Visibility Graph

1. Visibility Graph Team 10 NakWon Lee, Dongwoo Kim

2. Robot Motion Planning • Consider the case of point robot • The polygons in S are obstacles, and their total number of edges is denoted by n • The point robot can touch obstacles, because obstacles are open set. • Start position . Goal position . • Paths do not intersect the interior of any of the obstacles.

3. Robot Motion Planning • Use trapezoidal map of the free configuration space . • Configuration space is the parameter space of a robot . Denoted by . • For a point robot, was simply the empty space between obstacles.

4. Configuration Space

5. Shortest Paths • Construct the trapezoidal map for configuration space . • Construct the load map . • Find the trapezoids which have the start point or end point. • Find the shortest route in the load map using breadth first search algorithm.

6. Shortest Paths

7. Shortest Paths • Shortest path in road map is not a real shortest path • because some arcs are btw nodes that are far apart, whereas others are btw nodes that are close to each other. • It is just minimum number of hop. • To improve this problem, give each arc a weight corresponding to the Euclidean length, and use graph search algorithm that find the shortest path in a weighted graph. • Dijkstra's algorithm • But, it is stillnot ashortest path.

8. Shortest Paths • Think of this path as an elastic rubber band. • Fix the endpoints at the start and goal position. • Try to tighten the rubber band. • It will be stopped by the obstacles. • The new path will follow parts of the obstacle boundaries and straight line segments through open space.

9. Shortest Paths • Lemma 15.1 Any shortest path between and among a set of disjoint polygonal obstacles is a polygonal path whose inner vertices are vertices of • Inner vertex – A vertex that is not the begin or endpoint of the path.

10. Shortest Paths • Proof. • Suppose for a contradiction that a shortest path is not polygonal. • Point on that lies in the interior of the free space with the property that no line segment containing is contained in .

11. Shortest Paths • There is a disc of positive radius centered at that is completely contained in the free space. • Part of inside the disk, which is not a straight line segment, can be shortened by replacing it with the segment connecting the points which lie on circle. • Contradict the theorem “any shortest path must be locally shortest”.

12. Shortest Paths • Now consider a vertex on . It cannot lies in the interior of the free space. • Consider the disc centered at such that half of the disc is contained in the free space. • Can replace the sub path inside the disc with a straight line segment. • The only possibility left is that is an obstacle vertex. short cut

13. Shortest Paths • Construct a road map with this characterization. • This road map is visibility graph of , denoted by . • Its nodes are the vertices of . • There is an edge between vertices and if they can see each other. • The segment dose not intersect the interior of any obstacle in .

14. Shortest Paths • Two vertices that can see each other are called visible. • The segment connecting visible two vertices is called a visibility edge. • Endpoint of the same obstacle edge always see each other. • Hence, the obstacle edges form a subset of the edges of .

15. Shortest Paths • To make a complete road map to find shortest path, add start and goal point as vertices to . • So we consider the visibility graph of the set . • Corollary 15.2 The shortest path between and among a set of disjoint polygonal obstacles consists of arcs of the visibility graph , where .

16. Shortest Paths

17. Visibility Graph Definition: • The visibility graph of s and t and the obstacle set is a graph whose vertices are s and t the obstacle vertices, and vertices v and w are joined by an edge if v and w are either mutually visible or if (v, w) is an edge of some obstacle.

18. Visibility Graph

20. Visible Vertices

21. Visible Vertices

22. Visible Definition Two points p and q are mutually visible if the open line segment joining them doesn't intersect the interior of any obstacle.

24. Search Tree

25. Computing the Visibility Graph • VisibleVertices: – (sorting) • Operation on blanced search tree : • For every vertices with Visible Vertices. • VisibilityGraph –

26. Computingthe Visibility Graph • Give an procedure for constructing. • is the number of line segments in the plane. • No three vertices are collinear?? • The algorithm is not output sensitive. • An algorithm ( is the number of edges in visibility graph) is existent but quite complicated.

27. Computingthe Visibility Graph • The text’s algorithm operates by doing sweep one vertex at a time. • New algorithm does the sweep for all vertices simultaneously. • When we use the arrangement, angular sort can be performed for all vertices in time. • If we build the entire arrangement, this sorting algorithm will involve space. • However it can be implemented in space using an algorithm called topological plane sweep.

28. Computingthe Visibility Graph • First, recall the algorithm for computing trapezoidal maps. • Shoot a bullet up and down from every vertex until it hits its first line segment. • Angle continuously sweeps out all slopes from to . • All bullet lines attached to all the vertices begin to turn slowly counterclockwise.

29. Computingthe Visibility Graph • The question is what are the significant event points, and what happens with each event?

30. Computingthe Visibility Graph • It is useful to view the problem both in its primal and dual form. • ( is the number of segments) segment end points , its dual line . • Significant event occurs whenever a bullet path in the primal plane jumps from one line segment to another. • This occurs when reaches the slope of the line joining two visible endpoints and .

31. Computingthe Visibility Graph event event event

32. Computingthe Visibility Graph • To keep track of which endpoints are visible and which are not is complicated. • Instead we will take events to be all angles between two endpoints, whether they are visible or not. • By duality, the slope of such an event will correspond to the -coordinate of the intersection of dual lines and in the dual arrangement.

33. Dual Arrangement Dual of point D x-coordinate is the slope of dual segment Dual of point C

34. Find Events • By sweeping the arrangement of the dual lines from left-to-right, we will enumerate all the slope events in angular order.

35. Find Events Angular order • By using topological plane sweep, event do not need to be sorted.

36. Topological Plane Sweep • Provides a way to sweep an arrangement of lines using a “flexible” sweeping line. • Because events do not need to be sorted, we can avoid factor.

37. Topological Plane Sweep • Upper horizon tree • Lower horizon tree

38. Topological Plane Sweep • Data structure • – array of line equation. , if the th line of , , is . • – an array representing the upper horizon tree. is a pair of indices indicating the lines that delimit the segment of in the upper horizon tree to the left and to the right. • – represents the lower horizon tree. • – a set of integers, represented as a stack. If is in , then and share a common right endpoint. • – an array holding the current sequence of indices that form the lines of the cut. • – a list of pairs of indices indicating the lines delimiting each edge of the cut. thus encodes the endpoints of the edge on . The same convention as that above is used for missing endpoints.

39. Topological Plane Sweep If segment is the rightmost on , set If segment is the leftmost on , set

40. Topological Plane Sweep • Algorithm Initialization: • 1. Sort the lines of the arrangement by slope. • 2. Find the leftmost and the rightmost intersection point of the lines. Let the two points be and . • 3. Create vertical lines , as left boundary and right boundary. Determine the intersection points of lines with the boundaries. • 4. Create upper horizon tree: Insert . Assume have been inserted. These lines form an upper bay. To insert , begin at its endpoint on the left boundary. Walk in counter clockwise order around the bay till we find the intersection point of with an edge. • 5. Create lower horizon tree similarly by starting the travers at endpoints on the right boundary. • 6. Initialize : Let and . If intersects to the left of the intersection point of and then the right delimiting line of is . Otherwise, the right delimiting line of is . • 7. Initialize by scanning .

41. Topological Plane Sweep • Algorithm Elementary Step • While • 1. Pop from • 2. Swap , /*lines are going to cross, after the elementary step*/ • 3. , /*the point of elementary step becomes the left endpoint of the two new cut edges */ • 4. Update , . • 5. , /* find the new right endpoints */ • 6. If then push into . If then push into . /* push valid vertices formed after sweeping into I if there is any */

42. What Happens at Each Event • For each vertex , there are two bullet paths growing from along the line with slope . • Let and denote the line segments that forward bullet path and backward bullet path hit. • If either path does not hit any segment then we store a special null value. • Each slope is determined by exactly two lines.

43. What Happens at Each Event • Possible scenarios

44. What Happens at Each Event • Same segment • If and are endpoints of the same segment, then they are visible, and we add the edge to the visibility graph. • How can I check this?

45. What Happens at Each Event • Invisible • Consider the distance from to . • Compute the contact point of the bullet path shot from in direction with segment or . • If is longer than , and are not visible. Calculate the distance of Calculate the distance of

46. What Happens at Each Event • Segment entry • If we are entering the segment, then we set or to this segment. • can be a visibility edge.

47. What Happens at Each Event • Segment exit • The bullet path will need to shoot out and find the next segment that it hits. • It is available in time. • Since we are sweeping over at the same time that we are sweeping over . • We know that the bullet extension from hits . • So, new is same as .

48. Thank You Q&A