## On Separators, Segregators and Time versus Space

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Citations: | 6 - 0 self |

### BibTeX

@MISC{Santhanam_onseparators,,

author = {Rahul Santhanam and To Paul},

title = {On Separators, Segregators and Time versus Space},

year = {}

}

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### Abstract

We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that NT IME(n p log (n)) 6= DT IME(n p log (n)). We show that if the class of multi-pushdown graphs has (o(n); o(n=log(n))) segregators, then NT IME(nlog(n)) 6= DT IME(nlog(n)). We also show that atleast one of the following facts holds - (1) P 6= L , (2) For all polynomially bounded constructible time bounds t, NT IME(t) 6= DT IME(t). We consider the problem of whether NT IME(t) is distinct from NSPACE(t) for constructible time bounds t. A pebble game on graphs is defined such that the existence of a "good" strategy for the pebble game on multi-pushdown graphs implies a "good" simulation of nondeterministic time bounded machines by nondeterministic space-bounded machines. It is shown that there exists a "good" strategy for the pebble game on multi-pushdown graphs i the graphs have sublinear separators. Finally, we show that nondeterministic time bounded Turing machines can be simulated by 4 machines with an asymptotically smaller time bound, under the assumption that the class of multi-pushdown graphs has sublinear separators.

### Citations

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(Show Context)
Citation Context ...game such that a \good" strategy for this game implies a fast simulation of non-deterministic time by nondeterministic space and hence a separation, by the nondeterministic space hierarchy theore=-=m of [14]. Unfortun-=-ately, we are not able to prove that there exists a \good" strategy for this game on the class of pushdown graphs. Instead, we show that the existence of a \good" strategy is equivalent to a... |

111 | The method of forced enumeration for nondeterministic automata - Szelepcsényi - 1988 |

54 |
On time versus space
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(Show Context)
Citation Context ...ally bounded t. The major result in this connexion was the separation of nondeterministic linear time and deterministic linear time by Paul, Pippenger, Szemeredi and Trotter [24], building on work by =-=[12-=-] and [25]. They show how to simulate deterministic Turing machines with 4 machines that use less time, and then use a collapse lemma and a hierarchy theorem to prove their result. Gupta [10] showed ... |

39 |
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(Show Context)
Citation Context ...ertices has a p(n) segregator of size q(n). [24] shows that the class H r (n) of multi-pushdown graphs has (n=(log (n)); n=(log (n))) segregators, using path-compression techniques due to [6] and [2=-=8-=-]. This result is the cornerstone of their proof that deterministic linear time is distinct from nondeterministic linear time. Their speed-up of deterministic time by 2 time, and hence our result als... |

25 | On the complexity of SAT - Lipton, Viglas - 1999 |

23 | Nondeterministic polynomial time versus nondeterministic logarithmic space: Time-space tradeoffs for satisfiability - Fortnow - 1997 |

23 | Black-white pebbles and graph separation - Lengauer - 1981 |

23 | Time-Space Tradeoffs for Nondeterministic Computation - Fortnow, Melkebeek - 2000 |

18 | Speedups of Deterministic Machines by Synchronous Parallel Machines - Dymond, Tompa - 1985 |

15 |
On nontrivial separators for k-page graphs, and simulations by nondeterministic one-tape Turing machines
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(Show Context)
Citation Context ...a \good" strategy is equivalent to a longstanding open problem - the existence of sublinear separators for the class of multi-pushdown graphs. The best results known for the latter problem are du=-=e to [9-=-], and show that any upper bound on the size of separators has to be very close to linear. We also show that there exists a fast simulation of nondeterministic time by 4 time under the hypothesis tha... |

11 |
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(Show Context)
Citation Context ...aphs, has o(n) separators i for each graph in the family, there is a strategy for the BWR game that uses o(n) space. Proof Immediate from Proposition 4.2, Lemma 4.3 and Lemma 4.4. It is mentioned in [=-=22]-=- that the existence of sublinear separators for multi-pushdown graphs implies that nondeterministic space is more powerful than nondeterministic time. 2-pushdown graphs are planar and these have subli... |

10 |
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(Show Context)
Citation Context ...with n vertices has a p(n) segregator of size q(n). [24] shows that the class H r (n) of multi-pushdown graphs has (n=(log (n)); n=(log (n))) segregators, using path-compression techniques due to [6=-=-=-] and [28]. This result is the cornerstone of their proof that deterministic linear time is distinct from nondeterministic linear time. Their speed-up of deterministic time by 2 time, and hence our r... |

5 | Time-space tradeos for nondeterministic computation - Fortnow, Melkebeek - 2000 |

3 |
Alternating Time Versus Deterministic Time: A Separation
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(Show Context)
Citation Context ...work by [12] and [25]. They show how to simulate deterministic Turing machines with 4 machines that use less time, and then use a collapse lemma and a hierarchy theorem to prove their result. Gupta [=-=1-=-0] showed that the simulation could in fact be done by 2 machines. Unfortunately, these proofs, and specically the collapse lemma that is common to them, cannot be used to separate NT IME(t) and DT I... |

3 |
separating nondeterminism from determinism
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(Show Context)
Citation Context ...re able to prove that the disjunction of these two statements holds. Such a result lends support to our intuition that both of these statements are true. Our technique is similar to that of Kannan in =-=[15]-=-, where he proved that P 6= L or DT IME(n) 6= NT IME(n). Since we now know that the second of these hypotheses is true(this was not known when the theorem was proved), the theorem lacks content for se... |

3 |
Some results on relativized deterministic and nondeterministic time hierarchies
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(Show Context)
Citation Context ... and 4 T IME(t). An important feature of the techniques we use is that they are non-relativizing. Hence there is a possibility that they can be extended to prove deeper results. By results of Moran [=-=20-=-], NT IME(t) ? = DT IME(t) cannot be decided by relativizing techniques for time-constructible bounds t. 2 Preliminaries We assume the standard denitions of deterministic and nondeterministic Turing m... |

3 |
E.Szemeredi and W.Trotter. On determinism versus nondeterminism and related problems
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(Show Context)
Citation Context ...ace Rahul Santhanam, Department of Computer Science, University of Chicago. E-mail:rahul@cs.uchicago.edu Abstract We give thesrst extension of the result due to Paul, Pippenger, Szemeredi and Trotter =-=[2-=-4] that deterministic linear time is distinct from nondeterministic linear time. We show that NT IME(n p log (n)) 6= DT IME(n p log (n)). We show that if the class of multi-pushdown graphs has (o(n)... |

2 |
On sparse graphs with dense long paths. Computers and Mathematics with Applications
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(Show Context)
Citation Context ...F with n vertices has a p(n) segregator of size q(n). [24] shows that the class Hr(n) of multi-pushdown graphs has (n/(log ∗ (n)), n/(log ∗ (n))) segregators, using path-compression techniques due to =-=[6]-=- and [28]. This result is the cornerstone of their proof that deterministic linear time is distinct from nondeterministic linear time. Their speed-up of deterministic time by Σ2 time, and hence our re... |

1 | and J.Gabarro. Structural Complexity 1. Volume 11 - Balcazar - 1988 |

1 |
and J.Gabarro. Structural Complexity 2. Volume 22
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(Show Context)
Citation Context ...nistic and nondeterministic Turing machines and of time and space bounded complexity classes [13]. We also assume the denitions of alternating Turing machines making a bounded number of alternations [=-=-=-2]. A computation of an alternating Turing machine on an input is any correctsnite sequence of congurations of the machine beginning with the initial conguration and ending with an accepting or reject... |

1 |
Tape-Bounded Turing Acceptors and AFLs
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(Show Context)
Citation Context ...tmost r tapes besides the read-only input tape. Lemma 2.1 For any k, k T IGU(t(n); LIN) 2 k T IGU(t(n); LIN ). Proof Our simulation of r tapes by 2 tapes is analogous to the standard simulation [3]. Given an r-tape guess-bounded machine M that makes k 1 alternations, we build a machine M 0 to simulate it. M 0srst guesses a computation of M on one tape. It then veries the computation by using i... |

1 |
A.R.Meyer and D.Weise. On time versus space 3
- Halpern
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(Show Context)
Citation Context ...rministic space hierarchy theorem, it follows that space is more powerful than time for deterministic Turing machines. The main theorem of [12] was extended to RAMs by [26] and to pointer machines by =-=[11-=-]. But the related question for nondeterministic machines remains open. The proof in [12] proceeds by dening a pebble game on graphs, such that a good strategy for the pebble game implies a good simul... |

1 |
Unraveling k page graphs
- Kannan
- 1985
(Show Context)
Citation Context ...sublinear(in fact, O( p n)) separators [18]. The question of existence of sublinear separators for 3-pushdown graphs is equivalent to the question for general r-pushdown graphs, by a result of Kannan =-=[16]-=-. The question is still open, the most recent attack on it being the paper by Galil, Kannan and Szemeredi [9] in which they prove nearly linear lower bounds on the sizes of separators of certain pushd... |

1 | E.Prauss and R.Reischuk. On alternation - Paul |

1 |
On alternation 2
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(Show Context)
Citation Context ...cessors by comparing the starting congurations of the vertex with the ending congurations of its immediate predecessors. If all the checks succeed, it accepts, otherwise it rejects. Proposition 5.2([25]) For any k, k T IME(t) 6 k T IME(o(t)). Corollary 5.3 If the class of pushdown graphs has sublinear separators, for all t, NT IME(t) 6= 4 T IME(t) Proof By Theorem 5.1, under the given hypothe... |

1 |
On time versus space 2
- Paul, Reischuk
- 1981
(Show Context)
Citation Context ...ACE(t(n)=log(t(n))). By the deterministic space hierarchy theorem, it follows that space is more powerful than time for deterministic Turing machines. The main theorem of [12] was extended to RAMs by =-=[26-=-] and to pointer machines by [11]. But the related question for nondeterministic machines remains open. The proof in [12] proceeds by dening a pebble game on graphs, such that a good strategy for the ... |