Dynamic Algorithms for Planar Point Location

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Dynamic Algorithms for Planar Point Location
Kanat Tangwongsan in collaboration with Guy
Blelloch, Umut Acar, Srinath Sridhar.
Aladdin Center - Summer 2004
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Planar Point Location

• Classical geometric retrieval problem.
• Subdivide a Euclidean plane into polygons by line
  segments. (These segments intersect only at
  their endpoints).
• Identify which polygon the query point (red spot)
  is in.

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Static vs Dynamic

• Static fixed structure
• Dynamic possible insertion/deletion of segments

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Motivation

• Applications of Point Location
• Geographic Information Systems (GIS)
• User Interface (Mouse/Windows relation)
• Graphics and CAD/CAM
• etc.
• Why dynamic?
• Interesting
• Heavy use in circuit layout, computer graphics,
  etc.

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Some Previous Work

• Many solutions to this problem
• Kirkpatrick 1983 Edelsbrunner, Guibas 1986
• Overmars 1985 Sarnak, Tarjan 1986, etc.
• Various approaches and techniques persistent
  trees, segment trees, etc.
• See Chiang and Tamassias Dynamic Algorithms in
  Computational Geometry for pointers to these.
• Close look at Sarnak-Tarjan solution.

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Sarnak-Tarjan Solution

• Idea (partial) persistence
• O(log n)-query-time, O(n)-space solution.
• Static unable to insert/delete segments.
• Relies on Coles observation and Dobkin-Lipton
  construction.

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Dobkin-Lipton Construction

• Draw a vertical line through each vertex,
  splitting the plane into vertical slabs.

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• Locate a point with two binary searches - O(log
  n)-query time.

• Nice but space inefficient can cause O(n2).

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• ?(n) segments in each slabs, and ?(n) slabs.

• If storing a tree for each slab, order n2 space
  usage.

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Coles Observation

• Sets of line segments intersecting contiguous
  slabs are similar.

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• Reduces the problem to storing a persistent
  sorted set.

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Some Notions of Persistency

• Ephemeral data structure cannot go back in time.
• Persistent data structure keeps all previous
  versions accesible.
• Partial persistence
• Various solutions see e.g. Driscoll et al.,
  Overmars, etc.
• Trees
• path-copying is simple and powerful
• Native to functional programming languages e.g.
  ML, Scheme, etc.

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Path-copying illustrated
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Version Dictionary
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Friendly Dynamic/Stability 101

• What is a Dynamic Algorithm?
• Maintains its input/output relationship as the
  input changes.
• Some Words of the Day
• to dynamize an algorithm is to make it dynamic.
• dynamization is the process of making an
  algorithm dynamic.
• stable a change to the input does not change
  the execution trace (function call tree) too
  dramatically.

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Dynamizing Persistent Data Structure

• Acar et al (SODA 2004) shows a way to dynamize
  static algorithms.
• The efficiency depends on stability.

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Digression 1 Skip Lists

• Traditional linked-list is bad for look-up
  O(n).
• An additional layer saves a lot of probing.
• Achieve O(log n) by adding a few more layers.
• Say with n/4, n/8 and so forth nodes.
• Sadly, maintaining this perfect structure is hard.

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Digression 1 Skip Lists

• Pugh1990 invented skip lists.

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• The level of a node is drawn from the geometric
  distribution i.e. a node has level k -1 with
  probability 1/2k
• Skip Lists expected O(n)-space, and expected
  O(log n) insert/delete/look-up.

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Our Unified Goals

• Dynamize Sarnak-Tarjan
• use the idea of persistent data structure and be
  able to support insertion/deletion/lookup
  on-the-fly.
• stability is the key to efficiency.
• Treaps and skip lists looks promising.
• path-copying as the starting point.
• Benchmark against existing techniques.

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Questions?
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Thank you!