Introduction to Geomtric Computing

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Introduction to Geomtric Computing

• D. T. Lee
• Institute of Information Science
• Academia Sinica
• October 14, 2002

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What is Computer Science about?

• Not about programming
• But about problem solving
• Basic disciplinary areas include
• Software Programming-related
• Hardware-related
• Computer systems
• Theory
• Mathematics of computer science
• Applications
• etc.

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MinMax Problem 4

• MinMax Problem
• Input A set S of N numbers on the real line
• Output The Minimum and Maximum of S
• Simple method Finding the min. or max. of N
  numbers requires N-1 comparisons.
• Reduce MinMax to 2 Min or Max problems!

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MinMax Problem 3
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MinMax Problem 2

• But finding both Maximun and Minimum can be done
  with ? 3N/2 ? ? 2 comparisons

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MinMax Problem 1
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Sorting Problem

• Sorting N numbers
• Simple method find Min repeatedly
• (N-1) (N-2) 2 1 (N-1)N/2
• Selection method
• Smallest can be found in (N-1) comparisons
• 2nd smallest found in log2 N additional comp.
• 3rd smallest found in log2 N additional comp.
  

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Knockout Tournament
min
Total number of matches for 8 competitors is
7 N/2 N/4 4 2 1 N-1
Total number of rounds is log2 N
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Geometric Computing

• Design and Analysis of Algorithms for geometric
  problems
• Theory Computational Complexities
• Time, Space, Degree
• Lower and Upper Bounds of Computation
• Applications
• VLSI CAD, Computer-Aided Geometric Design,
  Computer Graphics, Operations Research, Pattern
  Recognition, Robotics, Geographical Information
  Systems, Statistics, Cartography, Metrology, etc.

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Theory of Computation

• Models of Computation
• Real arithmetic, RAM
• Computation Tree Model - Unit Cost
• Off-line, On-line, Real-time
• Parallel Computation vs Serial Computation
• Problem Complexity vs Algorithm Complexity
• Decision vs Computation Problems
• Output Sensitive Algorithms

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Checking if a Polygon is Convex 4

• A polygon P is represented by a sequence of
  points v0,v1,v2, ,vn-1 where (vi, vi1) is an
  edge, i0,1,..,n-1, vn v0.
• Theorem A polygon P is convex if segments
  connecting any two points inside the polygon
  always lie inside P.

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Checking if a Polygon is Convex 3
Problem There are infinitely many
possible Segments to check!
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Checking if a Polygon is Convex 2

• Theorem A polygon is convex if every diagonal
  lies inside the polygon.
• It takes O(N3) time to check O(N2) diagonals
  against every edge of P.
• Very inefficient!
• Theorem A polygon is convex if every interior
  angle of each vertex is less than 180o.

• This result would imply an O(N) time algorithm!

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Checking if a Polygon is Convex1
Consider three vertices vi,vj,vk and determinant
where (xi yi ) are the x- and y-coordinates
of vi
det(vi,vj,vk ) gt 0 when vi,vj,vk form a left turn
det(vi,vj,vk ) 0 when vi,vj,vk are collinear
det(vi,vj,vk ) lt 0 when vi,vj,vk form a right turn
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Checking if a Polygon is Simple
v0
v3
v1
v5
v4
v2
How do you check if the input polygon is simple??
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Art Gallery Problem

• Consider an art gallery room whose floor plan is
  modeled by a simple polygon with N sides.
• How many stationary guards are needed to guard
  the room?
• Assumption Each guard is considered to have 360o
  unlimited visibility.
• Two points p and q are said to be visible to each
  other, if the line segment (p,q) does not
  intersect any edge of the polygon.

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Art Gallery Problem
p and q are visible
r and q are not visible
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Art Gallery Problem
Visibility polygon of q r
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Art Gallery Theorem

• Theorem Given a simple polygon with N sides,
  stationary guards are sufficient, and
  sometimes necessary to cover the entire polygon.

Theorem Given a simple polygon with N sides, the
problem of finding a minimum number of stationary
guards to cover the entire polygon is
intractable.
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Lower Bound of Art Gallery Problem
A comb with 8 prongs (N24) requires ?N/3? 8
guards
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Upper Bound of Art Gallery Problem
Triangulation of a Simple Polygon
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Coloring of a Graph

• Coloring of a graph an assignment of colors to
  nodes of the graph such that no two adjacent
  nodes are assigned the same color.
• Chromatic number The minimum number of colors
  required to color a graph.
• Theorem Every triangulation of a simple polygon
  is 3-colorable.

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Art Gallery Problem Example
Select red color to place guards 3 ?
?N/3? 4
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Art Gallery Theorem

• Theorem In a 3-coloring of a triangulation of a
  simple polygon with N vertices, there exists a
  color that is used no more than ?N/3? times.
• Proof Pigeon-hole principle
• If n pigeons are placed into k pigeon-holes, at
  least one of the holes must have no more than n/k
  pigeons.
• Since N is an integer, there exists at least one
  color that is used in no more than ?N/3? times.

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Variations of Art Gallery Problem

• Shapes of Art Gallery
• Rectilinear polygon
• Polygon with holes
• Capabilities of Guard
• Mobile guards moving along a track
• Guards with limited visibility (angle and
  distance)

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Rectilinear Polygon
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Polygons with Holes
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Guard Capabilities
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Map Coloring 2

• Given a planar graph (planar map), find a
  coloring of the regions so that no two adjacent
  regions are assigned the same color.
• How many colors are sufficient?
• 4-color theorem!
• Find an algorithm that colors the map.

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Map Coloring 1
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Proximity

• Closest Pair Problem Consider N points in the
  plane, find the pair of points whose Euclidean
  distance is the smallest.
• Brute force Enumerate all N2 pairwise distances
  and find the minimum.
• Can one do better?

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Closet Pair on the Real Line

• Consider N numbers x0, x1, x2, , xn-1 on the
  real line, find the two numbers xi and xj such
  that xi xj is the smallest.
• Lemma The two numbers xi and xj such that xi
  xj is the smallest must be adjacent to each
  other, when these numbers are sorted.

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Closest Pair in the Plane
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Voronoi Diagram
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Closest Pair Problem

• Theorem Given N points in k-dimensional space,
  the closest pair of points can be determined in
  O(N log N) time, which is asymptotically optimal.

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Geometric Optimization

• Shortest Path Problem
• Given a polygonal domain (obstacles), and two
  points s and t, find a shortest path between s
  and t avoiding these obstacles.
• Homotopic Routing
• Given a layout and the homotopy of k 2-terminal
  nets, find a detailed routing of minimum wire
  length.

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Homotopic Routing

• Input A sketch S(F, M, W), a planar drawing of
  a physical layout of modules M in the grid and a
  finite set W of nets connecting point features F,
  each represented by a vertex-disjoint simple
  path.
• Output A detailed routing on the grid without
  overlapping of these nets of minimum total length
  among those that are homotopic to the given
  sketch.
• Net p is homotopic to net q, if they have the
  same terminals, and one can be transformed
  continuously into the other without crossing
  features.

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After Compaction
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Topological Via Minimization 4

• Given a set of n 2-terminal nets in a two-layer
  bounded routing region, find a layer assignment
  of wire segments so as to minimize the number of
  vias used.

layer1
layer2
via
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Topological Via Minimization 3
of Vias 4
of Vias 1
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Topological Via Minimization 2
of Vias 1
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Topological Via Minimization 1

• Lemma There exists an optimal solution to the
  two-layer bounded region TVM problem such that
  each net uses at most one via.

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Circle Graph 2

• Definition A set of chords is said to be
  2-independent if it can be partitioned into two
  subsets of independent chords

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Circle Graph 1

• Theorem Given n chords in a circle and an
  integer k, the problem of deciding if there
  exists a 2-independent set of chords of size k or
  more (2-independent set in circle graphs) is
  NP-complete.
• Proof (Sketch)
• The 2-independent set problem is in NP.
• Planar 3SAT is polynomially reducible to the
  2-independent set problem.

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2-Independent Set Problem

• Circle graphs
• Given n chords in a circle, find a 2-
  independent set of chords of maximum cardinality.

NP-hard
O(n log n) time

• Permutation graphs
• Given n chords in a two-sided channel in
  which each chord connects one point in each side,
  find a 2-independent set of chords of maximum
  cardinality.

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Permutation Graph
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Geometric Optimization

• Minimum Covering
• Given a set of N points, find the smallest disk
  covering the entire set.
• Smallest enclosing circle, rectangle, square,
  annulus, parallel strip, etc.
• P-center problem
• minimum covering with fixed-size disk,
  square,etc.
• weighted version

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Minimum Covering 3

• 1-center problem (smallest circle)

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Minimum Covering 2

• 1-center problem (smallest rectangle)

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Minimum Covering 1

• p-center problem (fixed-size disk covering)

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Matching 2

• Pattern matching Given a target pattern P, and A
  test pattern Q, find if Q is a sub-pattern in P.

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Matching 1
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Inexact Matching
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Bipartite Matching

• Bi-chromatic matching Given two sets, P and Q of
  points, find a one-to-one mapping such that the
  sum of distances of matched pairs of points is
  minimized.

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Searching

• Searching Problems
• Given a set S of objects (points, line segments,
  etc.), determine if there exists any object with
  certain properties.
• Given a set S of objects, report objects that
  satisfy certain properties.
• With preprocessing allowed, how fast can we find
  objects in S satisfying certain properties?
• What if S is subject to updates?
• Memory requirement, query time, update time

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Searching and Retrieval

• Exact Match (membership problem)
• Given a set S of objects pi(x1,x2,,, xn), is
  (q1,q2,,, qn) in S?
• Range Search
• Given a set S of objects pi(x1,x2,,, xn),
  find those whose xk lies within a range interval
  qk1,qk2 for each k1,2,..,n.
• In 2D, the problem is solvable in O(log2 N K)
  optimal time
• In higher dimensions, it is solvable in O((log2
  N)d-1 K) time, where K is the output size.

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Example of Range Search
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Conclusion

• Computer Science is a discipline addressing
  areas ranging from theory to applications.
• Mathematics is a fundamental to theoretical
  computer science.
• Geometric computing is a very rich area with many
  applications
• Cryptography (one-way function, security)
• Number theory, combinatorics
• Packing Problem, Trapezoid Problem, etc.

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Trapezoid Construction Problem
c
Given d, b, e, f construct a trapezoid as shown
Given a,c, e, f construct a trapezoid as
shown Solved.
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The End

• Thank you for your attention!