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AALG, lecture 10, © Simonas Šaltenis, 2007 2
Voronoi diagrams
Main goals of the lecture: to understand the concept of Voronoi
diagrams and their key properties; to understand the doubly-connected edge
list data structure used to represent space subdivisions;
to understand the main ideas of the Fortune's algorithm, an example of an advanced plane sweep algorithm
AALG, lecture 10, © Simonas Šaltenis, 2007 3
Nearest-Neighbor Queries
Nearest-neighbor query: Given a set of multi-dimensional points S and a
query point q, return a point p ∈ S, closest to q:• That is ∀ a∈S d q , pd q ,a
q
p
AALG, lecture 10, © Simonas Šaltenis, 2007 4
Nearest-Neighbor Queries
Applications: Find a closest gas station Similarity search: find an image or a song that
is most similar to a given one.• Represent each image (song) as a feature vector
(multi-dimensional point)
How can we process NN queries in R-trees, kd-trees or similar structures? Note: both structures were designed with range
queries in mind
AALG, lecture 10, © Simonas Šaltenis, 2007 5
NN Querying in R-trees
Two algortihtms can be used: Best-first:
• Uses O(n) memory in worst-case• Is optimal for a given instance of the data structure,
but runs in O(n) in worst-case
Depth-first:• Uses only O(log n) memory• Runs in O(n) in worst-case
What if we want a O(log n) worst case performance for NN queries?
AALG, lecture 10, © Simonas Šaltenis, 2007 6
Voronoi Region, Diagram
Consider a set S of objects S = {s1, s2, ..., sn}
Voronoi cell of si:
Voronoi diagram Vor(S): partition of the plane into cells V(si) Named after: Georgy Voronoi (1908), but can be
traced back to Descartes (XVII century) a.k.a Dirichlet tesselation
V si={p : d p , sid p , s j ,∀ j≠i}
AALG, lecture 10, © Simonas Šaltenis, 2007 7
Voronoi Diagram: Example
AALG, lecture 10, © Simonas Šaltenis, 2007 8
Voronoi Region: Closer Look
hij: perpendicular bisector of segment (si, sj)
Hij: half-plane delimited by hij and containing si
Hij = {p : p is closer to si than sj}
V si=∩H ij
AALG, lecture 10, © Simonas Šaltenis, 2007 9
Voronoi Diagram: Properties
Voronoi diagram is connected, its edges are segements or halflines Except for the degenerate case. Which?
Each vertex is a center of a circle going thorough three (or more) adjacent sites.
AALG, lecture 10, © Simonas Šaltenis, 2007 10
Voronoi Diagram: Size
Max size of one cell: n – 1 edges... But the total number of vertices ≤2n–5 and
the total number of edges ≤3n–6 Proof using the Euler's formula – for any planar
ambeded graph:• mv – me + mf = 2
AALG, lecture 10, © Simonas Šaltenis, 2007 11
Using Voronoi Diagrams
Nearest-neighbor queries can be answered in worst-case O(log n) time using point location data structres on top of Voronoi diagrams
NN queries using R-trees storing Voronoi cells (still O(n) in worst case).
Closest-pair in linear time: Members of the closest-pair can be found in
adjacent Voronoi cells A lot of applications in GIS, etc.
AALG, lecture 10, © Simonas Šaltenis, 2007 12
Relations to Other Concepts
Convex Hull: Sites in unbounded cells of the Voronoi diagram
define the convex hull of all the sites! (Why?)
AALG, lecture 10, © Simonas Šaltenis, 2007 13
Relations to Other Concepts
Delaunay triangulation – the dual of Voronoi diagram: Has a number of nice properties such as that the
triangles are “not thin”
AALG, lecture 10, © Simonas Šaltenis, 2007 14
Exercise
Draw a Voronoi diagram
2
3
5
4
1
AALG, lecture 10, © Simonas Šaltenis, 2007 15
Doubly-Connected Edge List
Data structure for Voronoi diagrams:• Each edge is represented as two half-edges – for both
directions • Vertex entry:
• Coordinates• IncidentEdge – pointer to an arbitrary half-edge that has this
vertex as an origin
• Face (cell) entry:• Boundary – pointer to some half-edge on its boundary
• Half-edge entry:• Origin – pointer to an origin vertex• Twin – pointer to a corresponding oposite half-edge• IncidentFace – pointer to the bounded face (cell) to the left• Next – pointer to the next half-dege on the boundary• Prev – pointer to the previous half-edge on the boundary
• The whole diagram has to be put into a bounding box...
AALG, lecture 10, © Simonas Šaltenis, 2007 16
Computing Voronoi Diagrams
How do we compute a Voronoi Diagram? Naïve: compute each voronoi cell by doing a
bunch of half-plane intersections• Running time?
Divide-and-conquer• O(n log n)
Fortune's algorithm, a plane sweep algorithm• O(n log n)
AALG, lecture 10, © Simonas Šaltenis, 2007 17
Fortune's Algorithm
Sweeps the line from top to bottom What part of the Voronoi diagram can we be
sure of when the sweepline is at a specific position? Above the beach line:
• An arc Ai is defined by a visited site si:
• For any point a on Ai: d(a,si) = d(a,l) where l is the sweepline
• A breakpoint between two arcs traces an edge of a voronoi diagram
AALG, lecture 10, © Simonas Šaltenis, 2007 18
Site Events
What are the events? When does the combinatorial structure of the beach line change? When does a new arc appear? Only when a new site is reached: site event
AALG, lecture 10, © Simonas Šaltenis, 2007 19
Site Events
There can be at most 2n–1 arcs in the beachline (Why?)
Actions on a site event: Update a beachline (one new arc, two new
break-points) Add an edge to the Voronoi diagram (not
connected to anything yet...)
AALG, lecture 10, © Simonas Šaltenis, 2007 20
Circle Events
When does an arc disappear from the beachline? When it's length becomes zero: a circle event For each new triplet of adjacent arcs we have to
compute when such event will take place and insert it into the event queue
AALG, lecture 10, © Simonas Šaltenis, 2007 21
Circle Events
Actions on Circle Events: Update a beachline (remove an arc)
• Check for new triplets of arcs
Add a vertex, connect two edges corresponding to the two merging breakpoints to this vertex and add an edge to the Voronoi diagram• The only time when a vertex is added to the Voronoi
diagram
AALG, lecture 10, © Simonas Šaltenis, 2007 22
Circle Events
Does every new triplet result in a definite circle event? Diverging breakpoints – no circle event is
generated
False alerts – the event is inserted into the queue but is removed before being processed
AALG, lecture 10, © Simonas Šaltenis, 2007 23
Data Structures
How do we represent the beach line?
s1
s2
s3
s1 s2 s3 s2
s1,s2 s3,s2
s2,s3 Balanced binary search tree T Leaf node represents an arc:
• Site generating an arc• Pointer to a future circle event in
the event queue or nil
Internal node represents a brake point:• Pair of points• Pointer to a half-edge being
traced out in the Voronoi diagram
AALG, lecture 10, © Simonas Šaltenis, 2007 24
Event Queue
All site events are inserted when the algorithm starts
All new triples of arcs are investigated when processing any event, to look for circle events. They are inserted into the queue
Circle event: The lowest point of the circle Pointer to a corresponding arc (leaf node in T)
AALG, lecture 10, © Simonas Šaltenis, 2007 25
Running Time, Space
Running time: Processing each event: O(log n) Number of events:
• Site events: n• Processed circle events: at most 2n – 5
Total time O(n log n)
Space (beach line + queue): O(n)
AALG, lecture 10, © Simonas Šaltenis, 2007 26
Exercise
Show event queue and beach line tree
Mark edges and vertices of the Voronoi diagram that have been created
What happens, when site event (5) is processed?
2
3
5
4
1
