Generating Golomb Rulers using Genetic Algorithms

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Generating Golomb Rulersusing Genetic Algorithms

• Sharat S. Chikkerur
• ssc5_at_eng.buffalo.edu

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Organization

• Golomb Rulers
• Genetic Algorithms
• Naïve fixed length approach
• Fixed mark approach
• BDD Chromosomes
• Comparative Results
• Conclusion

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Introduction to Golomb Rulers

• Common Ruler
• Constructed with a set of m marks placed at
  equal distances
• The length of the ruler is the distance between
  the terminal marks
• In a common ruler, there are more than one ways
  to measure each distance
• Golomb Ruler
• Golomb ruler is a ruler with m marks placed
  such that no two marks have the same distance
  between them
• A Golomb ruler is also a sequence of numbers such
  that the difference between any pair of numbers
  is unique
• Named after Solomon W. Golomb, USC who first
  studied such sequences

Common Ruler
Golomb Ruler
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Applications of Golomb Rulers 1

• Radio Communications
• Frequencies of carriers can be located at mark
  positions(scaled)
• Non linear effects produce minimum distortion2
• Up to mark 10 rulers were used
• Orthogonal Codes
• Used in error detection and correction 3
• Based on difference triangle sets (DTS)
• Computer communications
• Each distance simultaneously specified the source
  and destination
• Other applications
• X-ray crystallography
• Linear arrays minimum redundancy arrays
• Pulse phase modulation

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Perfect Golomb Rulers

• Perfect Golomb Rulers
• A ruler with m marks and length N is said to
  be perfect if it can measure all distances from 1
  to N
• No perfect rulers exists for (m gt 4) 1
• Perfect rulers place an upper bound on the
  density of markers

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Optimal Golomb Rulers

• Optimal Golomb Rulers
• No other ruler with m marks can have a shorter
  length N
• Less dense than perfect rulers

• Example
• M5, N 11
• No other ruler of length 11 has M gt 5
• All distances except 6 can be measured

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Sub optimal Golomb Rulers

• Sub-optimal Golomb Rulers
• Neither perfect nor optimal
• Less dense than optimal ruler
• A sub-optimal ruler with m marks can have any
  length greater than the optimal ruler with m
  marks (up to infinity)
• A sub-optimal ruler can be generated by removing
  one or more marks from the optimal ruler

• Example
• (1) M4, N 11. Can measure only 6 distances, 4
  distances are missing compared to optimal ruler
• (2) M 2,N 11. Can measure only 1 distance, 9
  distances are missing compared to optimal ruler

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Properties of Golomb Rulers

• Closed expression for upper bound of m
• Consider a ruler with m marks
• Distance triangle
• Recursive relation
• Each mark adds m additional distances
• N(m1) m N(m)
• The upper bound for marks is bounded by a perfect
  ruler

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Properties of Golomb Rulers

• Upper bound on length N
• No closed form expression exists for the upper
  bound on the length N of an optimal ruler for a
  given number of markers

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Known Optimal Golomb Rulers

• Optimal Golomb Rulers of up to 23 marks have been
  found
• Resources on the internet
• http//members.aol.com/golomb20/
• http//www.hewgill.com/ogr/
• http//www.distributed.net/ogr/
• http//www.research.ibm.com/people/s/shearer/gropt
  .html
• http//library.thinkquest.org/C007645/english/2-go
  lomb-0.htm

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Generating Golomb Rulers

• Deterministic approaches
• Naïve approach
• Place the mark at 2Kn where K1,K2..are integers.
• Further, we can construct a Golomb ruler by
  placing a mark at the terms of any geometric
  progression.
• Scientific American Algorithm/EXHAUST 4
• Based on backtracking. Consists of two processes
  Generator and Checker
• Requires both number of marks m and length N.
  Highly dependent on N
• Token passing(A.Dollas) 1
• Shifting (McCracken, W.T., Rankin et. al)1
• Fastest of the existing algorithms
• Tree pruning1
• Exponential time
• Problems
• Exponential Time
• Computationally Hard problem
• Advantages
• Exponential but bounded. Can be solved by
  distributed/parallel computing
• Can prove optimality

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Solving hard problems

• Heuristics
• Based on domain knowledge
• Highly application specific
• Gradient Descent/Hill Climbing
• Requires smooth cost function surface
• Always guaranteed to converge
• Can get stuck at a local minima
• Simulated Annealing
• Always converges
• Search space is successively narrowed
• Genetic Algorithms
• Based on the paradigm of evolutionary computing
• Neural Networks
• Function estimators
• Require lot of training data

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Genetic Algorithms

• Introduced by John Holland, University at
  Michigan, 1970
• A new computational paradigm suitable for solving
  non parametric and NP complete optimization
  problems
• It is biologically motivated
• Arrives at an optimal solution through natural
  selection of sub optimal approaches mimicking
  evolution
• Process
• Start with a random population of solutions
• Evaluate the fitness of each individual in the
  population
• Select the healthy individuals from the
  populations
• Produce offspring by breeding the healthy
  individuals/solutions
• Repeat the procedure with the new generation
• Introduce mutation to avoid local minima

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Chromosomes

• Each solution or approximation is encoded as a
  string (usually a bit string) called a
  chromosome
• Each combination of alphabets in the string
  (genotype) represents a solution to the problem
  (phenotype)
• Each parameter in the solution can be encoded as
  a bit string, using any of the encoding schemes
  such as fixed point, floating point or plain
  binary scheme.
• The fitness of each individual is determined by
  an objective function
•  Example

Objective function
Phenotype
Genotype
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Genetic Algorithm Operations

• Natural Selection
• Selects the individuals who will survive to the
  next generation.
• Survival probability is proportional to the
  fitness of the individual
• Methods
• Roulette Wheel Selection
• Cross Over
• Produced two offspring from two parents
• Methods
• Single point, multipoint
• Mutation
• Produces diversity in the solution space
• Prevents from getting stuck in a local minima

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Example

• Objective Function
• Optimal solution
• Chromosome
• Each parameter is represented as a 4 bit binary
  string
• Results
• Simulation done using GAToolbox for Matlab5
• 10 out of 15 chromosomes have optimal solution
• Exhaustive search will require 65536 combinations

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Naïve fixed length approach

• The length of the ruler is fixed(N).
• Encoding
• The phenotype is identical to the genotype
• Each bit string represents a candidate ruler
• The candidates can have variable number of marks
  m
• Fitness function

• M number of marks in the ruler
• R The autocorrelation sequence (Rn
  sns-n)
• onumber of ones in the autocorrelation sequence
  Rn

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Results
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Fixed mark approach

• The number of marks m of the ruler is fixed
• Encoding
• Each chromosome is made up of (m-1) segments that
  indicate the distance between the marks
• Each segment encodes the index of the available
  distance
• The available distance is pruned after each
  segment is selected
• Fitness function
• The length of the ruler
• Advantages
• Each chromosome encodes a valid Golomb Ruler(100
  survival)
• Disadvantages
• Produces very sparse rulers

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Distance pruning example

• Consider a mark m ruler.
• The maximum distance(N) is chosen to be m2(ub)
• Each segment is made up on log2(N/2) bits
• Mark at position 0 is assumed to exists
• The list of available distances is therefore
  1N/2-1

Mark list
Distance list
Chromosome
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Distance pruning example

• Consider a mark m ruler.
• The maximum distance(N) is chosen to be m2(ub)
• Each segment is made up on log2(N/2) bits
• Mark at position 0 is assumed to exists
• The list of available distances is therefore
  1N/2-1

Mark list
Distance list
Chromosome
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Distance pruning example

• Consider a mark m ruler.
• The maximum distance(N) is chosen to be m2(ub)
• Each segment is made up on log2(N/2) bits
• Mark at position 0 is assumed to exists
• The list of available distances is therefore
  1N/2-1

Mark list
Distance list
Chromosome
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Distance pruning example

• Consider a mark m ruler.
• The maximum distance(N) is chosen to be m2(ub)
• Each segment is made up on log2(N/2) bits
• Mark at position 0 is assumed to exists
• The list of available distances is therefore
  1N/2-1

Mark list
Distance list
Chromosome
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Results
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BDD Approach

• Based on J.P.Robinsons approach for Golomb
  Arrays6
• Always produces a valid Golomb Ruler
• Encoding
• Each bit in the chromosome specifies if the next
  available position is to be chosen or skipped
• Cost function
• Number of marks in the ruler

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BDD Approach

• Based on J.P.Robinsons approach for Golomb
  Arrays6
• Always produces a valid Golomb Ruler
• Encoding
• Each bit in the chromosome specifies if the next
  available position is to be chosen or skipped
• Cost function
• Number of marks in the ruler

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BDD Approach

• Based on J.P.Robinsons approach for Golomb
  Arrays6
• Always produces a valid Golomb Ruler
• Encoding
• Each bit in the chromosome specifies if the next
  available position is to be chosen or skipped
• Cost function
• Number of marks in the ruler

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Results
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Comparative Results

• Soliday and Hofimar et. al7
• Chromosome encodes the distance between the marks
• Produces very spare arrays and low survival rate
• Perira and Costa et. al8
• Based on random keys to encode the distance
  permutation
• Parameters are determined heuristically
• Low survival rate
• Fixed length approach
• Search space too large for higher marks
• Fixed mark approach
• Chromosomes encode the index of the distance
• Always produces a valid Golomb Ruler
• Greedy approach (deterministic)
• Places a mark at the next available position
• BDD approach
• Based on encoding for Golomb arrays
• Always produces a valid ruler

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Comparative Results
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Summary

• Genetic algorithms can be successfully used to
  produce sub optimal Golomb rulers in bounded time
• There are multiple ways to encode the phenotype
• The efficiency of the solutions lies in the
  encoding of the chromosomes
• Fixed mark approach produces very sparse rulers
• Adapted BDD approach for Golomb rulers
• BDD approach is the fastest to converge with
  optimal solutions obtained in less than 500
  generations

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References

• Rankin W.T, Optimal Golomb Rulers An Exhaustive
  Parallel Search Implementation, M. S. Thesis,
  Duke University ,1993
• Babcock, W.C, Intermodulation Interference in
  Radio Systems, Bell Systems Technical Journal,
  pp. 63-73 1953
• Torleiv Klove, Bounds on the size of optimal
  difference triangle sets, IEEE Transaction on
  Information Theory, 1988
• A. K. Dewdney.Computer Recreations,Scientific
  American, pp. 16-26 December 1985.
• Genetic Algorithm Toolbox, University of
  Sheffield, UK, http//www.shef.ac.uk/gaipp/ga-too
  lbox/
• J.P.Robinson, Genetic Search for Golomb Arrays,
  IEEE Trans. On Information Theory, Vol.46, No.3,
  May 2000
• Soliday, Hofimar, Genetic Algorithm Approach to
  Search for Golomb Rulers, ICGA-95, pp.528-535,
  1995
• Perira, Tavares and Costa, Golomb Rulers The
  Advantage of Evolution

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