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1Institute of Information Science Academia Sinica
1
Introduction to Geomtric Introduction to Geomtric ComputingComputing
D. T. LeeInstitute of Information Science
Academia Sinica
October 14, 2002
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What is Computer Science about?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 MinMax Problem 44
Min&Max 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 Min&Max to 2 Min or Max problems!
2N-3 comparisons
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min
min
maxSimple comparator
MIN min
MinMax Problem MinMax Problem 33
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MinMax Problem MinMax Problem 22
But finding both Maximun and Minimum can be done with 3N/2 2 comparisons
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min
max
MIN min
MAX max
MinMax Problem MinMax Problem 11
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Sorting ProblemSorting 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.
(N-1)+ (N-1)log2 N
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8
min
1st round 3rd round2nd round
Total number of matches for 8 competitors is 7N/2 + N/4 + … + 4 + 2 + 1 = N-1
Total number of rounds is log2 N
Knockout TournamentKnockout Tournament
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Geometric ComputingGeometric 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 ComputationTheory 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 Checking if a Polygon is Convex 44
A polygon P is represented by a sequence of points v0,v1,v2, …,vn-1 where (vi, vi+1) is an edge, i=0,1,..,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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Problem: There are infinitely many possibleSegments to check!
Checking if a Polygon is Convex Checking if a Polygon is Convex 33
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Checking if a Polygon is Convex Checking if a Polygon is Convex 22
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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Consider three vertices vi,vj,vk and determinant
det(vi,vj,vk ) = xi yi 1xj yj 1xk yk 1
where (xi yi ) are the x- and y-coordinates of v i
det(vi,vj,vk ) = 0 when vi,vj,vk are collinear
det(vi,vj,vk ) < 0 when vi,vj,vk form a right turn
det(vi,vj,vk ) > 0 when vi,vj,vk form a left turn
vi
vj
vk
left turn
Checking if a Polygon is ConvexChecking if a Polygon is Convex11
vi
vj
vk
right turnvi
vj
vk
collinear
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Checking if a Polygon is SimpleChecking if a Polygon is Simple
v1
v2
v3
v4
v0
v5
How do you check if the input polygon is simple??
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Art Gallery Problem 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 ProblemArt Gallery Problem
p and q are visible
q
r
r and q are not visible
pq
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Art Gallery ProblemArt Gallery Problem
Visibility polygon of q & r
q r
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Art Gallery TheoremArt Gallery Theorem Theorem: Given a simple polygon with
N sides, stationary guards are sufficient, and sometimes necessary to cover the entire polygon.
N/3
Theorem: Given a simple polygon with N sides, the problem of finding a minimumnumber of stationary guards to cover the entire polygon is intractable.
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Lower Bound of Art Gallery ProblemLower Bound of Art Gallery Problem
A comb with 8 prongs (N=24) requiresN/3 = 8 guards
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Upper Bound of Art Gallery ProblemUpper Bound of Art Gallery ProblemTriangulation of a Simple PolygonTriangulation of a Simple Polygon
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Coloring of a GraphColoring 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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N = 14
Art Gallery Problem: ExampleArt Gallery Problem: Example
Triangulation
Select red color to place guards
3 N/3 = 4
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Art Gallery TheoremArt 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 ProblemVariations 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 dist
ance)
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Rectilinear PolygonRectilinear Polygon
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Polygons with HolesPolygons with Holes
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Guard CapabilitiesGuard Capabilities
Mobile GuardLimited visibility
Mobile guard with limited visibility
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Map Coloring Map Coloring 22
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 Map Coloring 11
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ProximityProximity
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 LineCloset 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.
xi+1xi
Closest pair
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Closest Pair in the Plane Closest Pair in the Plane
2
1
= 2
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Voronoi DiagramVoronoi Diagram
Delaunay TriangulationClosest pair must beVoronoi neighbors
Circumcircle of a triangle
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Closest Pair ProblemClosest 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 OptimizationGeometric 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 RoutingHomotopic 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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AfterCompaction
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Topological Via Minimization Topological Via Minimization 44
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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Layer 1
Layer 2
Via
Layer 1
Layer 2
Via
# of Vias = 4
# of Vias = 1
Topological Via Minimization Topological Via Minimization 33
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Layer 1
Layer 2
Via
# of Vias : 1
Topological Via Minimization Topological Via Minimization 22
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Topological Via Minimization Topological Via Minimization 11
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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Definition: A set of chords is said to be 2-independent if it can be partitioned into two subsets of independent chords
Circle Graph Circle Graph 22
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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.
Circle Graph Circle Graph 11
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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.
2-Independent Set Problem2-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
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Permutation GraphPermutation Graph
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Geometric OptimizationGeometric Optimization
Minimum Covering• Given a set of N points, find the smallest di
sk covering the entire set.• Smallest enclosing circle, rectangle, squar
e, annulus, parallel strip, etc.• P-center problem• minimum covering with fixed-size disk,
square,etc. • weighted version
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Minimum Covering Minimum Covering 33
1-center problem (smallest circle)
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Minimum Covering Minimum Covering 22
1-center problem (smallest rectangle)
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Minimum Covering Minimum Covering 11
p-center problem (fixed-size disk covering)
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Matching Matching 22
Pattern matching: Given a target pattern P, and A test pattern Q, find if Q is a sub-pattern in P.
PQ
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Matching Matching 11
P
Q
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Inexact MatchingInexact Matching
PQ
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Bipartite MatchingBipartite 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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SearchingSearching
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 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 k=1,2,..,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 SearchExample of Range Search
salary
Date of birth
19,500,000 19,559,999
3,000
4,000
G. Ometer
born: Aug 19, 1954
Salary: 3500
19,500,000
19,559,999
3,000
4,000
24
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ConclusionConclusion
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 ProblemTrapezoid Construction Problem
c
a
bde f
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 EndThe End
Thank you for your attention!