagram.It solves the minimum spanning circle problem!

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCVORONOI DIAGRAM AND INTERSECTION OF HALFSPACES IN 3DConsider the point set P as being embedded in the plane z = 0.Consider the paraboloid z = x 2 + y 2 .For each point p i ∈ P let p ∗ i be its vertical projection of p i onto the paraboloid, i.e.:if p i = (a i , b i , 0), then p ∗ i = (a i , b i , a 2 i + b 2 i ).For each point p ∗ i consider the plane which is tangent to the paraboloid at p∗ i .

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCVORONOI DIAGRAM AND INTERSECTION OF HALFSPACES IN 3DConsider the point set P as being embedded in the plane z = 0.Consider the paraboloid z = x 2 + y 2 .For each point p i ∈ P let p ∗ i be its vertical projection of p i onto the paraboloid, i.e.:if p i = (a i , b i , 0), then p ∗ i = (a i , b i , a 2 i + b 2 i ).For each point p ∗ i consider the plane which is tangent to the paraboloid at p∗ i .The (farthest point) Voronoi diagram of P is the orthogonalprojection onto the plane z = 0 of the polyhedralconvex region obtained when intersecting the upper(lower) halfspaces defined by these planes.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCThe (farthest point) Voronoi diagram of P is the orthogonalprojection onto the plane z = 0 of the polyhedralconvex region obtained when intersecting the upper(lower) halfspaces defined by these planes.

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTHREE BIG QUESTIONS1. How to build the Voronoi diagramof P ?2. How to store the Voronoi diagramof P ?3. How to use the Voronoi diagramto improve the solutions of theproximity problems?

PROXIMITYComputational Geometry, Facultat d’Informàtica de Barcelona, UPCTWO ADDRESSES TO PLAYWITH VORONOI DIAGRAMShttp://www.pi6.fernuni-hagen.de/GeomLab/VoroGlide/http://www.dma.fi.upm.es/docencia/segundociclo/geomcomp/voronoi.htmlAND A BOOK WITH MUCH MORE INFORMATIONA. Okabe, B. Boots, K. Sugihara, S. N. ChiuSpatial Tessellations2nd ed., J. Wiley & Sons, 2000.

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