### Abstract

We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1*. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let l^{s}(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, l^{s}+1(n)=log_{2}(l^{s}(n)) for s≥0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/l^{s}(n)) vertices. 2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n^{2}/l^{s}(n)) time for any positive integer.

Original language | English |
---|---|

Pages (from-to) | 134-149 |

Number of pages | 16 |

Journal | Journal of Computer and System Sciences |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1989 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

**On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape turing machines.** / Galil, Zvi; Kannan, Ravi; Szemerédi, E.

Research output: Contribution to journal › Article

*Journal of Computer and System Sciences*, vol. 38, no. 1, pp. 134-149. https://doi.org/10.1016/0022-0000(89)90036-6

}

TY - JOUR

T1 - On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape turing machines

AU - Galil, Zvi

AU - Kannan, Ravi

AU - Szemerédi, E.

PY - 1989

Y1 - 1989

N2 - We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1*. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let ls(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, ls+1(n)=log2(ls(n)) for s≥0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/ls(n)) vertices. 2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n2/ls(n)) time for any positive integer.

AB - We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1*. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let ls(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, ls+1(n)=log2(ls(n)) for s≥0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/ls(n)) vertices. 2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n2/ls(n)) time for any positive integer.

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U2 - 10.1016/0022-0000(89)90036-6

DO - 10.1016/0022-0000(89)90036-6

M3 - Article

AN - SCOPUS:0024605867

VL - 38

SP - 134

EP - 149

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 1

ER -