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Kinetic Algorithms

Kinetic Algorithms. Data Structure for Mobile Data. Plan. Motivation Introduction Illustrative example Applications 2-D Convex Hull Closest pair in 2-D Others Issues. Motivation. Maintain an attribute of interest in a system of geometric objects undergoing continuous motion.

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Kinetic Algorithms

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  1. Kinetic Algorithms Data Structure for Mobile Data

  2. Plan • Motivation • Introduction • Illustrative example • Applications • 2-D Convex Hull • Closest pair in 2-D • Others Issues

  3. Motivation • Maintain an attribute of interest in a system of geometric objects undergoing continuous motion. • Take advantage of the coherence present in continuous motion to process a minimal number of combinatorial event.

  4. Introduction • Problems such as convex hull and closest pair maintenance have been extensively studied: • Static objects • Insertion and deletion operations • Wasteful of computation • Compute form scratch

  5. IntroductionKDS? • KDS: Kinetic Data Structure • Process of discrete events associated with continuously changing data. • Kinetisation • Process of transforming an algorithm on static data into a data structure which is valid for continuously changing data.

  6. Sweep line Time IntroductionThe Big Picture • Resemble to a sweep-line/plane algorithm • Use of a global event queue as interface between KDS and object in motion.

  7. IntroductionFlight plan • Use of flight plans • The motion of the objects is known in advance • BUT: • Can be imprecise (use of intervals) • Can be updated

  8. B A B C A C A C B IntroductionHow does it work? • The correctness of whatever configuration can be guaranteed with a conjunction of low-degree algebraic conditions involving a bounded number of objects each. To Collinear Inverse

  9. Introduction Certificates • A certificates will guarantee the correctness of any configurations

  10. IntroductionTimeline • Event queue contains KDS events corresponding to times (precise / interval) when certificates might change sign. • IF ( flight plan of O updated ) THEN All the certificates involving O must be recalculated and updated.

  11. Certificates Time Update Recapitulation • Attribute certificates for any configurations • Keep track of the change in the certificates and update when necessary

  12. Definitions • Certificates • Guarantee the correctness • Complexity • Number of points moving in a system. • Small cost • If it running time of the order or

  13. Definitions • Internal events (IE) • Events process by the structure for its internal needs • External events (EE) • Events affecting the configuration we are maintaining. • Lower bound for IE

  14. Definitions • Efficiency • If the ratio IE / EE is small. • Range: 1 to Infinity • Responsiveness • Worst-case cost of processing a certificate • Size • Maximum number of events it needs to schedule in the event queue

  15. Definitions • Compact • If size linear to the number of moving objects • Local • If the maximum number of events in the event queue that depend on a single object is small

  16. Y Illustrative example • Consider the following1D situation • Given a set of n points moving continuously along the y-axis, we are interested to know which is the topmost point. • Constant speed • Arbitrary initial configuration

  17. Solution 1Trivial Case • Draw the lines in the ty-plane • Compute the upper envelope of the set of lines. • Efficient algorithm but does not support update of the flight plan.

  18. Solution 2Schedule-Deschedule • Maintain, on-line, the sorted order along the y-axis • Schedule event that is the first time when two consecutive points cross. • Destroy and create 2 adjacencies • Schedule and deschedule up to 2 new events • Not efficient: process up to events.

  19. Solution 3Heap • For each link in the heap, a certificate guarantees that the child point is below the parent point. • We associate an event at the time these points meet.

  20. Solution 3Heap • For an event we may interchange parent and child

  21. Solution 4Kinetic Tournament • Consider a divide-and-conquer algorithm, like a tournament from bottom to up for computing a global leader. comparisons • As long as each of the comparison remain valid the identity of the maximum remains valid.

  22. Solution 4Kinetic Tournament • Imagine that a particular comparison involved flip and precolated up the tree. • For a balanced tree the computation is made in time and can affect at most certificates.

  23. Solution 4Kinetic Tournament • For points with constant speed, similar to the computation of the upper-envelope in the ty-plane. • In the worst case we have new leaders, each computed in time, for a total worst case running time of

  24. Solution 4Kinetic Tournament • KDS • efficient • compact • local Winner

  25. 2D Convex Hull • Problem • Maintain the convex hull of a set of moving points in 2D. • Dual form • Point (p,q) to the line y=px+q

  26. 2D Convex Hull • Goal: Maintain the upper envelope of the set of lines

  27. 2D Convex Hull • Perform a kinetic tournament. • From a red and a blue chain maintain the purple upper envelope of the 2 chains

  28. 2D Convex HullKinetisation • We keep a record of the entire computation in a balanced binary tree • Each node is in charge of maintaining the upper envelope of two upper envelopes computed by its children • If a event creates a change, the event is process through the tree.

  29. 2D Convex HullOperators • <x X value comparison • <y Y value comparison • <s Slope comparison • Ce(ab) Contender edge • Color of the vertex

  30. 2D Convex HullCertificates

  31. 2-D Convex HullCertificates

  32. 2D Convex HullExample

  33. 2-D Convex Hull • Lemma 2.1 Consider a configuration C of two convex piecewise linear functions and the certificate list L for their upper envelope as defined earlier. Let C’ be a configuration for which all the certificates of L hold. Then the upper envelope of C’ has the same combinatorial description as that of C.

  34. 2D Convex HullProof • The x-certificates prove the correctness of the contender edge pointer. • Any vertex that has a y-certificate in L is also guaranteed to be placed in C’ and in C.

  35. 2D Convex HullProof • We can show that the vertices without y-certificates can not be placed differently in C and C’

  36. Certificate updates Lemma 2.2 The following procedure correctly updates the certificate list when the configuration events happen 2D Convex Hull

  37. 2D Convex HullAnalysis • Theorem 2.3 • The KDS for maintaining the convex hull is efficient, responsive, compact and local • Proof • Responsive: • Compact : • Local: • Efficient: in the worst case

  38. 2D Convex Hull Demo • Demo

  39. Closest Pair in 2DProblem definition • Given a set S of n moving points find the two closest points in S. • Traditional static algorithms are ill-adapted for the kinematisation.

  40. Closest Pair in 2DStatic Algorithm • We divide the space around each vertex into six 60º wedges, the nearest neighbor of each point is the closest of the nearest neighbor in the six wedges. • ADD PICTURE

  41. Closest Pair in 2DDefinitions • Dom(p): The right extending wedge that make ±30º angles with the x axis. • Circ(p, r): The circle of radius r centered at p. • We denote the closest pair in S, (a,b)

  42. Closest Pair in 2DProblem definition • Lemma 3.1 • Point b is not contained in Dom(p) for any third point p with a to the left of b. • Lemma 3.2 • The leftmost point in Dom(a) is b.

  43. Closest Pair in 2D Contradictions Lemma 3.1 Lemma 3.2

  44. Closest Pair in 2DAlgorithm • The plane sweep algorithm performs a set of operation three times, for S rotated by 0 and ±60º.

  45. Closest Pair in 2DAlgorithm • For each point p in S from right to left • Set Cands(p) = Maxima Dom(p) • Set lcand(p) to be the leftmost element of cands(p) • Delete points of Cands(p) from Maxima • Insert p into Maxima at its proper place in y-order • Repeat for the three directions

  46. Closest Pair in 2DAlgorithm • Analysis • Sorting: • Compute Cands(p): • Finding lcand(p): in the worst case • Splitting and inserting: • Total:

  47. Closest Pair in 2DKinetisation • Certificates • The projection of the point on x for S rotated by 0 and ±60º. • Each point belong to a maximum of six certificates, involving is two neighbors in each of the three sorted order.

  48. Closest Pair in 2DEvents • Three types of event: • Change of order in x • Change of order in ±60º

  49. Closest Pair in 2DUpdates

  50. Closest Pair in 2DUpdates

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