search algorithms winter semester 2004/2005 10 jan 2005 11th lecture
DESCRIPTION
Search Algorithms Winter Semester 2004/2005 10 Jan 2005 11th Lecture. Christian Schindelhauer [email protected]. Spatial Searching. Prolog: Searching with some help Searching with total Uncertainty Nearsighted Search The Cow Path Problem The Concept of Competitive Analysis - PowerPoint PPT PresentationTRANSCRIPT
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HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Search AlgorithmsWinter Semester 2004/2005
10 Jan 200511th Lecture
Christian [email protected]
Search Algorithms, WS 2004/05 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Spatial Searching
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– Searching without help– Probabilistic Solution – The Wall Problem
Farsighted Search– The Watchman Problem– How to Learn your Environment
Search Algorithms, WS 2004/05 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
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Searching with some Help
From “Searching with Uncertainty”, Kranakis, Krizanc, SIROCCO, 1999, 194-203
Problem:–In Manhattan you are looking for a restaurant–You ask a policeman at every crossing for directions–This policeman stays at his crossing–There is a constant error probability p<1/2 that this information is wrong
Task:–Find the restaurant as fast as possible–with respect to the start distance d
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Search Algorithms, WS 2004/05 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Main Problem
If you rely all policemen there is a constant probability that you get stuck
If you ignore the advice of the policemen you need much more time
– Then you can use a spiral approach taking O(d2) steps
target
start
Search Algorithms, WS 2004/05 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Solution for the 1-Dimensional Case
Consider a ring of length nSearch-Ring(k)1. dir direction at start position2. for i1 to n or target is reached do3. move in direction dir4. od5. if the majority of the policeman
met so far agree then6. continue in direction dir until
target is found7. else8. reverse direction dir and
continue until target is found
9. fi
Observation: The probability that Search-Ring continues in the correct direction after k steps is
Search Algorithms, WS 2004/05 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The probability to make a wrong decision is given by
Observe
To describe Xm m/2 we set
This leads to
Let m = c log n and c’=min(,2) then the error probability is
Analysis of the Ring Case
TheoremSearch-Ring(k) for k=(log n) solves the search problem within d+O(log n) steps with high probability, i.e. 1-n-O(1).
Proof Consider Bernoulli experiment of Xi of
choosing direction with probability p for outcome 1 and probability 1-p for outcome 0.
Chernoff bound:– For independent Bernoulli variable Xi
and with
Search Algorithms, WS 2004/05 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The 2-Dimensional Case
Assumptions– Torus street map
– Policemen point to east/west bound direction before showing to north/south bound direction
• (if their advice is correct).
Search Algorithms, WS 2004/05 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Block-Wise Approach
Consider Blocks of size (log n) x (log n) Properties (with high probability)
– West/east bound directions can be decided correctly with high probability
– If the target is north/south bound from a block this can be decided correctly by scanning all the columns of a block
Kranakis-Krizanc-Search1. Apply horizontal Search-Ring(c log n) to find the
west/east bound direction2. Follow this direction until a majority of policemen
in a block advises to go into the opposite direction3. Find the correct vertical column in the current
block and the block visited before4. Move along the vertical direction until the target is
found or the majority of policemen points backwards
5. If the target has not been found by now, perform a spiral search
Search Algorithms, WS 2004/05 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Performance of Kranakis-Krizanc-Search
TheoremWith high probability the Kranakis-Krizanc-Search finds the target in d + O(log2 n) steps.
Proof:Follows by applying Chernoff bounds
Discusssion:
Prefering horizontal directions over vertical ones is an (unnecessary) simplification
The block-wise approach can solve this problem as well, since either the horizontal or the vertical direction can be detected by a walk through a block
– This results in an algorithm with same performance (exercise!!)Policemen help a lot
Search Algorithms, WS 2004/05 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Cow-Path Problem
Given–A near-sighted cow–A fence with a gate–The cow does not know the direction
Task–Find the exit as fast as possible
???
Search Algorithms, WS 2004/05 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Competitive Analysis
How to evaluate the online solutionClassical approach:
– Worst-case time• This is always n for a fence of length n
– Average case• This is not better
Competitive Analysis– Compare the cost of the solution of an
instance x• CostAlg(x)
– to the best possible offline solution (unknown to the cow)
• Costoffline(x)Minimize the competitive ratio
=
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Search Algorithms, WS 2004/05 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Rental Ski Problem
Problem–Buying a pair of skis costs 100 Euro–Renting a pair of skis one day costs 10 Euro
How many days do we have snow?Shall we buy or rent?
1st Solution: Buy on the first snow day
Buy one the first day of snow–CostBuy(1) = 100
Worst case = 1 day of snow –Costoffline(1) = 10 (1 day renting)
2nd Solution: Always rentRent every day
–For x snow days–CostBuy(x) = 10 x
Worst case = always snow–best strategy: buy on the first day–Costoffline(1) = 100
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Search Algorithms, WS 2004/05 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
A Better Solution for the Rental Ski Problem
Rent&Buy1. Rent for 10 days2. Then Buy on the 11th day
Cost for x<11 days of snow:– CostRent&Buy (x) = 10x
Cost for x>10 days of snow:– CostRent&Buy (x) = 200
Best strategy for x<11 days of snow:– Rent for x days– Costoffline (x) = 10x
Best strategy for x>10 days of snow:– Buy on the first day– Costoffline (x) = 100
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Search Algorithms, WS 2004/05 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Solution of the Cow Fence Problem
Deterministic Cow-Path1. dir left2. for i 0 to log n do3. go 2i steps to direction dir4. go 2i steps back to the origin5. revert direction dir6. od
Theorem [Baeza-Yates, Culberson, Rawlins, 1993]
The deterministic Cow-Path algorithm has a competitive ratio of 9.This competitive ratio is optimal.
Search Algorithms, WS 2004/05 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Exit
Performance of the Cow-Path Algorithm
Performance of the best (offline) strategy: d
– where d is the shortest way to the exit Worst case of the Cow-Path Algorithm
– d = 2x+1– Let d’=d-1
Number of steps before finding the exit:1+1+2+2+4+4+...+d’/2+d’/2+d’+d’+2d’+2d’+d’+1 = 9 d’-1 = 9 d - 10
d’2d’d’2d’d’+1
d’/2d’/4 ...
Search Algorithms, WS 2004/05 16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Shoreline Problem
Problem description A boat is lost in a half ocean with a linear
shoreline No compass on board No sight because of dense fog The distance to the shoreline is unknown
Task Find the coast as fast as possible
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Dekompressor „TIFF (Unkomprimiert)“ benötigt.
Zur Anzeige wird der QuickTime™ Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Zur Anzeige wird der QuickTime™ Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Zur Anzeige wird der QuickTime™ Dekompressor „TIFF (Unkomprimiert)“ benötigt.?
Search Algorithms, WS 2004/05 17
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Spiral Solution for the Shoreline ProblemBaeza-Yates, Culberson, Rawlins, 1993
Solution: Use logarithmic spiral obeying
– where r is the polar radius from the starting point
– and is the polar angle
Numerical optimization leads to a competitive optimal ratio for k=1.250...
The shoreline problem can be solved using the logarithmic spiral method with competitive ratio 13.81...
1
k2
Search Algorithms, WS 2004/05 18
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Searching for a point in a Grid
Problem:– Find a spot in a grid without knowing
the coordinates– (finding the restaurant in New York
without policemen) Solution:
– Use a spiral covering all points in Hamming distance 1,2,3,4,...
Theorem [Baeza-Yates, Culberson, Rawlins, 1993]
– Using the spiral method this problem can be solved with competitive ratio 2d, where d is the Hamming distance between start and target.
– This competitive ratio is optimal.
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HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Thanks for your attentionEnd of 11th lecture
Next lecture: Mo 17 Jan 2005, 11.15 am, FU 116Next exercise class: Mo 10 Jan 2005, 1.15 pm, F0.530 or We 12 Jan 2005, 1.00 pm, E2.316