Summary
Using modular arithmetic we obtain the following improved bounds on the time and space complexities for n × n Boolean matrix multiplication: O(n log 2 7 lognlogloglognloglogloglogn) bit operations and O(n 2loglog n) bits of storage on a logarithmic cost RAM having no multiply or divide instruction; O(n log 2 7(logn)2−1/2log 2 7(loglog n)1/2log 2 7−1) bit operations and O(n 2log n) bits of storage on a RAM which can use indirect addressing for table lookups. The first algorithm can be realized as a Boolean circuit with O(n log 2 7lognlogloglognloglogloglogn) gates. Whenever n×n arithmetic matrix multiplication can be performed in less than O(n log 2 7) arithmetic operations, our results have corresponding improvements.
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Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Reading, Mass.: Addison-Wesley 1974
Arlazarov, V.L., Dinic, E.A., Kronrod, M.A., Faradžev, I.A.: On economical construction of the transitive closure of an oriented graph. Soviet Math. Dokl. 11, 1209–1210 (1970)
Fischer, M.J., Meyer, A.R.: Boolean matrix multiplication and transitive closure. IEEE 12th Annual Symposium in Switching and Automata Theory, pp. 129–131, 1971
Furman, M.E.: Application of a method of fast multiplication of matrices in the problem of finding the transitive closure of a graph, Soviet Math. Dokl. 11, 1252 (1970)
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: University Press 1956
Knuth, D.E.: The art of computer programming-errata et addenda. STAN-CS-71-194, Stanford University, pp. 25–26, 1971. The relevant information also appears in the later versions of Knuth's second volume (beginning with the second printing). The art of computer programming, Vol.2. Seminumerical algorithms, pp. 274–275. Reading, Mass.: Addison-Wesley 1969
Mairson, H.G.: Some new upper bounds on the generation of prime numbers. Comm. ACM 20, 664–669 (1977)
Munro, I.: Efficient determination of the transitive closure of a directed graph. Information Processing Lett. 1, 56–58 (1971)
Pan, V.: Strassen's algorithm is not optimal. IEEE 19th Annual Symposium on Foundations of Computer Science, 1978 (to appear)
Paterson, M.S., Fischer, M.J., Meyer, A.R.: An improved overlap argument for on-line multiplication. SIAM-AMS Proceedings 7, 97–11 (1974)
Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7, 281–292 (1971)
Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)
Tarjan, R.E.: Reference machines require non-linear time to maintain disjoint sets. Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, pp. 18–29, 1977
Weicker, R.: The influence of the machine model on the complexity of context-free language recognition. In: Lecture Notes in Computer Science, Vol. 53. Mathematical Foundations of Computer Science 1977, pp. 560–569. Berlin-Heidelberg-New York: Springer 1977. An abstract for these results appears as R. Weicker, Context free language recognition by a RAM with uniform cost criterion in time n 2 logn. In: Symposium on New Directions and Recent Results in Algorithms and Complexity (J.F. Traub, ed.). New York-London: Academic Press 1976
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This work was supported in part by the Office of Naval Research under contract N00014-67-0204-0063, by the National Research Council of Canada under grant A4307, and by the National Science Foundation under grants MCS76-17321 and GJ-43332
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Adleman, L., Booth, K.S., Preparata, F.P. et al. Improved time and space bounds for Boolean matrix multiplication. Acta Informatica 11, 61–70 (1978). https://doi.org/10.1007/BF00264600
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DOI: https://doi.org/10.1007/BF00264600