Geometric Algorithms Suman Sourav Paramasiven Appavoo Anuja Meetoo Appavoo Li Jing Lu Bingxin...

Geometric Algorithms Suman Sourav Paramasiven Appavoo Anuja Meetoo Appavoo Li Jing Lu Bingxin Suhendry Effendy Dumitrel Loghin

Introduction & Motivation Suman Sourav

Introduction Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Deep connections with classical mathematics and theoretical computer science on the one hand, and many ties with applications on the other. Focus is on discrete nature of geometric problems as opposed to continuous issues.

Introduction Any problem that is to be solved using a digital computer has to be discrete in form. For CG techniques to be applied to areas that involves continuous issues, discrete approximations to continuous curves or surfaces are needed. People deal more with straight or flat objects or simple curved objects, than with high degree algebraic curves.

Introduction Programming in CG is a little difficult. Libraries like LEDA and CGAL implement various data structures and geometric algorithms. CG algorithms suffer from the curse of degeneracies. So, certain simplifying assumptions at times like no three points are collinear, no four points are cocircular, etc are needed to be made.

Motivation & Application There are many areas that give rise to geometric problems. Computer graphics Computer vision and image processing Robotics Computer-aided designing (CAD) Geographic information systems (GIS) Telecommunication VLSI design (chip layout)

Example To find the nearest phone booth on campus

Example The motion planning problem

Example To build a model of the terrain surface, we can start with a number of sample points where we know the height.

How do we interpolate the height at other points? Nearest neighbor interpolation Piecewise linear interpolation by a triangulation Moving windows interpolation Natural neighbor interpolation

For interpolation, it is good if triangles are not long and skinny.

Example Each time you click inside the editing area of a diagram or inside a web page, the application must know inside which object the mouse pointer is. Given the high frequency of such a query, a fast algorithm must be used.

Facility location Given a set of houses and farms in an isolated area. Can we place a helicopter ambulance post so that each house and farm can be reached within 15 minutes? Where should we place an antenna so that a number of locations have maximum Reception?

Art gallery problem How many cameras are needed to guard a given art gallery so that every point is seen?

A Real Life Application

Voronoi Voronoi diagram - Properties - Construction methods Fortunes Algorithm - How does it work? - Implementation aspects - Data structures - Pseudocode - Complexity (time & space) Paramasiven Appavoo & Anuja Meetoo Appavoo

Voronoi edges: Each point on an edge of the Voronoi diagram is equidistant from its two nearest neighbors p i and p j. Voronoi vertices: It is the center of the circle passing through these sites, and this circle contains no other sites in its interior. Degree: If we assume that no four points are cocircular, then the vertices of the Voronoi diagram all have degree three. Voronoi diagram Properties

Convex hull: A cell of the Voronoi diagram is unbounded if and only if the corresponding site lies on the convex hull. Size: If n denotes the number of sites, then the Voronoi diagram is a planar graph (if we imagine all the unbounded edges as going to a common vertex infinity) with exactly n faces. Voronoi diagram Properties

Voronoi Construction methods Perpendicular bisector Divide and conquer Fortunes algo

Voronoi Fortunes algo Time complexity of perpendicular bisector Extra computation at the merge of divide and conquer Steven Fortune developed a plane sweep algorithm called the fortunes algorithm that uses: o a sweep line, o a beach line, o site event, and o circle event to generate the diagrams Steven Fortune, Bell Laboratories

Fortunes Algo

Fortunes algo - Concepts Sweep line The idea behind algorithms of this type is to imagine that a line is swept or moved across the plane, stopping at some points. A sweep line splits the problem domain into two regions, an explored region and an unexplored region Operations are restricted to objects that are either intersecting or are in the immediate vicinity of the sweep line whenever it stops The complete solution is available once the line has passed over all objects. It also applies an ordering to the problem

Fortunes algo - Concepts Site events A site event happens when the sweep line hits a site Property of the line perpendicular to the sweep line at the site event o All points are equidistant to: The site The closest point on the sweep line (the site itself, at this stage) site event What happens when the sweep line moves further away?

Fortunes algo - Concepts What happens The set of points that are equidistant to both the site and the closest point on the sweep line degenerate from a line to form a parabola... The parabola defines the set of points... Set of points that belong to this sites voronoi cell Important property implying that the cross section of such parabolas, from a number of sites will define the edges of a voronoi cell.. Why are we doing this?

Fortunes algo - Concepts 2 sites - Crossing parabolas p 1 and p 2 are two sites At the site event of p 2, point q is equidistant from p 1 and p 2 As the sweep line goes down, the intersection of the parabolas, breakpoints, are the endpoints for a partial edge... P1P1 P2P2 P1P1 P 2 site event q P1P1 breakpoints part of voronoi diagrams edge

Fortunes algo - Concepts More sites & beach line This dividing line is termed the beach line The algorithm for computing the Voronoi diagram of a set of points depends entirely on how this beach line changes as the sweep moves through the space. The beach lines topology changes when a new arc is added or another is absorbed. breakpoints parts of the edges of the voronoi diagrams... What is the stopping condition for an edges growth?

One endpoint is formed with a circle event o a vertex in the voronoi diagram. Each relating site is equidistant from the vertex. Vertex is the centre of a circle on which the three points lie Either by cross sections of parabolas or simply by using the intersection of each or the perpendicular bisector of the inner triangle Fortunes algo - Concepts Circle event... vertex circle event

Fortunes algo - Concepts From the vertex... vertex q If a circumcircle is empty in its interior then, in a Voronoi diagram: a, b, c would be Voronoi sites q would be a Voronoi vertex The perpendicular bisectors of abc would be Voronoi edges. a b c Now from the vertex...

Fortunes algo Implementation Aspects... Parabolas defining beach line not computed as sweep line moves through the problem space o Computationally expensive and unnecessary o Calculations required only when the beach line changes topology Site events Circle events Sweep line makes discrete steps, rather than a continuous sweep

Fortunes algo Data Structures Priority queue o State of sweep line o Indicates when topology of beach line can change Site events Circle events Doubly connected edge list (DCEL) o State of Voronoi diagram o Stores Vertex records Edge records Face records Balanced binary search tree ( AVL or red-black tree) o State of beach line

Fortunes algo Priority Queue of Events Priority based on their y-coordinates Site events are represented by their coordinates Circle events o Computed on the fly o Represented by coordinates of lowest point of empty circle touching 3 sites o Anticipated and may be false s1 (x1, y1) s2 (x2, y2) s3 (x3, y3) s5 (x5, y5) s4 (x4, y4) c1 (x, y) circle event added to queue

Fortunes algo DCEL VertexCoordinatesIncident edge v1(x 1, y 1 )e12 v2(x 2, y 2 )e23... v8(x 8, y 8 )e87 Vertex records Half edge Source vertex TwinIncident face Prev edge Next edge e23v2e32s3e52e37..... Edge records FaceIncident edge s1e32 s2e56... Face records

Fortunes algo Balanced Binary Search Tree Internal nodes o Represent breakpoints between two arcs (tuple) o Contains pointer to record of edge being traced Leaf nodes o Stores site defining arc on beach line o Contains a pointer to a potential circle event

Fortunes algo Pseudocode 1. Initialise data structures Event queue, Q all site events Binary search tree, T DCEL, D 1. while(queue not empty) 2. event = pop first event from queue 3. process(event, T, D) 4. finish all edges in binary search tree How to process site event? - Add the event and breakpoints to BST - Add new edge in DCEL - Delete false alarms for circle events - Add potential circle event(s) to queue How to process circle event? - Update breakpoints and leaf in BST - Record new vertex - Delete false alarms - Add possible circle event to queue

Fortunes algo Processing Site Event Locate existing arc (if any) that is above the new site o x-coordinate of new site is used for binary search o x-coordinate of each breakpoint along root to leaf path is computed on the fly s5 s4 Q

Fortunes algo Processing Site Event Break the arc o Replace the leaf node with a sub tree representing the new arc and its breakpoints Different arcs can be identified by the same site!!! s4 Q

Fortunes algo Processing Site Event Add new edge record in DCEL Create new face record with pointer to new edge New half-edge record s4 Q

Fortunes algo Processing Site Event Check for potential circle events o Scan for triple of consecutive arcs and determine if breakpoints converge Potential circle event Add potential circle event to to priority queue!!! Store a pointer to circle event in leaf node for s1!!! c1 s4 Q

Fortunes algo Processing Site Event Converging breakpoints may not always yield a circle event o Appearance of a new site before the circle event makes the potential circle non-empty Original circle event becomes a false alarm!

Fortunes algo Processing Circle Event Create new vertex record Add vertex to corresponding edge record Delete disappearing arc s4 Q

Fortunes algo Processing Circle Event Delete disappearing arc s4 Q

Fortunes algo Processing Circle Event s4 Q

Fortunes algo Processing Circle Event New edge being traced by new breakpoint New half-edge record Create new edge record Check new triple edges for potential circle events

Fortunes algo Algorithm Termination When Q is empty, beach line and its breakpoints continue to trace edges o Terminate half-infinite edges via a bounding box

Fortunes algo Complexity Locate leaf representing existing arc above new site o Delete potential circle event from queue Break arc by replacing leaf node with a sub tree representing new arc and break points Add new edge and face records to DCEL Check for potential circle event(s) o Add event to queue if they exist o Store pointer to event in proper leaf Running Time O(log n) O(1) Steps in handling site events O(1) O(log n)

Fortunes algo Complexity Delete disappearing BST leaf node and its associated circle events from event queue Add vertex record in DCEL Create new edge record in DCEL Check for potential circle event(s) Running TimeSteps in handling circle events O(log n) O(1)

Fortunes algo Complexity Time complexity for each event: O(log n) How many events are there? o Number of site events + Number of circle events o How many site events? n o How many circle events? Each circle event corresponds to a vertex 2n - 5 (Eulers formula) False alarms deleted before they are processed Total number of events = 3n - 5 Overall running time: O(n log n) Storage complexity: O(n)

Delaunay Triangulation Li Jing & Lu Bingxin Delaunay and Voronoi Delaunay properties Randomized Algorithm - Idea - Implementation aspects - Pseudocode - Data structure - Complexity (time & space)

Delaunay and Voronoi An intuitive conception General position assumption: no 4 points are co-circular (a) Voronoi diagram (b) Delaunay triangulation

Delaunay and Voronoi Delaunay and Voronoi complexes are dual to each other Dual correspondence Voronoi complexes Delaunay triangulation cells (regions) vertices edges edges vertices faces

Delaunay Triangulation (DT) Whats the difference?

Properties Circumcircle property The circumcircle of any triangle in DT{S) is empty. (It contains no points of S in its interior.) Proof By general position assumption, the degree of all Voronoi vertex is 3, edge vu exists Consider S 4, S 1 S 4 is perpendicular to l and divided half by l. S 4 is outside the circle In fact, one definition of Delaunay Triangulation is: Delaunay Triangulation is a triangulation that circumcircle of each triangle is empty.

Properties Empty circle property Two points are connected by an edge in the Delanuay triangulation. There is an empty circle passing through these two points. Proof Trivial from the circumcircle property. There are a series of circumcircles that pass through S i and S j Note down the centres of these circumcircles until they pass through another point The track is the bound for both cell(S i ) and cell(S j ) S i S j is an edge in the DT(S)

Locally Delaunay edge An edge ab is locally Delaunay if it belongs to only one triangle, or it belongs to two triangles, abc and abd, and d lies outside the circumcircle of abc. If an edge is locally Delaunay,we also call it legal, otherwise illegal.

Delaunay Lemma If every edge in T s is locally Delaunay, then T s is the Delaunay triangulation of S. Proof Let x be an arbitrary point in abc Let abc= 0, A 1, A 2, A k be the sequence of triangles that intersect xp Let d i (p) = |p a i | 2 r i 2 Because the edges along xp are locally delaunay, d 0 (p)> d 1 (p)> > d k (p) d k (p) = 0, so d 0 (p)>0

Edge Flipping Flip all edges in a triangulation until they are all locally Delaunay edges

Randomized Incremental Algorithm Randomized the points as p 1, p 2 , p n Find a sufficiently large triangle that contains P Insert p 1, then p 2... and finally p n suppose we have computed DT(P i-1 ) insert p i which splits a triangle into three perform edge flips until no illegal edge remains we have just computed DT(P i ) Repeat the process until i = n Discard the initial large triangle

First step Find a sufficiently large triangle that contains P

First step

Example insert p

Example split abc into abp, bcp, and acp

Example check edges ab, bc, and ac

Example edge ab is illegal flip it

Example edge ab is flipped into pd edge ad and bd are to be checked edge ad is legal, keep it

Example edge bd is illegal flip it

Example edge bd is flipped into pe edge ed and be are legal, keep them

Example edge bc is illegal flip it

Example edge bc is flipped into fp edge bf and cf are legal, keep them

Example edge ac is illegal flip it

Example edge ac is flipped into pg no more edge to flip: we are done

Pseudocode Algorithm DelaunayTriangulation(P) Input: a suitably shuffled (permuted uniformly at random) set of points P = (p 1, p 2, p 3,. , p n ) Output: DT(P) (* use a global DCEL to store DT(P) *) 1. Find a sufficiently large triangle T(p -3 p -2 p -1 ) containing P 2. for i = 1 to n do 3. Insert(p i ) 4. Endfor 5. Discard the triangle T(p -3 p -2 p -1 ) Algorithm Insert(p) Input: a point p, a set of point P and T = DT(P) Output: DT(P u {p}) 1. Find the triangle T(abc) of DT(P) containing p (* use conflict lists *) 2. Insert edges pa,pb and pc (* update conflict lists *) 3. SwapTest(ab) 4. SwapTest(bc) 5. SwapTest(ca) Algorithm SwapTest(ab) 1. if ab is an edge of the exterior face of DT(P) 2. do return 3. d

Time to update triangulation Consider the i th step of the algorithm P i : the set of the first i points (p 1, p 2, p 3,. , p i ) in the whole point set P, i >3, or the set of points in DT(P i ) Run the i th step backward (deleting a random point p in P i ) Note that each new edge added in DT(P i ) due to the insertion of p is incident to p The total number of edge changes made in the triangulation for the insertion of p is proportional to the degree of p after the insertion is complete.

Time to update triangulation The total degree of the vertices in P i is less than 6i There are at most 3i edges in DT(P i ) (planar graph) The sum of vertex degrees is twice the number of edges in a graph The expected degree of a random point in P i is at most 6 The expected number of edge changes for inserting a random point p is O(1) Summing over all the n steps, the time for updating the triangulation is O(n)

Update conflict lists Each non-inserted point is assigned to a triangle/bucket For a non-inserted point, if the triangle which it lies within is destroyed, we have to find another new triangle containing this non-inserted point The expected time to update conflict lists is the expected time to rebucket non-inserted points is proportional to the expected number of non- inserted points required to be rebucketed

Consider the i th step of the algorithm P i : the set of the first i points (p 1, p 2, p 3, , p i ) in the whole point set P, i >3, or the set of points in DT(P i ) P\P i : the set of non-inserted points (p i+1, p i+2, p i+3, , p n ) Run the i th step backward (deleting a random point in P i ) some triangles in DT(P i ) are destroyed some points in P\P i are required to be rebucketed Rebucket points

Time to rebucket points Assume a random point p in P i is deleted For a random point q in P\P i, suppose that q is bucketed in the triangle T(abc) of DT(P i ) If p is one of the three vertices of T(abc), T(abc) will be destroyed and q will be rebucketed Pr (p is deleted) = 1/i Pr (T(abc) is destroyed) = 3/i Pr (q needs to be rebucketed) = 3/i The expected number of points in P\P i required to be rebucketed is 3(n-i)/i

Time to rebucket points Summing over all the n steps The total expected number of non-inserted points required to be rebucketed is O(nlogn) The expected time to update conflict lists is O(nlogn)

Time complexity Major steps in the algorithm Running time o Find a sufficiently large triangleO(1) o Find the triangle containing a pointO(n) o Update the triangulationO(n) o Update conflict lists O(nlogn)

Complexity Expected time complexity: O(nlogn) Expected space complexity: O(n) DCEL, Conflict list

Trapezoidal Decomposition Dumitrel Loghin & Suhendry Effendy Concepts Randomized Algorithm Motivation - Point Location Complexity Analysis

Defining the problem Given a set S of n segments in the plane, with no two distinct end-points having the same x coordinate (general position) These segments intersect in k points There is a bounding rectangle R containing all n segments If we draw vertical lines through each end-point or intersection point, till they intersect a segment or R, we obtain a set of trapezoids T(S) which is the trapezoidal decomposition of S

Example p1p1 p2p2 p3p3 q1q1 q2q2 q3q3 R k1k1 k2k2 k3k3

Details

Construction Example

Randomized Algorithm Start with a random permutation S = { s 1, , s n }, with a bounding rectangle R and with T(S)= FOR i = 1 TO n DO # add segment s i and update T(S) Get the trapezoid which contains left end-point of s i and draw a vertical line. Move to the right on the planar graph and get all the intersected trapezoids: If the segment intersect the bottom or top segment of the trapezoid, the point is one of the k intersection points, hence, draw a vertical line. Draw a vertical line at the right end-point. Rescan the intersected vertical lines and trim them based on their starting point; merge the trapezoids which share the deleted line.

Point Location Given a query point q, find in which trapezoid it lies Data structure DAG internal nodes are end-points, intersection points or segments leaves are trapezoids and they may be shared (in-degree of a leaf may be more than one) Can be constructed simultaneously with trapezoidal decomposition Algorithm key idea: At step i, some of the leaves (trapezoids) become internal nodes (end-points of the segment s i or intersection points) and new trapezoids are added.

Point Location Example s1s1 s2s2 p1p1 q2q2 q1q1 p2p2 11 44 33 22 55 66 77 88 99 10 k1k1 q

Point Search in DAG p1p1 q1q1 p2p2 11 44 s1s1 q2q2 10 k1k1 k1k1 33 s2s2 99 33 s2s2 66 55 88 77 s1s1 s2s2 p1p1 q2q2 q1q1 p2p2 11 44 33 22 55 66 77 88 99 k1k1 q

Complexity Analysis Assumption: points (end of segment) are in general position = no two points lies in a same vertical line. = all x-coordinate are different. Later we will see how to lift this assumption.

Complexity Analysis What is the complexity of adding one segment s? find the left end point of segment s. trace (and update) all trapezoids intersected by s.

Adding 1 Segment

How to find this trapezoid?

Finding Trapezoid How to find trapezoid which contain the left end-point of segment s? TrapezoidSegment T1T1 s 5, s 7,... T2T2 s 8,... T3T3 s 6, s 9, s 10,...... Segment Trapezoid s5s5 T1T1 s6s6 T3T3 s7s7 T1T1 s8s8 T2T2 s9s9 T3T3 s 10 T3T3 bi-directional pointer couple of ways

Bi-directional pointer bi-directional pointer query find in O(1) need to update the pointer for each changed trapezoid (only update trapezoids which are intersected by new segment s).

Adding 1 Segment

Complexity Analysis For each segment s, we need to: 1.find the left end-point of s. 2.trace intersected trapezoids. 3.update the trapezoids. 4.update the bi-directional pointers. For each trapezoid:Overall: t(f) : the complexity of trapezoid f. p(f) : update the bi-directional pointer for trapezoid f.

Backward Analysis Imagine the algorithm run backwards, deleting segment one at a time. When we delete a segment s from H i, only trapezoids in H i which adjacent to s will be affected.

Complexity Analysis Since we insert (or delete in backward analysis) segment s in random order, then every remaining segment is equally likely to be chosen. E(i): the expected complexity of deleting the i th segment. (in backward analysis)

Observation Each trapezoid is adjacent to at most 4 segments. (using general position assumption)

Observation Trapezoid can have an arbitrary number of adjacent segments in non general position. We will deal with this case later.

Complexity Analysis Each trapezoid is adjacent to at most 4 segments. = Each trapezoid appears in at most 4 segments adjacency list

Complexity Analysis = all the left endpoints of segment that have yet to be added (or have been deleted in backward analysis).

Complexity Analysis =proportional to the number of vertices in all decomposition k = the number of intersection

Complexity Analysis

Randomized vs. Deterministic

Non General Position How to handle non general position? Rotation or Transformation Shear Transformation x = x + y

Summary Voronoi Diagrams -- Use Fortune Algorithm Delaunay Triangulation -- Randomized Incremental Construction Dual of Voronoi Trapezoidal Decomposition -- Randomized Incremental Construction

Conclusion Widely used in various other areas. We use it sometimes without even realising it. Lot of potential of further development. Numerous interesting open problems. http://compgeom.cs.uiuc.edu/~jeffe/open/

References Voronoi Derek Johns, "An Optimal Algorithm for Computing 2D Voronoi Diagrams Fortune's Sweep Algorithm", Available at http://cgm.cs.mcgill.ca/~mcleish/644/Projects/DerekJohns/Sweep.htm [Accessed February 2014] Dheeraj Kumar Singh, "Lecture 20: Voronoi Diagrams and Fortunes Algorithm", Available at http://intinno.iitkgp.ernet.in/courses/91/wfiles/37906 [Accessed February 2014] Voronoi Diagrams, Available at http://ima.udg.edu/~sellares/ComGeo/Vor2D_1.ppt [Accessed February 2014] Delaunay Triangulation http://www.cs.umd.edu/~mount/754/Lects/754lects.pdf http://www.cs.uu.nl/geobook/interpolation.pdf http://www.comp.nus.edu.sg/~hcheng/academic/courses/cs5237/notes/04.pdf http://www.comp.nus.edu.sg/~hcheng/academic/courses/cs5237/notes/05.pdf http://www.comp.nus.edu.sg/~tantc/ioi_training/CG/l9cs4235.pdf http://www.comp.nus.edu.sg/~tantc/ioi_training/CG/l10cs4235.pdf http://groups.csail.mit.edu/graphics/classes/6.838/F01/lectures/Delaunay/Delaunay2D.ppt Trapezoidal Decomposition Rajeev Motwani, Prabhakar Raghavan, Randomized Algorithms, 1995 Subhash Suri, Point Location, Available at http://www.cs.ucsb.edu/~suri/cs235/Location.pdf

Geometric Algorithms Suman Sourav Paramasiven Appavoo Anuja Meetoo Appavoo Li Jing Lu Bingxin...

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