Abstract
In this paper we extend a technique introduced in [14] for dynamic matrix functions. We present dynamic algorithms for computing matrix determinant and matrix adjoint over commutative rings. These algorithms are then used to construct an algorithm for dynamic shortest distances in unweighted graph. Our algorithm supports updates in O(n 1.932) randomized time and queries in O(n 1.288) randomized time. These bound improve over the previous results and solve a long-standing open problem if sub-quadratic dynamic algorithms exist for computing all pairs shortest distances.
Research supported by KBN grant 4T11C04425
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References
Bunch, J., Hopcroft, J.: Triangular Factorization and Inversion by Fast Matrix Multiplication. Math. Comp. 28, 231–236 (1974)
Coppersmith, D.: Rectangular Matrix Multiplication Revisited. J. Complex. 13(1), 42–49 (1997)
Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. In: Proceedings of the nineteenth annual ACM symposium on Theory of Computing, pp. 1–6. ACM Press, New York (1987)
Demetrescu, C., Italiano, G.F.: Fully Dynamic All Pairs Shortest Paths with Real Edge Weights. In: Proceedings of 42th annual IEEE Symposium on Foundations of Computer Science, pp. 260–267 (2001)
Demetrescu, C., Italiano, G.F.: Improved Bounds and New Trade-Offs for Dynamic All Pairs Shortest Paths. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 633–643. Springer, Heidelberg (2002)
Demetrescu, C., Italiano, G.F.: A new Approach to Dynamic all Pairs Shortest Paths. In: Proceedings of the thirty-fifth annual ACM Symposium on Theory of Computing, pp. 159–166. ACM Press, New York (2003)
Frandsen, G.S., Hansen, J.P., Miltersen, P.B.: Lower Bounds for Dynamic Algebraic Problems. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 362–372. Springer, Heidelberg (1999)
Greene, D.H., Knuth, D.E.: Mathematics for the Analysis of Algorithms. Birkhäuser, Basel (1982)
Henzinger, M.R., King, V.: Fully Dynamic Biconnectivity and Transitive Closure. In: Proceedings 36th annual IEEE Symposiumon Foundations of Computer Science, pp. 664–672 (1995)
Huang, X., Pan, V.Y.: Fast Rectangular Matrix Multiplication and Applications. Journal of complexity 14(2), 257–299 (1998)
King, V.: Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs. In: Proceedings of 40th annual IEEE Symposium on Foundations of Computer Science, pp. 81–91 (1999)
Kung, H.T., Traub, J.F.: All Algebraic Functions Can Be Computed Fast. J. ACM 25(2), 245–260 (1978)
Reif, J.H., Tate, S.R.: On Dynamic Algorithms for Algebraic Problems. J. Algorithms 22(2), 347–371 (1997)
Sankowski, P.: Dynamic Transitive Closure via Dynamic Matrix Inverse. In: Proceedings of the 45th annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2004)
Schonhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7, 281–292 (1971)
Schwartz, J.T.: Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. Algorithms 10, 701–717 (1980)
Strassen, V.: Vermeidung von Divisionen. J. reine u. angew. Math. 264, 182–202 (1973)
Ullman, J., Yannakakis, M.: High-probability Parallel Transitive Closure Algorithms. In: Proceedings of the 1990 ACM Symposium on Parallel Algorithms and Architectures, July 1990, pp. 200–209 (1990)
Zippel, R.E.: Probabilistic Algorithms for Sparse Polynomials. In: Ng, K.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)
Zwick, U.: All Pairs Shortest Paths in Weighted Directed Graphs Exact and Almost Exact Algorithms. In: Proceedings of the 39th annual IEEE Symposium on Foundations of Computer Science, pp. 310–319 (1998)
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Sankowski, P. (2005). Subquadratic Algorithm for Dynamic Shortest Distances. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_47
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DOI: https://doi.org/10.1007/11533719_47
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