Kinetic Algorithms

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
Introduction KDS?

• 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
Introduction The Big Picture

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

7
Introduction Flight 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
Introduction How 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
Introduction Timeline

• 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.

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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

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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

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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
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

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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.

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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.

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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.

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Solution 3Heap

• For an event we may interchange parent and child

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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.

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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.

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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

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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 ypxq

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2D Convex Hull

• Goal Maintain the upper envelope of the set of
  lines

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2D Convex Hull

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

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2D Convex Hull Kinetisation

• 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.

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2D Convex Hull Operators

• ltx X value comparison
• lty Y value comparison
• lts Slope comparison
• Ce(ab) Contender edge
• Color of the vertex

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2D Convex Hull Certificates
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2-D Convex Hull Certificates
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2D Convex Hull Example
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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.

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2D Convex Hull Proof

• 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.

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2D Convex Hull Proof

• We can show that the vertices without
  y-certificates can not be placed differently in C
  and C

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2D Convex Hull

• Certificate updates
• Lemma 2.2
• The following procedure correctly updates the
  certificate list when the configuration events
  happen

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2D Convex Hull Analysis

• 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

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2D Convex Hull Demo

• Demo

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Closest Pair in 2D Problem 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.

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Closest Pair in 2D Static 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

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Closest Pair in 2D Definitions

• 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)

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Closest Pair in 2D Problem 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.

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Closest Pair in 2D Contradictions
Lemma 3.1
Lemma 3.2
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Closest Pair in 2D Algorithm

• The plane sweep algorithm performs a set of
  operation three times, for S rotated by 0 and
  60º.

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Closest Pair in 2D Algorithm

• 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 2D Algorithm

• Analysis
• Sorting
• Compute Cands(p)
• Finding lcand(p) in the worst case
• Splitting and inserting
• Total

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Closest Pair in 2D Kinetisation

• 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.

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Closest Pair in 2D Events

• Three types of event
• Change of order in x
• Change of order in 60º

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Closest Pair in 2D Updates
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Closest Pair in 2D Updates
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Closest Pair in 2D Analysis

• The KDS for the closest pair problem is
  efficient, responsive, compact and local.
• Efficient The number of events process by the
  KDS is a logarithmic factor more then the maximum
  number of external events.

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Closest Pair in 2D Analysis

• Compact
• There is events in the queue at any time,
  for the certificates and for the kinetic
  tournament.
• Local
• Each point belongs to ar most six certificates
  and to at most three pair
• and in pair participate to at most
  events in the kinetic tournament

53
Closest Pair in 2D Demo

• Demo

54
Summary

• The way it works

55
Further Issues

• Avoid the problem related to the finite precision
  arithmetic. (sequencing of the events)
• Change the definition of efficiency to make it
  proportional to the number if external events
• Adapt the algorithm to real time computation

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References

• J. Basch and J.Guilbas. Data Structures for
  Mobile Data. J. of Algorithms 31, 1-28 (1999).
• J. Basch and al. Data Structures for Mobile Data.
  SODA 1997, 756-767.
• http//graphics.stanford.edu