## Determinism versus Non-Determinism for Linear Time RAMs with Memory Restrictions (1998)

Venue: | In Proc. of 31st STOC |

Citations: | 34 - 2 self |

### BibTeX

@INPROCEEDINGS{Ajtai98determinismversus,

author = {Miklos Ajtai},

title = {Determinism versus Non-Determinism for Linear Time RAMs with Memory Restrictions},

booktitle = {In Proc. of 31st STOC},

year = {1998},

pages = {632--641}

}

### Years of Citing Articles

### OpenURL

### Abstract

Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is sufficiently large then the elements distinctness problem cannot be solved in time kn with epsilon n bits of read and write memory, that is, there is no machine with this values of the parameters which decides whether there are two different input registers whose contents are identical. We also show that there is a simple decision problem that can be solved in constant time (actually in two steps) using non-deterministic computation, while there is no deterministic linear time algorithm with epsilon n log n bits read and write memory which solves the problem. More precisely if we allow kn time for some fixed constant k, then there is an epsilon > 0 so that the problem cannot be solved with epsilon n log n bits of read and write memory if n is sufficiently large. The decision problem is the following: "Find two different input registers, so that the Hamming distance of their contents is at most c log n".

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Citation Context ...he RAM model. We can solve the element distinctness problem with bucket sorting in our RAM in linear time with c # n log n bits of read and write memory, where c # is a suitably choosen constant (see =-=[AHU]-=-). This is a determinstic (non-probabilistic) algorithm. Our lower bounds are also about non-probabilistic algorithms. For the element distinctness problem we give a probabilistic algorithm which solv... |

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Citation Context ...dge which is labeled by the content of the register associated with the node. The number of outgoing edges is R, in our case R = n c . This computational model was introduced by Borodin and Cook (see =-=[BC]-=-) and they gave a time space tradeo# for sorting n integers. They introduced a technique for proving lower bounds for R-way branching problems where the number of output bits is relatively large compa... |

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Citation Context ...ry large so we do not try to give a review of them. The results that are closest to our present problem, are lower bounds on the computation of explicit functions given by Beame, Saks and Thathachar (=-=[BST]-=-). There is also a strong similarity between the proof techniques of [BST] and the present paper. Two combinatorial properties of the function f to be computed are introduced in [BST] (called P (#) an... |

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Citation Context ...s. This problem is of great practical and theoretical interest, it has been studied in great detail in various computational 1 models, particularly in the comparison model (see [BFKLT], [BFMUW], [K], =-=[Y]-=-). A time space trado# TS = ## n 2 ) for the elements distinctness problem on comparisonbased branching programs was conjectured by Borodin et al in [BFKLT]. A. Yao [Y] proved a tradeo# TS = ## n 2-#(... |

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Citation Context ... with identical contents. This problem is of great practical and theoretical interest, it has been studied in great detail in various computational 1 models, particularly in the comparison model (see =-=[BFKLT]-=-, [BFMUW], [K], [Y]). A time space trado# TS = ## n 2 ) for the elements distinctness problem on comparisonbased branching programs was conjectured by Borodin et al in [BFKLT]. A. Yao [Y] proved a tra... |

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Citation Context ...utput bits is relatively large compared to the time allowed. This method was successfully used for proving several lower bounds and space time tradeo#s for problems of similar nature. (See e.g. Beame =-=[Bea]-=-) Our problem does not belong to this category since the output is a single bit. Still the same very high level idea, cutting the time into short intervals and knowing that all we can use about the pa... |

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Citation Context ...ntical contents. This problem is of great practical and theoretical interest, it has been studied in great detail in various computational 1 models, particularly in the comparison model (see [BFKLT], =-=[BFMUW]-=-, [K], [Y]). A time space trado# TS = ## n 2 ) for the elements distinctness problem on comparisonbased branching programs was conjectured by Borodin et al in [BFKLT]. A. Yao [Y] proved a tradeo# TS =... |

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Citation Context ...e beginning of the interval, is applicable. The strongest known separation theorem between deterministic and nondeterministic computation is the theorem of Paul, Pippinger, Szemeredi and Trotter (see =-=[PPST]-=-) stating that non-deterministic linear time is more powerful than deterministic linear time for multitape Turing machines. The proof of this theorem is also using a segmentation of the time. In this ... |

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Citation Context ...Y] proved a tradeo# TS = ## n 2-#(n) ), where #(n) = O(1/(log n) 1 2 ), which is very close to optimal since TS = O(n 2 ) is achievable even for sorting in the range c 1 log n # S # c 2 n/ log n (See =-=[PR]-=-). The best upper bounds for the element distinctness problem are given in the RAM model. We can solve the element distinctness problem with bucket sorting in our RAM in linear time with c # n log n b... |

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Citation Context ...se the following three well-known facts. A (m) will denote the set of all 0, 1 sequences of length m. The first is the following theorem of Harper. (For a proof see e.g. Bollobas, Combinatorics, 16. (=-=[Bo-=-])). Theorem A (Harper). Assume that for all X # A (m) , N(X) is the set of sequences whose Hamming distance is at most one from at least on point of X. Then for all X # A (m) , |X| # P r i=0 m i imp... |

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Citation Context ...ntents. This problem is of great practical and theoretical interest, it has been studied in great detail in various computational 1 models, particularly in the comparison model (see [BFKLT], [BFMUW], =-=[K]-=-, [Y]). A time space trado# TS = ## n 2 ) for the elements distinctness problem on comparisonbased branching programs was conjectured by Borodin et al in [BFKLT]. A. Yao [Y] proved a tradeo# TS = ## n... |