Shifting Graphs and Their Applications
Journal of the ACM (JACM)
The Complexity of Computing
The complexity of control structures and data structures
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Switching functions whose monotone complexity
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
On the parallel evaluation of multivariate polynomials
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Graphs and Hypergraphs
Some Exact Complexity Results for Straight-Line Computations over Semirings
Journal of the ACM (JACM)
Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors
Journal of Computer and System Sciences
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Among the most remarkable algorithms in algebra are Strassen's algorithm for the multiplication of matrices and the Fast Fourier Transform method for the convolution of vectors. For both of these problems the definition suggests an obvious algorithm that uses just the monotone operations + and ×. Schnorr [18] has shown that these algorithms, which use &thgr;(n3) and &THgr;(n2) operations respectively, are essentially optimal among algorithms that use only these monotone operations. By using subtraction as an additional operation and exploiting cancellations of computed terms in a very intricate way Strassen showed that a faster algorithm requiring only O(n2.81) operations is possible. The FFT method for convolution achieves O(nlog n) complexity in a similar fashion. The question arises as to whether we can expect even greater gains in computational efficiency by such judicious use of cancellations. In this paper we give a positive answer to this, by exhibiting a problem for which an exponential speedup can be attained using {+,−,×} rather than just {+,×} as operations. The problem in question is the multivariate polynomial associated with perfect matchings in planar graphs. For this a fast algorithm is implicit in the Pfaffian technique of Fisher and Kasteleyn [6,8]. The main result we provide here is the exponential lower bound in the monotone case.