TU/e computational geometry introduction Mark de Berg

TU/e IPA Basic Course—Algorithms introduction what is CG applications randomized incremental construction easy example: sorting a general framework applications to geometric problems and more …

TU/e Computational Geometry Computational Geometry: Area within algorithms research dealing with spatial data (points, lines, polygons, circles, spheres, curves, … )  design of efficient algorithms and data structures for spatial data  elementary operations are assumed to take O(1) time and be available focus on asymptotic running times of algorithms computing intersection point of two lines, distance between two points, etc

TU/e Map overlay land usage annual rainfall overlay

TU/e Computational Geometry example: compute all intersections in a set of line segments naïve algorithm for i = 1 to n for j =i+1 to n do if s i intersects s j then report intersection Running time: O(n 2 ) Can we do better ?

TU/e Computational geometry: applications (1) geographic information systems (GIS) Mississippi delta Memphis Arlington 30 cities: 4 30 = possibilities

TU/e Computational geometry: applications (2) computer-aided design and manufacturing (CAD/CAM) 3D model of power plant ( triangles)  motion planning  virtual walkthroughs

TU/e Computational geometry: applications (3) 3D copier surface reconstruction 3D scanner 3D printer

TU/e Computational geometry: applications (4) Other applications of surface reconstruction  digitizing cultural heritage  custom-fit products  reverse engineering

TU/e Computational geometry: applications (5)  geographic information systems  computer-aided design and manufacturing  robotics and motion planning  computer graphics and virtual reality  databases  structural computational biology  and more … age salary

TU/e Computational Geometry: algorithmic paradigms Deterministic algorithms  plane sweep  geometric divide-and-conquer Randomized algorithms  randomized incremental construction  random sampling today’s focus

TU/e EXERCISES  Let S be a set of n line segments in the plane.  Design an algorithm that decides in O (n log n) time if there are any intersections in S. Hint: Sort the endpoints on y-coordinates, and handle them in that order in a suitable manner.  Extend the algorithm so that it reports all intersections in time O ((n+k) log n), where k is the number of intersections. For the experts:  Let u (n ) be the maximum number of pairs of points at distance 1 in a set of n points in the plane. Prove that u(n) = Ω (n log n).  Let D be a set of n discs in the plane. Prove that the union complexity of D is O (n ).

TU/e Plane-sweep algorithm for line-segment intersection sweep line insert segment 1 insert segment 2 insert segment 3 insert segment 4 insert segment 5 delete segment 1 When two segments intersect, then there is a horizontal line L such that  the two segments both intersect L  they are neighbors along L Ordering along L: 1,23,1,23,1,4,23,5,1,4,213,5,4,2

TU/e Plane sweep SegmentIntersect 1.Sort the endpoints by y-coordinate. Let p 1,…,p 2n be the sorted set of endpoints. 2.Initialize empty search tree T. 3. for i = 1 to 2n 4. do { if p i is the left endpoint of a segment s 5. then Insert s into the tree T 6. else Delete s from the tree T 7. Check all pairs of segments that become neighbors along the sweep line for intersection, report if there is an intersection. } Running time ? Extension to reporting all intersections ?

TU/e COFFEE

TU/e Lecture II: Randomized incremental construction ParanoidMax (A) (* A [1..n ] is an array of n numbers *)  Randomly permute the elements in A  max := A[1]  for i := 2 to n  do if A [i ] > max  then { max := A[i ]  for j = 1 to i -1  do if A[i ] > max then error }  return max Warm-up exercise: Analyze the worst-case and expected running time of the following algorithm.

TU/e Worst-case running time Worst-case = O(1) + time for lines 2—7 = O(1) + ∑ 2≤i≤n (time for i-th iteration) = O(1) + ∑ 2≤i≤n ∑ 1≤j≤i-1 O(1) = O(1) + ∑ 2≤i≤n O(i) = O(n 2 )

TU/e Expected running time E [ looptijd ] = O(1) + E [ time for lines 2—7 ] = O(1) + E [ ∑ 2≤i≤n (time for i-th iteration) ] = O(1) + ∑ 2≤i≤n E [ time for i-th iteration ] = O(1) + ∑ 2≤i≤n { Pr [ A[i] ≤ max { A[1..i-1 ] } ] ∙ O(1) + Pr [ A[i] > max { A[1..i-1 ] } ] ∙ O(i) } = O(1) + ∑ 2≤i≤n { 1 ∙ O(1) + (1/i ) ∙ O(i) } = O(n )

TU/e EXERCISE 1.Give an algorithm that puts the elements of a given array A[1..n] into random order. Your algorithm should be such that every possible permutation is equally likely to be the result.

TU/e Sorting using Incremental Construction Input: numbers x 1,…,x n Output: partitioning of x-axis into a collection T of empty intervals defined by the input points x 3 x 1 x 5 x 4 x 2 A geometric view of sorting Incremental Construction: “Add” input points one by one, and maintain partitioning into intervals x 3 x 1 x 5 x 4 x 2

TU/e Sorting using Incremental Construction x 3 x 1 x 5 x 4 x 2 IC-Sort (S) 1.T := { [ -infty : +infty] } 2. for i := 1 to n 3. do { Find interval J in T that contains x i and remove it. 4. Determine new intervals and insert them into T. }  for each point, maintain pointer to interval that contains it  for each interval in T, maintain conflict list of all points inside list is empty list: x 4, x 5 How?

TU/e Sorting using Incremental Construction IC-Sort (S) 1.T := { [ -infty : +infty] } 2. for i := 1 to n 3. do { Find interval J in T that contains x i and remove it. 4. Determine new empty intervals and insert them into T. 5. Split conflict list of J to get the two new conflict lists. }  for each point, maintain pointer to interval that contains it  for each interval in T, maintain conflict list of all points inside TOTAL TIME: O ( total size of conflict lists of all intervals that ever appear) WORST-CASE: Time analysis: O(n 2 )

TU/e Sorting using Incremental Construction RIC-Sort (S) 1.Randomly permute input points 2.T := { [ -infty : +infty] } 3. for i := 1 to n 4. do { Find interval J in T that contains x i and remove it. 5. Determine new empty intervals and insert them into T. 6. Split conflict list of J to get the two new conflict lists. }  for each point, maintain pointer to interval that contains it  for each interval in T, maintain conflict list of all points inside TOTAL TIME: O ( total size of conflict lists of all intervals that are created) EXPECTED TIME: ?? Time analysis:

TU/e An abstract framework for RIC S = set of n (geometric) objects; the input C = set of configurations D: assigns a defining set D(Δ) S to every configuration Δ є C K: assigns a killing set K(Δ) S to every configuration Δ є C 4-tuple (S, C, D, K) is called a configuration space configuration Δ є C is active over a subset S’ S if: D(Δ) S’ and K(Δ) S’ = o T(S) = active configurations over S; the output ∩ ∩ ∩ ∩ ∩ /

TU/e Sorting using Incremental Construction RIC(S) 1.Compute random permutation x 1,…x n of input elements 2.T := { active configurations over empty set } 3. for i := 1 to n 4. do { Remove all configurations Δ with x i є K(Δ) from T. 5. Determine new active configurations; insert them into T. 6. Construct conflict lists of new active configurations. }  for each x i, maintain pointers to all configurations Δ such that x i є K(Δ)  for each configuration Δ in T, maintain conflict list of all x i є K(Δ) Theorem: Expected total size of conflict lists of all created configs: O( ∑ 1≤i≤n (n / i 2 ) ∙ E [ # active configurations in step i ] )

TU/e EXERCISE 1.Use the theorem to prove that the running time of the RIC algorithm for sorting runs in O( n log n ) time.

TU/e LUNCH

TU/e Lecture III RIC: more applications Voronoi diagrams and Delaunay triangulations

TU/e Principia Philosophiae (R. Descartes, 1644). Voronoi diagram The universe according to Descartes

TU/e Construction of 3D height models (1)

TU/e height? Construction of 3D height models (2)

TU/e Construction of 3D height models (3) Better: use interpolation to determine the height triangulation 983

TU/e Construction of 3D height models (4) long and thin triangles are bad  try to avoid small angles What is a good triangulation?

TU/e Construction of 3D height models (5) Voronoi diagram Delaunay triangulation: triangulation that maximizes the minimum angle !

TU/e Voronoi diagrams and Delaunay triangulations P : n points (sites) in the plane V(p i ) = Voronoi cell of site p i = { q є R 2 : dist(q,p i ) < dist(q,p j ) for all j ≠i } Vor(P ) = Voronoi diagram of P = subdivision induced by Voronoi cells DT(P) = Delaunay triangulation of P = dual graph of Vor(P)

TU/e EXERCISES  Prove: a triangle p i p j p k is a triangle in DT(P) if and only if its circumcircle C (p i p j p k ) does not contain any other site  Prove: number of triangles in DT(P) is at most 2n-5 Hint: Euler’s formula for connected planar graphs: #vertices — #edges + #faces = 2  Design and analyze a randomized incremental algorithm to construct DT(P).

TU/e COFFEE

TU/e RIC – Limitations of the framework Robot motion planning What is the region that is reachable for the robot? Lecture IV: wrap-up

TU/e RIC – Limitations of the framework (cont’d) The single-cell problem: Compute cell containing the origin Idea: compute vertical decomposition for the cell using RIC ?!

TU/e Spatial data structures  store geometric data in 2-, 3- or higher dimensional space such that certain queries can be answered efficiently  applications: GIS, graphics and virtual reality, … Examples:  range searching  nearest-neighbor searching  point location more computational geometry

TU/e  plane sweep  parametric search  arrangements  geometric duality, inversion, Plücker coordinates, …  kinetic data structures  core sets  Davenport-Schinzel sequences  … and more …