Voronoi Diagrams. 7.0 Voronoi Diagrams for points. Libraries for Computational Geometry. D7013E Lecture 7

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1 Fortune s Algorithm D7013E Lecture 7 Voronoi Diagrams Voronoi Diagrams Plane-sweep Proximity problems Voronoi Diagrams for points Input: A set of n points P = {p 1, p 2,, p n } called sites Output: A subdivision Vor(P) of the plane into n cells such that: 1. Each cell contains exactly one (1) site 2. The cell V( ) of contains all points closer to than any other site Boundaries of cells consists of points equidistant from two or more sites V( ) Libraries for Computational Geometry LEDA Library of Efficient Data Structures C++ CGAL Computational Geometry Algorithms Library C++ CORE They state: In fact, simplest usage is to take a standard C/C++ program, and insert this 2-line preamble: #define Level <level_number> #include "CORE.h where <level_number> sets the desired accuracy. 3 4

2 Some definitions We use Euclidean distance in the plane Slight abuse of notation: Sometimes Voronoi diagram will refer to the edges and vertices that bound the cells rather than the cells themselves The bisector of a line segment: Basic Properties Observation 7.1: Each cell V( ) is the intersection of n-1 halfplanes defined by bisectors So, at most n-1 vertices and n-1 edges Cells are open sets Separate from edges and vertices Edges are open line segments Separate from vertices (the end points of the edges) 5 6 Basic Properties Basic Properties Theorem 7.2: If all sites are collinear, Vor(P) consists of n-1 parallel lines Argue using bisectors If not, Vor(P) is connected and its edges are either line segments or half-lines Proof by contradiction Theorem 7.3: For n 3, the number of vertices of a Vor(P) is at most 2n-5 and the number of edges is at most 3n-6 Vor(P) is a planar graph, if we add a virtual vertex to which all unbounded lines are connected Euler s formula Note: A single cell could have O(n) vertices By Theorem 7.3, there are just a constant number of such cells 7 8

3 Basic Properties Computing a Voronoi Diagram Vertices of Vor(P) are intersections between bisectors The number of bisectors is O(n 2 ) But Vor(P) has only linear size..? The largest empty circle C P (q) of q with respect to P is the largest circle centered at q that does not contain any point of P Theorem 7.4: a) The point q is a vertex of Vor(P) if and only if C P (q) contains 3 or more sites on its boundary b) The bisector L between and define an edge of Vor(P) if and only if there is a point q on L such that C P (q) has and on its boundary but no other sites 9 A first solution: For each site : For all other sites, where j i: Compute the bisector L i of [, ]» These bisectors (lines) bound half-planes Compute the intersection of the half-planes of all L i This is V( ) (Connect all V(pi) into Vor(P)) Computing n intersections of n-1 half-planes takes O(n 2 log n) time as explained in Chapter 4 Can we do better? 10 A lower bound Fortune s Algorithm Well, computing a Voronoi Diagram must take Ω(n log n) time because it can be reduced to the problem of sorting: Map all numbers x i to points (x i, 0), compute the Voronoi diagram (see below), and find the minimum number x min (x min, 0) lies in the left-most vertical Voronoi cell ((x 2, 0) below) The rest of the numbers can now be extracted one after another from left to right by observing that all vertical Voronoi edges lie exactly in the middle between the points (x i, 0) x 2 x 4 x 7 x 1 x 9 x 10 x 3 x 5 x 6 x 11 x 8 and there is a better algorithm Steve Fortune published an optimal algorithm in 1987 Works in Θ(n log n) time and O(n) space Based on plane sweep However, slightly more complicated than what we have seen so far The sweep line is a line as before The status, however, is not the same A concatenation of parabolas above the sweep line Called a beach line There are two kinds of event points where the sweep line stops: Site events, and Circle events 11 12

4 Demo The Beach Line Acts as our status information The edges of Vor(P) are traced out by the breaks in the beach line as the sweep is carried out When the sweep line moves, the beach line will also move/change Invariant: Above the beach line, the Voronoi diagram has been computed Note: Not everything above the sweep line(!) The Beach Line The Voronoi diagram of a point and a line is a parabola A beach line is the lower envelope of the Voronoi diagrams of each site above the sweep line and the sweep line The point-wise minimum of all the parabolas An event happens when an arc [of a parabola] appears on, or disappears from, the beach line Site Events Creating Voronoi edges Arcs appear in the beach line at site events, which is when the sweep line hits a site: The break has traced out a line segment, a part of the Voronoi edge between V( ) and V( ) V( ) V( ) 15 16

5 Site Events Lemma 7.6: The only way in which a new arc can appear on the beach line is through a site event Essentially, the further above the sweep line a site lies, the wider the parabola Corollary: There are at most 2n-1 arcs in the beach line Each new arc might split at most one old arc Since Voronoi edges are traced out by break points, and break points are introduced when arcs are inserted, site events are where new Voronoi edges are starting to take form starting with the second site event (we need at least two arcs to have a break point) Circle events Creating Voronoi vertices An arc disappears at a circle event A circle event is the lowest point on the circle through three sites defining consecutive arcs in the beach line Two break points meet <=> two Voronoi edges are joined by a Voronoi vertex Representation The Voronoi diagram under construction: A DCEL with a bounding box large enough to hold the final Voronoi diagram Representation The beach line: A balanced binary search tree T in which leaves correspond to arcs (ordered by x- coordinate), and (internal) nodes correspond to breakpoints. Takes O(log n) time to find the arc above a new site Pointers from leaves into the event queue (to the circle events that removes the arcs, if they exist) nodes into the DCEL (the edge being traced out by the breakpoint) The event queue: Events ordered by y-coordinate Circle events are represented by the lowest point on the circle Pointers from circle events to their leaves in T 19 20

6 False alarms (false circle events) Complexity Circle events are added immediately when detected during the sweep Two concerns: Not all three points defining consecutive arcs in the beach line give rise to a circle event Just skit A circle event might not take place; it must then be cancelled ahead of time A new arc splits the three consecutive arcs (a site event occurs) so that the triple disappears (Note that this might introduce other, new, circle events) Remove it by using the pointers between the circle event in the event queue and the leaf in T of the arc hit by the new arc There are two special cases to consider: Two or more events on the same y- coordinate Example: Four or more sites on a circle Would be several coinciding circle events Use zero-length Voronoi edges during the computation; filter them away afterwards A site event occurs exactly under a break point Also here, a zero-length edge is created and everything works fine Theorem 7.10: The Voronoi diagram of n points can be computed in O(n log n) time and O(n) storage Voronoi diagrams of line segments If the sites are not points but objects, we use as metric the distance to the closest point on the object The Voronoi diagram for disjoint line segments consists of line segments and parabolas pend pstart A problem If line segments share end points, some bisectors become strange. They are no longer curves but regions The remedy here is to move line segments a (very) small distance away from each other so they do not intersect at all We just consider them moved 23 24

7 An algorithm An algorithm It turns out essentially the same plane sweep still works, with some adjustments However, the beach line now contains both parabolas and line segments s 1 s 2 s 3 s 4 s 5 Figure 7.5 The beach line for a set of line segment sites. The breakpoints trace the dashed arcs, which include the Voronoi edges l The input segments act as sites End points of these are site event points There are 5 kinds of circle event points The beach line can be maintained in the spirit of how we first did it but, of course, involving more cases Read on your own s1 s2 Figure 7.5 The beach line for a set of line segment sites. The breakpoints trace the dashed arcs, which include the Voronoi edges s3 s4 s5 l Result 7.4 Furthest-point Voronoi diagrams The number of events turns out to be O(n) Theorem 7.11: The Voronoi diagram of a set of n disjoint line segments can be computed in O(n log n) time using O(n) storage Application: Motion planning compute an obstacle avoiding path for a circle Move along the Voronoi diagram Remove edges at too narrow passages Do a depth-first search So, the path can be computed in O(n log n) time The cell of a site consists of those points that lie further away from than any other site The intersection of the other sides of the bisectors Vor fp (P) Observation 7.13: Only sites on the convex hull of the sites have cells Can be computed in O(n log n) time O(n) time if the points are all in convex position Bonus: We can get the convex hull in O(n) time from Vor fp (P) cell of cell of 27 28

8 7.4 Furthest-point Voronoi diagrams Application: Roundness of manufactured goods via surface sampling Measured by computing a smallestwidth annulus of the sampled points P Using both the Vor(P) and the Vor fp (P) Theorem 7.15: A smallestwidth annulus can be computed in O(n 2 ) time using O(n) space 29