Heuristics for Minimum Brauer Chain Problem

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Heuristics for Minimum Brauer Chain Problem

• Fatih Gelgi Melih Onus

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Outline

• Problem Definition
• Binary Method
• Factor Method
• Heuristics
• Experimental Results
• Conclusions

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Brauer Chain

• A Brauer chain for a positive integer n is a
  sequence of integers 1 a0, a1, a2, , ar n
  such that ai ai-1 ak for some 0 k lt i and 1
  i n
• Example
• 1
• 1, 112
• 1, 2, 224
• 1, 2, 4, 426
• 1, 2, 4, 6, 6612
• 1, 2, 4, 6, 12, 12214
• is a Brauer chain for 14 with length 5

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Binary Method

• Write the number n in binary form
• Replace each 1 with DA and each 0 with D
• Remove the leading DA
• Begin from 1, follow the sequence from left to
  right
• For each D, double the current number
• For each A, add 1 to the current number
• Example
• Binary representation of 19 is 10011
• Sequence is DDDADA
• 1, 112, 224, 448, 819, 9918, 18119

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Factor Method

• Let n pq where n, p, q?Z. First calculate
  Brauer chain for p and q
• Let 1 a1, a2, , ak p is Brauer Chain for p
  
• Let 1 b1, b2, , bk q is Brauer Chain for q
  
• 1 a1, a2, , ak p pb1, pb2, , pbm pq
  is a Brauer chain for npq
• Example
• lt1, 2 ,4, 5gt lt1, 2, 3gt
• lt1, 2 ,4, 5, 10, 15gt

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Heuristics

• Binary Heuristic
• Factorization Heuristic
• Dynamic Heuristic
• It uses the previous solutions to obtain the best
  solution for current n value
• We store only one solution for each number
• The dynamic formula is, l(n) minl(k)1 where
  kltn and the solution sequence of k must contain
  n-k
• 2-3-5 Heuristic
• This algorithm always begins with numbers 1, 2,
  3, 5, .
• For the next element, it selects as maximum as
  possible
• 2-3-6 Heuristic
• 2-4-8 Heuristic

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Experiments

• We did 3 types of experiments
• We calculated all n values up to about 4500
• We calculated only one randomly chosen n value
  within each interval of 200 up to 10000
• We calculated all n values up to 20000 (using
  factorization and dynamic heuristics)
• To calculate optimum value, we used exhaustive
  search with branch and bound technique
• Although our pruning conditions make the search
  quite faster, we were able to calculate the
  optimum values up to 4500 since the running time
  is O(n!)

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Optimum and 1.5 optimum values for n up to 4500
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The performance of heuristics
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Experiment II

• To obtain optimum values for larger ns and
  increase the quality of our experiments, for each
  interval with size 200 we chose a random n value
• All the solutions of heuristics are clearly
  smaller than 1.5 optimum

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Experiment II
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Approximation Ratios

• All the heuristics obviously has 1.5
  approximation ratios
• Better than 2-4-8, 2-3-6, 2-3-5 and binary,
  factorization has 1.25 approximation value
• Dynamic has an incredible approximation ratio
  with 1.1
• Empirically, factorization is even smaller than
  1.5 ?lg n? and dynamic is smaller than 1.4 ?lg n?

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Conclusion

• Several heuristics for approximating minimum
  Brauer chain problem is discussed
• The optimum function is not monotone we couldnt
  prove a theoretical approximation ratio better
  than 2
• Experimental results show that there is
  empirically an approximation with 1.1 which is
  incredible for the problem
• For approximation results, we also achieved 1.4
  ?lg n? length for any number n where the trivial
  lower bound is ?lg n?
• Providing a better lower bound, the approximation
  factor can be decreased
• With a good lower bound, approximation factor can
  be proved theoretically