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2 G ENETIC A LGORITHMS FOR FAST M ATRIX M ULTIPLICATION András Joó Anikó Ekárt Juan Neirotti United Kingdom GECCO 2011 HUMIES AWARDS 14/07/2011 T HE P ROBLEM : R ECURSIVE M ATRIX M ULTIPLICATION 3 Standard algorithm for multiplying two square matrices of size n n requires n3 multiplications and 2 dn n 1 additions Strassen’s algorithm reduces the number of required multiplications to n log 7 if n is a power of 2 (1969) 2 GECCO 2011 HUMIES AWARDS 14/07/2011 K NOWN L IMITS 4 For matrices of size 22 at least 7 multiplications needed For matrices of size 3 3 at least 19 multiplications needed Best known exact algorithm for size 3 3 contains 23 multiplications GECCO 2011 HUMIES AWARDS 14/07/2011 P RACTICAL S IGNIFICANCE 5 An exact algorithm using 22 multiplications on matrices of size 3 3 would be an improvement on the best known algorithm for this size An exact algorithm using 21 multiplications on matrices of size 3 3 would be an overall improvement on how recursive matrix multiplication is currently performed on large matrices As the search space has size 2.25e+180 for 21 multiplications and 8.71e+188 for 22 multiplications, respectively, it is highly unlikely that a human or a simple algorithm would discover a solution! GECCO 2011 HUMIES AWARDS 14/07/2011 O UR SOLUTION : PARALLEL GA 6 Parallel island model, with unidirectional ring topology and migration Steady-state elitist GA Continuous real-valued representation Variety of crossover and mutation operators Periodic explicit enforcing of diversity GECCO 2011 HUMIES AWARDS 14/07/2011 GA R ESULTS 7 On matrices of size 3 3 reproduced a solution with 23 multiplications found an approximate solution of fitness 0.9978 for 22 multiplications GECCO 2011 HUMIES AWARDS 14/07/2011 W HY H UMAN -C OMPETITIVE ? 8 In 1976, J. D. Laderman published his article “A noncommutative algorithm for multiplying 3 3 matrices using 23 multiplications” in the Bulletin of the American Mathematical Society . Others published equivalent algorithms. The theoretically proven lower bound is 19 multiplications, but no exact algorithm with less than 23 multiplications is known to date. Our GA approach could reproduce matrix multiplication algorithms using 23 multiplications and also led to an approximate algorithm requiring 22 multiplications. GECCO 2011 HUMIES AWARDS 14/07/2011 W HICH C RITERIA ? 9 B: The result is equal to or better than a result that was accepted as a new scientific result at the time when it was published in a peer-reviewed scientific journal. D: The result is publishable in its own right as a new scientific result independent of the fact that the result was mechanically created. F: The result is equal to or better than a result that was considered an achievement in its field at the time it was first discovered. G: The result solves a problem of indisputable difficulty in its field. GECCO 2011 HUMIES AWARDS 14/07/2011