HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Search Algorithms
Winter Semester 2004/2005
31 Jan 2005
13th Lecture
Christian Schindelhauer
schindel@upb.de
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HEINZ NIXDORF INSTITUTE
Spatial Searching
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Prolog: Searching with some help
Searching with total Uncertainty
Nearsighted Search
– The Cow Path Problem
– The Concept of Competitive Analysis
– Deterministic Solution
– Finding a Shoreline
– Probabilistic Solution
– The Wall Problem
Farsighted Search
– The Watchman Problem
– How to Learn your Environment
Search Algorithms, WS 2004/05
2
HEINZ NIXDORF INSTITUTE
The Art Gallery Problem
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Given:
– a simple polygon with n corners
 Question:
– How many watchmen must be placed
such that they can survey the
complete polygon
 Observations
– In a convex polygon one watchman
suffices
– It suffices to watch only all edges
• Then all other points are covered,
too
 Trival:
– In all polygons n watchmen suffice
 Also trivial:
– In all polygons n/2 watchmen suffice
Search Algorithms, WS 2004/05
3
HEINZ NIXDORF INSTITUTE
The Ears of a Simple Polygon
 A polygon P is said to be simple if the
only points of the plane belonging to two
polygon edges of P are the polygon
vertices of P.
– Such a polygon has a well defined
interior and exterior.
 A diagonal is a line segment lying entirely
inside polygon P and joining two nonconsecutive vertices
 A vertex is called a principal vertex
– if the diagonal between the
neighbored vertices intersects the
boundary of simple polygon P only at
the neighbors
 A principal vertex of a simple polygon
P is called an ear
– if the diagonal between the neighbors
lies entirely in P.
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
not a simple polygon
simple polygon
principal vertices
non principal vertices
All ears
Search Algorithms, WS 2004/05
4
HEINZ NIXDORF INSTITUTE
Two-Ear-Theorem
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Theorem
Except for triangles every simple polygon
has at least two ears.
Dual-tree:
Given a triangulated simple polygon,
the dual-tree is the graph generated
by plotting a vertex at each triangle
and edges joining vertices in
adjacent triangles,
i.e.triangles which share a diagonal
Proof: By Joseph O'Rourke
– It is known that a simple polygon can
always be triangulated.
– Leaves in the dual-tree of the
triangulated polygon correspond to
ears and every tree of two or more
nodes must have at least two leaves.
Search Algorithms, WS 2004/05
5
HEINZ NIXDORF INSTITUTE
The Art Galllery Theorem
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Theorem
Any simple polygon can be guarded by
n/3 cameras.
 Build up a triangulization by sequentially
removing an ears
– Note that the dual graph is a tree
 Color all nodes
– such that two neighbored nodes of the
triangulation have different colors
 Choose the color with the smallest number
of occurrences n/3
 Place watchman at theses nodes
 Now every triangle is observed
– therefore n/3 watchmen suffice
Search Algorithms, WS 2004/05
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HEINZ NIXDORF INSTITUTE
n/3 cameras are necessary
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Theorem
There is a simple polygon such that at least
n/3 cameras are necessary
Theorem
The decision problem whether k cameras
are sufficient for controlling an art gallery is
NP-complete
Search Algorithms, WS 2004/05
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The Complexity of the Art Gallery-Problem
Figures of the proof (from Chris Wu)
HEINZ NIXDORF INSTITUTE
http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2001/CW/noholeproof.html
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (Unkomprimiert)“
benötigt.
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Reduction from 3-Sat
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Search Algorithms, WS 2004/05
8
HEINZ NIXDORF INSTITUTE
Streets are special Polygons
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Definition
A street is a
• simple polygon P in the plane
• with two distinguished vertices s
and t
• the two boundary chains from s to
t are weakly mutually visible
From every side at least one point of
the other side of the street is visible
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Problem:
– Given an unknown street
– Find the shortest path from s to t
A strategy is c-competitive for
searching a goal in a street if its path
never is longer than c times the length
of the shortest path from s to t.
Search Algorithms, WS 2004/05
9
HEINZ NIXDORF INSTITUTE
Algorithms for Exploration of a Street
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Proposed by
–R. Klein. Walking an unknown street
with bounded detour. Comput. Geom.
Theory Appl., 1:325–351, 1992.
 1992: Klein: 5.72
 Lower Bound: 1.41.. = 21/2
 1992: Icking, Klein: 4.44
 ... Kleinberg,
 1996: Lopez-Ortiz, Schuierer: 1.73
 1997: Christian Icking, Alejandro
Lopez-Ortiz, Sven Schuierer, Ines
Semrau: 1.54
 1999: Icking, Klein, Langetepe and
independently Schuierer and Semrau:
Optimal = 1.41..
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Search Algorithms, WS 2004/05
10
The Idea for Optimal Street Walking
[Icking, Klein, Langetepe]
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Collect the left and right vertices of the
street
 Careful analysis reduces the problem to
the problem of walking a funnel
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
 Consider one (visibility) triangle (s,vl,vr)
– For all possible combinations optimize
the path
– yielding a curve obeying trigonometric
properties
 Related to the problem of looking around a
corner
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Search Algorithms, WS 2004/05
11
HEINZ NIXDORF INSTITUTE
Search Pattern for known Rectilinear Polygons
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 “On-Line Search in a Simple Polygon”
[Kleinberg]
 Given:
–a rectilinguar simple polygon
–a robot has a map of this polygon
–unknown position of the target
 Find:
–a search pattern that minimizes the
path until the target can be seen
 Definition:
–Extended edge is a prolongation of an
edge into the interior of the polygon to
another point of the polygon
–Essential cut is an extended edge
such that no other extended edge (of
same orientation) can be reached
through this cut
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Search Algorithms, WS 2004/05
12
The Lower Bound for Searching in a Rectilinear
Polygon
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Theorem [Kleinberg]
A polygon cannot be searched better than
O(m) competitive, where m is the number
of essential cuts
Algorithm:
 Build a tree from s to all essential cuts
 Use a spiral search strategy to find the
target on all the paths from s to the
essential cuts
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Theorem [Kleinberg]
This construction is optimal up to a
constant factor.
Search Algorithms, WS 2004/05
13
Looking around a Corner
An Optimal Competitive Strategy for Looking Around a
Corner Christian Icking, Rolf Klein, Lihong Ma, 1994
 Two halflines meet at the origin O
 The shaded wedge formed by the halflines
is opaque and has an angle less than 180◦
 Assume that on one of the halflines a
mobile robot is located at point W, outside
the wedge
– equipped with an on-board vision
system facing O.
 Its task is to “look around the corner”,
– i. e. to inspect the other halfline which
is invisible from W (but for its
endpoint, O).
 A strategy for this class of problems
should produce a path from W to any point
on the prolongation, M(φ), of the invisible
halfline.
– φ denotes the angle between the
invisible halfline and the prolongation
of the visible one.
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Search Algorithms, WS 2004/05
14
HEINZ NIXDORF INSTITUTE
Looking around a Corner
Competitive Ratio:
–Path of algorithm versus
–Shortest path to look around corner
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Theorem[Icking, Klein Ma, 1994]
There is an optimal strategy solving the
“Looking around a Corner problem” with
competive ratio
c = 1.21218...
See also
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
http://videos.compgeom.org/socg04video/
5. Searching with an Autonomous Robot
Search Algorithms, WS 2004/05
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HEINZ NIXDORF INSTITUTE
Summary of the Lecture
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
PART I (1st-6th lecture)
PART II (7th-13th lecture)
Pattern Search
–Rabin-Karp
–KMP
–Boyer-Moore
Pattern Search in Compressed Text
–Huffman Code
–Shannon Bound
–Kolmogorov-Complexity
–LZW-Compression
–Searching in LZW-Compressed Text
Searching the Web
–PageRank
–Kleinberg’s HITS-Algorithm
–Structure of the Web
–Searching in Power Law Networks
Spatial Search
–Searching with some Uncertainty
–Competitive Algorithms
–The Cow Problem
–Probabilistic Cows: Smart Cows
–The Wall Problem
–Watchman Problem
–Exploring a Street
–Exploriing a Polygon
Search Algorithms, WS 2004/05
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HEINZ NIXDORF INSTITUTE
The Exam is at 17/18 Feb 2005 or
16 Mar 2005
Register for the
oral exam (now)
1 hour prior to the oral exam:
Solve an exercise
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Learn!!
no
Have you presented
an exercise during the
second part?
yes
Do you accept your
grade from your
midterm exam?
yes
Answer question of the
second half
no
Answer question of the
complete lecture
Goto party on
Feb 18th, 7pm
Kachelöfchen
Search Algorithms, WS 2004/05
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Thanks for your attention
End of 13th lecture
Final exam:
Next exercise class:
Mo 17/18 Feb 2005
or We 16 Mar 2005
Mo 17 Jan 2005, 1.15 pm, F0.530 or
We 19 Jan 2005, 1.00 pm, E2.316
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