Geometric Data Structures

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Geometric Data Structures

• Dr. M. Gavrilova

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Lecture Plan

• Voronoi diagrams
• Trees and grid variants

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Part 1

• Voronoi diagram
• Delaunay triangulation

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Voronoi diagrams

• VD Thiessen polygons, Delaunay triangulations,
  tessellations

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Tessellation

• Tessellation is often obtained using Delaunay
  triangulation.

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Voronoi diagram

• Given a set of N sites (points) in the plane or a
  3D space
• Distance function d(x,P) between point x and site
  P is defined according to some metric
• Voronoi region Vor(P) is the set of all points
  which are closer to P than to any other site
• Voronoi diagram is the union of all Voronoi
  regions

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Voronoi Diagram
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Voronoi diagram
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Voronoi diagram
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Voronoi diagram properties

• Assumption 1. No four points from the set S are
  cocircular.
• Property 1. Voronoi vertex is the intersection of
  3 Voronoi edges and a common point of 3 Voronoi
  regions
• Property 2. Voronoi vertex is equidistant from 3
  sites. It lies in the center of a circle
  inscribed between 3 cites
• Property 3. Empty circle property This inscribed
  circle is empty, i.e. it does not contain any
  other sites
• Property 4. Nearest-neighbor property If Q is the
  nearest neighbor of P then their Voronoi regions
  share an edge (to find a nearest neighbor it is
  sufficient to check only neighbors in the VD)
• Property 5. Voronoi region Vor(P) is unbounded if
  and only if P belongs to the boundary of convex
  hull of S.

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VD properties
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VD properties
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VD properties
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VD properties
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VD properties
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Delaunay triangulation

• Definition 3. A Delaunay triangulation (DT) is
  the straight-line dual of the Voronoi diagram
  obtained by joining all pairs of sites whose
  Voronoi regions share a common Voronoi edge
  Delaunay 34.
• Follows from the definition
• If two Voronoi regions Vor(P) and Vor(Q) share an
  edge, then sites P and Q are connected by an edge
  in the Delaunay triangulation
• If a Voronoi vertex belongs to Vor(P), Vor(Q) and
  Vor(R), then DT contains a triangle (P,Q,R)

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Delaunay triangulation
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DT properties

• Assumption 2. No three points from the set S lie
  on the same straight line.
• Theorem. The straight-line dual of the Voronoi
  diagram is a triangulation of S Preparata and
  Shamos 85.
• Property 5. The circumcircle of any Delaunay
  triangle does not contain any points of S in its
  interior Lawson 77.
• Property 6. If each triangle of a triangulation
  of the convex hull of S satisfies the empty
  circle property, then this triangulation is the
  Delaunay triangulation of Lawson 77.

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VD and DT
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DT properties

• The edge of the quadrilateral satisfies the
  local min-max criterion if the following equation
  holds
• A triangulation satisfies the global min-max
  criterion if every internal edge of a convex
  quadrilateral in the triangulation satisfies the
  local min-max criterion.
• Property 8. The Delaunay triangulation satisfies
  the global min-max criterion Lawson 77.
• Property 9. If a triangulation of the convex hull
  of satisfies the global min-max criterion then
  it is the Delaunay triangulation of Lawson 77.
  

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VD and DT

• Both DT and VD effectively represent the
  proximity information for the set of sites. They
  can be easily transformed into each other.
• VD contains geometrical information, while DT
  contains topological information.

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Generalized Voronoi diagram

• Given a set S of n sites (spheres) in
  d-dimensional space
• Distance function d(x,P) between a point x and a
  site P is defined.

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Generalized Voronoi diagram

• A generalized Voronoi diagram (GVD) for a set of
  objects in space is
• the set of generalized Voronoi regions
• where d(x,P) is a distance function between a
  point x and a site P in the
• d-dimensional space.

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Generalized Delaunay tessellation

• A generalized Delaunay triangulation (GDT) is
  the dual of the generalized Voronoi diagram
  obtained by joining all pairs of sites whose
  Voronoi regions share a common Voronoi edge.

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General metrics

• Generalized distance functions
• Power
• Additively weighted
• Euclidean
• Manhattan
• supremum

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Power and Euclidean Voronoi diagrams
Euclidean bisector
Power bisector
Power diagram and Delaunay triangulation
Euclidean diagram and Delaunay triangulation
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Manhattan and Supremum VD
Manhattan bisectors
Supremum bisectors
Manhattan diagram and Delaunay triangulation
Supremum diagram and Delaunay triangulation
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Properties of Generalized VD and DT

• The vertex of generalized Voronoi diagram is a
  center of a sphere inscribed between d 1 Voronoi
  sites (spheres).
• The inscribed sphere is empty, i.e. it does not
  contain any other sites.
• One of the facets of the generalized Voronoi
  region Vor(P) defines a nearest-neighbor of P.
• A Voronoi region Vor(P) is unlimited if and only
  if site P belongs to the convex hull of S.
• The sphere inscribed between the sites comprising
  a simplex of generalized Delaunay tessellation is
  an empty sphere.
• The power Delaunay tessellation of a set of
  spheres S is a tetrahedrization.

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Algorithmic Strategies

• Incremental
• Divide-and-conquer
• Sweep-line/plane
• Dimension reduction
• Geometric decomposition

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Algorithmic Strategies

• Algorithm development techniques on example of
  Voronoi Diagrams
• Incremental construction
• Divide and conquer
• Sweep-plane
• Geometric transformations
• Dynamic data structures

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Incremental construction
INCIRCLE condition
power
Polynomial of 4th order in the plane
Euclidean
Polynomial of 8th order in the plane
System of linear equations and inequalities in
the plane
Supremum
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Incremental construction

• Incremental method outline
• Insert new site P
• Perform swap operations on quadrilaterals where
  the empty-sphere condition is not satisfied
• Method complexity is O(n2) in the plane.

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Sweep-plane algorithm

• Algorithm description
• Throw pebbles in the water
• Intersection of waves gives the Voronoi diagram
• Add time as 3rd dimension waves transform to
  pyramids.
• Sweep pyramids with the sweep-plane to get the
  Voronoi diagram

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Sweep-plane algorithm

• Properties
• The complexity of the sweep-plane algorithm in
  generalized Manhattan metric is O(n log n).
• Sweep-plane method is not applicable to the power
  diagram construction.

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Sweep-plane algorithm

• An example of a sweep-plane construction of a
  Voronoi diagram in L1 metric

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Sweep-plane algorithm
Cones in L1
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Sweep-plane algorithm
Sweeping the cones
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Sweep-plane algorithm
sweep direction
Parabolas
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Sweep-plane algorithm
Site event
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Sweep-plane algorithm
Site event
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Sweep-plane algorithm
two half-bisectors are created one is growing
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Sweep-plane algorithm
The half-bisector changes its direction
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Sweep-plane algorithm
Site event 2 half-bisectors created
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Sweep-plane algorithm
The half-bisector Changes its direction
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Sweep-plane algorithm
Circle event Triangle added to DT 2
half-bisectors deleted 1 half-bisector created
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Sweep-plane algorithm
Half-bisector changes its direction
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Sweep-plane algorithm
No more events
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Sweep-plane algorithm
Resulting Voronoi diagram and Delaunay
triangulation
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Sweep-plane algorithm
Waves in Manhattan metric
Waves in power metric
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Swap method from VD to power diagram

• Relationship between Voronoi bisector and power
  bisector
• The power bisector is moved relative to the
  Voronoi bisector by

Bisector transformation
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Swap method from VD to power diagram

• Algorithm overview
• Inflate VD sites
• Transform edges of VD
• Perform INCIRCLE test on quadrilaterals and
  perform swap operations if needed.
• The worst-case complexity of algorithm is O(n2).

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Swap method

• Voronoi to power

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Dynamic data structures swap application

• Approach
• construct and maintain a dynamic Delaunay
  triangulation for the set of moving disks (in
  some metric).
• Collisions checks are performed only along the
  Delaunay edges.

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Dynamic VD and DT

• Problem
• Given a set of N moving and/or changing (i.e.
  growing) sites
• Construct the VD (DT) for the set
• Maintain the VD (DT) dynamically

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Dynamic VD and DT
Dynamic Voronoi diagram
Dynamic Delaunay triangulation
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Dynamic VD and DT

• Dynamic DT is easier to maintain than dynamic VD
  in a sense that the coordinates of VD vertices
  must be calculated, while coordinates of DT
  vertices (the sites) are already known for any
  moment of time
• As sites move, the topological structure of DT
  (and VD) changes only at discrete moments of
  time, which are called topological events.
• Topological event involves swapping of the
  diagonal in a quadrilateral in the DT
• In the VD corresponding edge shrinks to zero and
  another edge starts to grow

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Swap condition
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INCIRCLE test
To determine the swap time we have to solve
1. Power metric
2. Euclidean metric
3. Manhattan metric (L)
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Dynamic DT maintenance

• Preprocessing
• Construct the static Delaunay triangulation for
  the original site distribution
• For every quadrilateral in DT calculate the
  potential topological event. Insert all such
  events into the priority event queue sorted
  according to the time order
• Iteration
• Take the next event from the event queue
• Perform the swap operation and update the data
  structure
• Delete topological events planned for the four
  disappearing quadrilaterals from the event queue.
• Compute the new topological events for the four
  new quadrilaterals and insert them into the queue.

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Algorithm complexity

• Time
• Constructing initial static DT - O(N log N)
• Each swap - O(1)
• Insertion into the priority queue - O(log N)
• Deletion from the queue can be performed in O(1)
• Number of topological events depends on the
  trajectories of moving sites. It can vary from
  from O(1) up to O(n3)
• and even to infinity (in the case of periodic
  trajectories)
• When sites do not move, but just grow, we
  conjecture that the number of topological events
  will not be greater than O(N) in general case.
• Space
• DT occupies O(N)
• Queue requires O(N) because only one event can be
  scheduled for each edge in the DT

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Geometric Data Structures

• Part 2 Grid files and tree variants

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Part 2

• Grid file
• Tree
• Variants

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Geometric data structures

• Voronoi diagrams
• Regular grid (mesh)
• Grid file
• Quad tree
• k-d tree
• Interval tree

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Trees

• BST search trees, O(n)
• AVL, IPR balanced O(log n)
• B-trees for indexing and searching in data
  bases
• Grow from the leaf level
• More compact faster search
• B, B - used for indexing, store data in leaves,
  nodes are more full

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Operations on spatial trees

• Spatial queries
• Point location
• Stabbing query (which intervals/polygons contain
  the point)
• Window query which objects (polygons, points)
  are intersecting the given window (polygon)

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Spatial queries (2D)

• Point query find an object containing a point
  (find Voronoi region containing a point)
• Window query find an object overlapping a
  rectangle
• Spatial join join parts of objects satisfying
  some relationship (intersection, adjacency,
  containment)

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Interval trees

• Geometric, 1-dimensional tree
• Interval is defined by (x1,x2)
• Split at the middle (5), again at the middle
  (3,7), again at the middle (2,8)
• All intervals intersecting a middle point are
  stored at the corresponding root.

(4,6) (4,8)
(6,9)
(2,4)
(7.5,8.5)
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Interval trees

• Finding intervals by finding x1, x2 against the
  nodes
• Find interval containing specific value from
  the root
• Sort intervals within each node of the tree
  according to their coordinates
• Cost of the stabbing query finding all
  intervals containing the specified value is O(log
  n k), where k is the number of reported
  intervals.

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SAM (Spatial Access Method)

• Constructs the minimal bounding box (mbb)
• Check validity (predicate) on mbb
• Refinement step verifies if actual objects
  satisfy the predicate.

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The grid

• Fixed grid
• Stored as a 2D array, each entry contains a link
  to a list of points (object) stored in a grid.

a,b
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Page overflow

• Too many points in one grid cell
• Split the cell!

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Rectangle indexing with grids

• Rectangles may share different grid cells
• Duplicates are stored
• Grid cells are of fixed size

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Grid file vs. grid

• In a grid file, the index is dynamically
  increased in size when overflow happens.
• The space is split by a vertical or a horizontal
  line, and then further subdivided when overflow
  happens!
• Index is dynamically growing
• Boundaries of cells of different sizes are
  stores, thus point and stabbing queries are easy

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The quadtree

• Instead of using an array as an index, use tree!
• Quadtree decomposition cells are indexed by
  using quaternary B-tree.
• All cells are squares, not polygons.
• Search in a tree is faster!

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Grid file

• Example of a grid file

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Linear quadtree

• B index actual references to rectangles are
  stored in the leaves, saving more space access
  time
• Label nodes according to Z or pi order

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Linear quadtree

• Level of detail increases as the number of
  quadtree decompositions increases!
• Decompositions have indexes of a form
• 00,01,02,03,10,11,12,13, 2,300
• 301 ,302 ,303 ,31 ,32 ,33
• Stores as Bplus tree

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Finer Grid

• R-tree
• Each object is decomposed and stored as a set of
  rectangles
• Object decomposition larger areas of a grid are
  treated as one element
• Raster decomposition each smaller element is
  stored separately

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R-trees

• R-tree
• Objects are grouped together according to
  topological properties not a grid.
• More flexibility.
• R tree- Optimizes
• Node overlapping
• Areas covered by the node
• R tree B tree, bounding rectangles do not
  intersect

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K-d tree

• Used for point location, k number of the
  attributes to perform the search
• Geometric interpretation to perform search in
  2D space 2-d tree
• Search components (x,y) interchange

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K-d tree
d
d
e
c
f
b
f
c
a
e
b
a
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Conclusions

• Numerous applications exist based on Voronoi
  diagram methodology and tree/grid based data
  structures.