Delaunay triangulations: properties, algorithms, and complexity
Olivier Devillers 1-1 Delaunay triangulations: properties, algorithms, and complexity
Extra question: what is the difference between an algorithm and a program?
Olivier Devillers 1-2 Algorithm
Recipe to go from the input to the output
Formalized description in some language
May use data structure
Proof of correctness
Complexity analysis 2 Delaunay Triangulation: definition, empty circle property
3-1 Delaunay Triangulation: definition, empty circle property
Point set
3-2 Delaunay Triangulation: definition, empty circle property
Point set
Query
3-3 Delaunay Triangulation: definition, empty circle property
Point set
Query
3-4 Delaunay Triangulation: definition, empty circle property
Point set
Query
3-5 Delaunay Triangulation: definition, empty circle property
Point set
Query
Nearest neighbor
3-6 Delaunay Triangulation: definition, empty circle property
Point set
Voronoi diagram
3-7 Delaunay Triangulation: definition, empty circle property
Point set
Voronoi diagram
3-8 Delaunay Triangulation: definition, empty circle property
Point set
Voronoi diagram
3-9 Delaunay Triangulation: definition, empty circle property
Point set
Voronoi diagram
3-10 Delaunay Triangulation: definition, empty circle property
Point set
Voronoi diagram
Delaunay triangulation
3-11 Delaunay Triangulation: definition, empty circle property
Point set
Delaunay triangulation
Empty circle property 3-12 Delaunay Triangulation: definition, empty circle property
Point set
Delaunay triangulation
Empty circle property 3-13 Delaunay Triangulation: size
4-1 Delaunay Triangulation: size
Vertices Edges Faces
Euler relation n e + t +1 =2
4-2 Delaunay Triangulation: size
Vertices Edges Faces
Euler relation n e + t +1 =2
triangular faces 3t + k =2e
4-3 Delaunay Triangulation: size
Vertices Edges Faces
Euler relation n e + t +1 =2
triangular faces 3t + k =2e
t =2n k 2 < 2n e =3n k 3 < 3n
4-4 Delaunay Triangulation: size
Vertices Edges Faces
Euler relation n e + t +1 =2
triangular faces 3t + k =2e
d (p)=2e =6n 2k 6 p S X2
1 E(d (p)) = n d (p) < 6 p S X2 average on the choice of point p in set of points S 4-5 Delaunay Triangulation: max-min angle
5-1 Delaunay Triangulation: max-min angle
Triangulation Delaunay
5-2 Delaunay Triangulation: max-min angle
Triangulation Delaunay
smallest angle
5-3 Delaunay Triangulation: max-min angle
Triangulation Delaunay
smallest angle
5-4 Delaunay Triangulation: max-min angle
Triangulation Delaunay
smallest angle
5-5 Delaunay Triangulation: max-min angle
Triangulation Delaunay
smallest angle second smallest angle 5-6 Delaunay Triangulation: max-min angle
Proof
6-1 Delaunay Triangulation: max-min angle Definition Delaunay edge
6-2 Delaunay Triangulation: max-min angle Definition Delaunay edge
empty circle 9
6-3 Delaunay Triangulation: max-min angle Definition locally Delaunay edge w.r.t. a triangulation
6-4 Delaunay Triangulation: max-min angle Definition locally Delaunay edge w.r.t. a triangulation
circle 9 not enclosing the two neighbors
neighbor = visible from the edge 6-5 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 ()
7-1 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 () Proof:
choose an edge
7-2 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 () Proof:
choose an edge
edges of the quadrilateral are locally Delaunay
7-3 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 () Proof:
choose an edge
edges of the quadrilateral are locally Delaunay
7-4 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 () Proof:
choose an edge
edges of the quadrilateral are locally Delaunay
Vertices visible through one edge are outside circle
7-5 Delaunay Triangulation: max-min angle Lemma ( edge: locally Delaunay) Delaunay 8 () Proof:
choose an edge
edges of the quadrilateral are locally Delaunay
Vertices visible through one edge are outside circle
Induction all vertices outside circle ! 7-6 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle ()
Two possible triangulation
8-1 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle () Case 1: smallest angle in corner
8-2 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle () Case 1: smallest angle in corner
<
asmallerangle other triangulation 8-3 9 2 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle () Case 2: smallest angle along diagonal
8-4 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle () Case 2: smallest angle along diagonal
8-5 Delaunay Triangulation: max-min angle Lemma For four points in convex position Delaunay maximize the smallest angle () Case 2: smallest angle < along diagonal
asmallerangle other triangulation 8-6 9 2 Delaunay Triangulation: max-min angle 6n 3k 4 Map: Triangulations R smallest angle ↵1 !
9-1 Delaunay Triangulation: max-min angle 6n 3k 4 Map: Triangulations R smallest angle ↵1 ! second smallest angle ↵2
9-2 Delaunay Triangulation: max-min angle 6n 3k 4 Map: Triangulations R smallest angle ↵1 ! second smallest angle ↵2 third smallest angle ↵3
9-3 Delaunay Triangulation: max-min angle 6n 3k 4 Map: Triangulations R smallest angle ↵1 ! second smallest angle ↵2 (↵1,↵2,↵3,...,↵6n 3k 4) third smallest angle ↵3
9-4 Delaunay Triangulation: max-min angle 6n 3k 4 Map: Triangulations R smallest angle ↵1 ! second smallest angle ↵2 (↵1,↵2,↵3,...,↵6n 3k 4) third smallest angle ↵3
sort triangulations in lexicographic order
9-5 Delaunay Triangulation: max-min angle Theorem: Delaunay maximizes minimum angles (in lexicographic order)
10 - 1 Delaunay Triangulation: max-min angle Theorem: Delaunay maximizes minimum angles (in lexicographic order) Proof: Let T be the triangulation maximizing angles
10 - 2 Delaunay Triangulation: max-min angle Theorem: Delaunay maximizes minimum angles (in lexicographic order) Proof: Let T be the triangulation maximizing angles
= convex quadrilateral (from 2 triangles T ) )8 2 the diagonal maximizes smallest angle (in quad)
10 - 3 Delaunay Triangulation: max-min angle Theorem: Delaunay maximizes minimum angles (in lexicographic order) Proof: Let T be the triangulation maximizing angles
= convex quadrilateral (from 2 triangles T ) )8 2 the diagonal maximizes smallest angle (in quad)
= edge, it is locally Delaunay )8
10 - 4 Delaunay Triangulation: max-min angle Theorem: Delaunay maximizes minimum angles (in lexicographic order) Proof: Let T be the triangulation maximizing angles
= convex quadrilateral (from 2 triangles T ) )8 2 the diagonal maximizes smallest angle (in quad)
= edge, it is locally Delaunay )8
= T = Delaunay )
10 - 5 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
11 - 1 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
project on parabola
11 - 2 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
project on parabola
compute Delaunay triang.
11 - 3 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
project on parabola
compute Delaunay triang.
find lowest point
11 - 4 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
project on parabola
compute Delaunay triang.
find lowest point
enumerate x coordinates in ccw CH order
11 - 5 Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
O(n) project on parabola
f(n) compute Delaunay triang.
O(n) find lowest point
O(n) enumerate x coordinates in ccw CH order
Lower bound on sorting
= f(n)+O(n) ⌦(n log n) 11 - 6 ) Delaunay triangulation Lower bound
A stupid algorithm for sorting numbers
O(n) project on parabola
f(n) compute Delaunay triang.
O(n) find lowest point
O(n) enumerate x coordinates in ccw CH order
Lower bound on sorting
= f(n)+O(n) ⌦(n log n) 11 - 7 ) Delaunay Triangulation: predicates
12 - 1 Delaunay Triangulation: predicates Orientation predicate
r pqr +? q 111 xq xp xr xp = xp xq xr > 0 yq yp yr yp yp yq yr q pqr -? p 111 xp xq xr < 0 r yp yq yr p pqr 0? q
111 r xp xq xr =0 yp yq yr p 12 - 2 Delaunay Triangulation: predicates Incircle predicate r s
q p
pqr ccw triangle
query s inside circumcircle
12 - 3 Delaunay Triangulation: predicates
r s
q p
pqr ccw triangle
query s cocircular
12 - 4 Delaunay Triangulation: predicates
r
s
q p
pqr ccw triangle
query s outside circumcircle
12 - 5 Delaunay Triangulation: predicates
r
q p
pqr ccw triangle
query s
12 - 6 Delaunay Triangulation: predicates
Space of circles
? 2 2 p =(x, y) p =(x, y, x + y )
12 - 7 Delaunay Triangulation: predicates
Space of circles
s inside/outside of circle through pqr ? ? ? plane through p q r above/below s?
incircle predicate 3D orientation predicate
12 - 8 Delaunay Triangulation: predicates
Space of circles
s inside/outside of circle through pqr ? ? ? plane through p q r above/below s?
incircle predicate 1111 xp xq xr xs 3D orientation predicate sign yp yq yr ys 2 2 2 2 2 2 2 2 x + y x + y x + y x + y p p q q r r s s 12 - 9 Delaunay Triangulation: data structure
Data structure for (Delaunay) triangulation
Representing incidences
Representing hull boundary
Representing user’s data put colors in triangles
13 Delaunay Triangulation: very naive algorithm
14 - 1 Delaunay Triangulation: very naive algorithm
For each triple of points (p, q, r) If pqr ccw Delaunay = true; For each point s If s in circle pqr Delaunay = false; Output pqr 14 - 2 Delaunay Triangulation: very naive algorithm
For each triple of points (p, q, r) If pqr ccw Delaunay = true; Correctness: easy For each point s If s in circle pqr Delaunay = false; Output pqr 14 - 3 Delaunay Triangulation: very naive algorithm
For each triple of points (p, q, r) If pqr ccw Delaunay = true; Correctness: easy For each point s Complexity: O(n4) If s in circle pqr Delaunay = false; Output pqr 14 - 4 Delaunay Triangulation: very naive algorithm
For each triple of points (p, q, r) If pqr ccw Delaunay = true; Correctness: easy For each point s Complexity: O(n4) If s in circle pqr Delaunay = false; Output pqr does not compute incidences 14 - 5 Delaunay Triangulation: incremental algorithm
15 - 1 Delaunay Triangulation: incremental algorithm
New point
15 - 2 Delaunay Triangulation: incremental algorithm
New point
Locate
15 - 3 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 4 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 5 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 6 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 7 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 8 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 9 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 10 Delaunay Triangulation: incremental algorithm
New point
Locate
e.g.: visibility walk 15 - 11 Delaunay Triangulation: incremental algorithm not unique New point
Locate
e.g.: visibility walk 15 - 12 Delaunay Triangulation: incremental algorithm not unique New point
Locate
e.g.: visibility walk 15 - 13 Delaunay Triangulation: incremental algorithm Visibility walk terminates
15 - 14 Delaunay Triangulation: incremental algorithm Visibility walk terminates
15 - 15 Delaunay Triangulation: incremental algorithm Visibility walk terminates
May loop
15 - 16 Delaunay Triangulation: incremental algorithm Visibility walk terminates
May loop
Not Delaunay 15 - 17 Delaunay Triangulation: incremental algorithm Visibility walk terminates
Delaunay Triangulation: pencils of circles
Power of a point w.r.t a circle 2 2 (x + y 2a0x 2b0y + c0) +(1 ) (x2 + y2 2ax 2by + c ) =0
blue yields smaller power
equal power black yields smaller power 15 - 18 Delaunay Triangulation: incremental algorithm Visibility walk terminates
15 - 19 Delaunay Triangulation: incremental algorithm Visibility walk terminates
Green power < Red power
15 - 20 Delaunay Triangulation: incremental algorithm Visibility walk terminates
Green power < Red power Power decreases 15 - 21 Delaunay Triangulation: incremental algorithm Visibility walk terminates
Green power < Red power Power decreases Visibility walk terminates 15 - 22 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 1 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 2 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 3 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 4 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 5 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 6 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 7 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 8 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 9 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 10 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 11 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 12 Delaunay Triangulation: incremental algorithm
New point
Locate
Search conflicts
16 - 13 Delaunay Triangulation: incremental algorithm
New point
16 - 14 Delaunay Triangulation: incremental algorithm
New point
16 - 15 Delaunay Triangulation: incremental algorithm
Proof of correctness
circle of new triangle 17 - 1 Delaunay Triangulation: incremental algorithm
Proof of correctness
circle of new triangle ⇢ [ 17 - 2 Delaunay Triangulation: incremental algorithm
Complexity
Locate
Search conflicts
18 - 1 Delaunay Triangulation: incremental algorithm
Complexity
Locate
Search conflicts ] triangles in conflict
] triangles neighboring triangles in conflict
18 - 2 Delaunay Triangulation: incremental algorithm
Complexity
Locate
Search conflicts ] triangles in conflict
] triangles neighboring triangles in conflict
degree of new point in new triangulation
18 - 3 Delaunay Triangulation: incremental algorithm Complexity Locate Walk may visit all triangles < 2n Search conflicts degree of new point in new triangulation 18 - 4 Delaunay Triangulation: incremental algorithm Complexity Locate O(n) per insertion Search conflicts 18 - 5 Delaunay Triangulation: incremental algorithm Complexity Locate O(n) per insertion Search conflicts O(n2) for the whole construction 18 - 6 Delaunay Triangulation: incremental algorithm Complexity Locate Search conflicts 18 - 7 Delaunay Triangulation: incremental algorithm Complexity Locate Search conflicts Insertion: ⌦(n) Whole construction: ⌦(n2) 18 - 8 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right 19 - 1 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right 19 - 2 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Certified Delaunay triangles 19 - 3 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Certified Delaunay triangles 19 - 4 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Certified Delaunay triangles 19 - 5 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Certified Delaunay triangles Certified Delaunay edges 19 - 6 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right 19 - 7 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right New point Locate vertically I I H H 19 - 8 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right New point Locate vertically Create edge I I H H 19 - 9 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Closing a triangle ? 19 - 10 Delaunay Triangulation: sweep-line algorithm Discover the points from left to right Next circle event Close triangle 19 - 11 Delaunay Triangulation: sweep-line algorithm Circle events Point events Complexity processed Number Triangulation List of events (x sorted) List of boundary edges (ccw sorted) 20 - 1 Delaunay Triangulation: sweep-line algorithm Circle events Point events Complexity processed Number 2n n create create 2 triangles one edge Triangulation per event per event 3 deletions 2 deletions 2 insertions 2 insertions List of events (x sorted) per event per event List of boundary edges replace locate, then (ccw sorted) 2edgesby1 insert 2 edges per event per event 20 - 2 Delaunay Triangulation: sweep-line algorithm Circle events Point events Complexity processed Number 2n n create create 2 triangles one edge TriangulationO(1) per operation per event per event 3 deletions 2 deletions per operation 2 insertions 2 insertions List(log of eventsn) (x sorted) O per event per event List of boundary edges replace locate, then n) per operation O(log (ccw sorted) 2edgesby1 insert 2 edges per event per event 20 - 3 Delaunay Triangulation: sweep-line algorithm Circle events Point events Complexity processed Number 2n n create create 2 triangles one edge TriangulationO(1) per operation per event per event 3 deletionsO(n log2 ndeletions) per operation 2 insertions 2 insertions List(log of eventsn) (x sorted) O per event per event List of boundary edges replace locate, then n) per operation O(log (ccw sorted) 2edgesby1 insert 2 edges per event per event 20 - 4 Delaunay Triangulation: predicates 21 - 1 Delaunay Triangulation: predicates x comparisons 21 - 2 Delaunay Triangulation: predicates x comparisons y comparisons 21 - 3 Delaunay Triangulation: predicates x comparisons y comparisons more intricate than orientation and incircle 21 - 4 Delaunay Triangulation: divide & conquer (sketch) 22 - 1 Delaunay Triangulation: divide & conquer (sketch) Divide 22 - 2 Delaunay Triangulation: divide & conquer (sketch) Divide Recurse 22 - 3 Delaunay Triangulation: divide & conquer (sketch) Divide Recurse Conquer 22 - 4 Delaunay Triangulation: divide & conquer (sketch) Divide Recurse Conquer O(n log n) 22 - 5 Delaunay Triangulation: divide & conquer (sketch) Divide Recurse Conquer O(n log n) balanced linear time easier conquer 22 - 6 Jump and walk Use randomness hypotheses 23 - 1 Jump and walk Use randomness hypotheses 23 - 2 Jump and walk Use randomness hypotheses 23 - 3 Jump and walk Use randomness hypotheses Hopefully shorter walk Designed for random points O(p3 n) expected location time 23 - 4 Jump and walk (no distribution hypothesis) Randomized 24 - 1 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= k 24 - 2 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= k n Walk length = O k choose k = p2 n 24 - 3 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= k Delaunay hierarchy n Walk length = O k choose k = p2 n 24 - 4 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= k n Delaunay hierarchy k1 n Walk length = O k choose k = p2 n 24 - 5 Jump and walk (no distribution hypothesis) Randomized n [] of in ]= k Delaunay hierarchy E k n + 1 k1 k2 n Walk length = O k choose k = p2 n 24 - 6 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= n k Delaunay hierarchy k + 1 + k2 + ... k1 k2 k3 n Walk length = O k choose k = p2 n 24 - 7 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= n k Delaunay hierarchy k + 1 + k2 + ... k1 k2 k3 n k Walk length = O k i = ↵ choose ki+1 choose k = p2 n 24 - 8 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= n k Delaunay hierarchy k + 1 + k2 + ... k1 k2 k3 n k Walk length = O k i = ↵ choose ki+1 choose k = p2 n point location in O (↵ log↵ n) 24 - 9 Jump and walk (no distribution hypothesis) Randomized n E [] of in ]= n k Delaunay hierarchy k + 1 + k2 + ... k1 k2 k3 n k Walk length = O k i = ↵ choose ki+1 choose k = p2 n point location in O (↵ log↵ n) point location in O (p↵ log↵ n) 24 - 10 Technical detail Walk length = O ] = O n ( of in ) k 25 - 1 Technical detail Walk length = O ] = O n ( of in ) k random point not a random point 25 - 2 Technical detail Walk length = O ] = O n ( of in ) k random point not a random point 1 1 E [d ]= d NN(q)= d v n n q q v=NN(q) X 1 X X1 = d v 6d v 36 n n 25 - 3 v v X q;v=XNN(q) X Technical detail d Walk length = O ] P = O n ( of in ) k random point not a random point 1 1 E [d ]= d NN(q)= d v n n q q v=NN(q) X 1 X X1 = d v 6d v 36 n n 25 - 4 v v X q;v=XNN(q) X Randomization Drawbacks of random order non locality of memory access data structure for point location Hilbert sort 26 27 - 1 27 - 2 27 - 3 27 - 4 Drawbacks of random order non locality of memory access data structure for point location Hilbert sort Walk should be fast Last point is not at all a random point no control of degree of last point 28 Biased Random Insertion Order (BRIO) 29 - 1 Biased Random Insertion Order (BRIO) 29 - 2 Biased Random Insertion Order (BRIO) 29 - 3 Biased Random Insertion Order (BRIO) 29 - 4 Biased Random Insertion Order (BRIO) 29 - 5 Biased Random Insertion Order (BRIO) 29 - 6 Algorithm Recipe to go from the input to the output Formalized description in some language May use data structure Proof of correctness Complexity analysis 30 - 1 Algorithm Program Recipe to go from the input to the output Implementation of an algorithm Formalized description in some language Translation in a programming language (C++) May use data structure May use software library Proof of correctness Debugging Complexity analysis 30 - 2 Running time Algorithm may be difficult to transform into Program Recipe to go from the input to the output Implementation of an algorithm Formalized description in some language too complicatedTranslation in a programming language (C++) May use data structure too complicated May use software library inexistant Proof of correctness Computation model Debugging O notation big Complexity analysis 30 - 3 Running time actual computer(cache, prediction. . . ) The31 end