## A non-linear time lower bound for boolean branching programs (1999)

### Cached

### Download Links

- [theoryofcomputing.org]
- [dimacs.rutgers.edu]
- [tocmirror.cs.tau.ac.il]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proc. of 40th FOCS |

Citations: | 59 - 0 self |

### BibTeX

@INPROCEEDINGS{Ajtai99anon-linear,

author = {Miklós Ajtai},

title = {A non-linear time lower bound for boolean branching programs},

booktitle = {In Proc. of 40th FOCS},

year = {1999},

pages = {60--70},

publisher = {IEEE}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract: We give an exponential lower bound for the size of any linear-time Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδ|logδ | −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.

### Citations

55 | A time-space trade-off for sorting on a general sequential model of computation
- Borodin, Cook
- 1982
(Show Context)
Citation Context ...ter can contain clog 2 n bits.) 1.2.2 Branching programs with many output bits, and the time segmentation method The computational model of R-way branching programs was introduced by Borodin and Cook =-=[8]-=-, who proved a time-space trade-off for sorting n integers. This work also introduced a method for proving lower bounds about R-way branching programs in the special case where the number of output bi... |

54 | On lower bounds for read-k-times branching programs
- Borodin, Razborov, et al.
- 1993
(Show Context)
Citation Context ...y independently run over large sets and now we want to guarantee that the x T By is not constant. Motivated by similar considerations, quadratic forms were studied by Borodin, Razborov, and Smolensky =-=[9]-=-, Jayram [13], and Beame, Saks, and Thatachar [6]. The result in this direction that we will use in our paper is the following. (This was proved in more general forms in [9], [13], and [6], and also f... |

49 |
and Oded Goldreich, Unbiased bits from sources of weak randomness and probabilistic communication complexity
- Chor
- 1985
(Show Context)
Citation Context ...aks, and Thatachar [6]. The result in this direction that we will use in our paper is the following. (This was proved in more general forms in [9], [13], and [6], and also follows from the results of =-=[10]-=-.) Suppose that the rank of the matrix B is r and x resp. y are taking values independently from m1 resp. m2 dimensional subspaces of an m dimensional vectorspace. If m1 + m2 + r > 2m then the quadrat... |

44 | Time-space tradeoffs for branching programs
- Beame, Saks, et al.
- 1998
(Show Context)
Citation Context ...e memory at the the beginning of the interval. 1.2.3 Lower bounds for explicit functions and decision problems Using the same high level proof structure, and other new ideas, Beame, Saks and Jayram 1 =-=[6]-=- gave a lower bound on the computational time for an explicitly given function with a Boolean branching program of size 2 o(n) . Namely they proved that there is an ε > 0 so that the question whether ... |

37 | Determinism versus Nondeterminism for Linear Time RAMs with Memory Restrictions - Ajtai |

31 | Time-space tradeoff lower bounds for randomized computation of decision problems
- Beame, Saks, et al.
(Show Context)
Citation Context ...f this paper was published in [3] containing all of the essential elements of the proofs presented here. Since then, the main result of this paper was further improved by Beame, Saks, Sun, and Vee in =-=[7]-=- by making the time/space lower bounds sharper and generalizing the theorem for the case of probabilistic branching programs. Their proofs use the results and techniques of the present paper (together... |

12 |
Time-space tradeoffs for algebraic problems on general sequential machines
- ABRAHAMSON
- 1991
(Show Context)
Citation Context ... the number of output bits is relatively large compared to the time allowed for the computation. Several other lower bounds and timespace trade-offs of similar nature were given, see e. g. Abrahamson =-=[1, 2]-=-, Beame [5], Karchmer [11], Reisch and Schnitger [12], and Yesha [14]. These lower bound proofs have a common high-level structure, namely the time is cut into short intervals and we use the fact that... |

12 | On separating the read-k-times branching program hierarchy - Thathachar - 1998 |

11 | A time-space tradeo for sorting on a general sequential model of computation - Borodin, Cook - 1982 |

8 |
Time-Space Tradeoffs for Branching Programs Contrasted with Those for StraightLine Programs
- Abrahamson
- 1986
(Show Context)
Citation Context ... the number of output bits is relatively large compared to the time allowed for the computation. Several other lower bounds and timespace trade-offs of similar nature were given, see e. g. Abrahamson =-=[1, 2]-=-, Beame [5], Karchmer [11], Reisch and Schnitger [12], and Yesha [14]. These lower bound proofs have a common high-level structure, namely the time is cut into short intervals and we use the fact that... |

7 | Time-space tradeos for branching programs - Beame, Saks, et al. - 1998 |

5 |
SCHNITGER: Three applications of Kolmogorov-complexity
- REISCH, G
- 1982
(Show Context)
Citation Context ... to the time allowed for the computation. Several other lower bounds and timespace trade-offs of similar nature were given, see e. g. Abrahamson [1, 2], Beame [5], Karchmer [11], Reisch and Schnitger =-=[12]-=-, and Yesha [14]. These lower bound proofs have a common high-level structure, namely the time is cut into short intervals and we use the fact that during such an interval any information that we can ... |

4 | A general sequential time-space tradeo# for finding unique elements - Beame - 1991 |

2 |
THATHACHAR: On separating the read-k-times branching program hierarchy
- S
- 1998
(Show Context)
Citation Context ...tly run over large sets and now we want to guarantee that the x T By is not constant. Motivated by similar considerations, quadratic forms were studied by Borodin, Razborov, and Smolensky [9], Jayram =-=[13]-=-, and Beame, Saks, and Thatachar [6]. The result in this direction that we will use in our paper is the following. (This was proved in more general forms in [9], [13], and [6], and also follows from t... |

1 |
AJTAI: A non-linear time lower bound for boolean branching programs
- unknown authors
- 1999
(Show Context)
Citation Context ...er uses the techniques of both of these directions. THEORY OF COMPUTING, Volume 1 (2005), pp. 149–176 153s1.3 Subsequent developments MIKLÓS AJTAI A preliminary version of this paper was published in =-=[3]-=- containing all of the essential elements of the proofs presented here. Since then, the main result of this paper was further improved by Beame, Saks, Sun, and Vee in [7] by making the time/space lowe... |

1 |
AJTAI: Determinism versus Non-Determinism for Linear Time RAMs with Memory Restrictions
- unknown authors
(Show Context)
Citation Context ...program which computes g(x1,...,xn) in depth kn. 1 T.S. Jayram, formerly Jayram S. Thathachar THEORY OF COMPUTING, Volume 1 (2005), pp. 149–176 151sMIKLÓS AJTAI The author of the present paper proved =-=[4]-=- that the element distinctness problem (where each “element” is the value of a variable) cannot be decided with an R-way branching program, for R = clog2 n, in length linear in n if the size of the pr... |

1 | BEAME: General Sequential Time-Space Tradeoff for Finding Unique Elements - unknown authors - 1991 |

1 |
KARCHMER: Two Time-Space Tradeoffs for Element Distinctness
- unknown authors
- 1986
(Show Context)
Citation Context ...s relatively large compared to the time allowed for the computation. Several other lower bounds and timespace trade-offs of similar nature were given, see e. g. Abrahamson [1, 2], Beame [5], Karchmer =-=[11]-=-, Reisch and Schnitger [12], and Yesha [14]. These lower bound proofs have a common high-level structure, namely the time is cut into short intervals and we use the fact that during such an interval a... |

1 | YESHA: Time-Space Tradeoffs for Matrix Multiplication and the Discrete Fourier Transform of Any General Sequential Random-Access Computer. Journal of Computer and System Sciences, 29:183–197 - unknown authors - 1984 |