## Directed planar reachability is in unambiguous logspace (2007)

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Venue: | In Proceedings of IEEE Conference on Computational Complexity CCC |

Citations: | 15 - 3 self |

### BibTeX

@INPROCEEDINGS{Bourke07directedplanar,

author = {Chris Bourke and Raghunath Tewari and N. V. Vinodchandran},

title = {Directed planar reachability is in unambiguous logspace},

booktitle = {In Proceedings of IEEE Conference on Computational Complexity CCC},

year = {2007}

}

### Years of Citing Articles

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### Abstract

We show that the st-connectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.

### Citations

252 |
Algorithmic Graph Theory
- Gibbons
- 1985
(Show Context)
Citation Context ...fied to give us a characterization of NL in terms of graph thickness. The usual graph-theoretic notion of thickness of a graph G is defined as the minimal number of planar subgraphs whose union is G [=-=Gibbons 1985-=-]. Intuitively, we can think of thickness as the minimal number of transparencies required to draw the graph so that no edges cross within any single transparency. Clearly, a graph is planar if and on... |

234 | Nondeterministic space is closed under complementation - Immerman - 1988 |

211 | Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1
- Barrington
- 1989
(Show Context)
Citation Context ... hence captures the power of nondeterminism in the context of logarithmic space. Various restricted versions of this problem characterize other low-level complexity classes such as L, AC 0 , and NC 1 =-=[14, 22, 8, 7]-=-. A natural and important restriction of the st-connectivity problem is when the graphs involved are planar, which we denote by PlanarReach in this paper. The complexity of this problem is not yet set... |

167 | Matching is as easy as matrix inversion - Mulmuley, Vazirani, et al. - 1987 |

143 |
Undirected ST-Connectivity in Log-Space
- Reingold
- 2005
(Show Context)
Citation Context ...exists a directed path from s to t in G. We will not be concerned with details about the representation of planar graphs. We note that the work of Allender & Mahajan [AM04], and subsequently Reingold =-=[Rei05]-=-, implies a deterministic logarithmic space algorithm that decides whether or not a given graph is planar, and if it is, outputs a planar embedding. We will use the following two results which require... |

142 | Complexity measures for public-key cryptosystems
- Grollmann, Selman
- 1988
(Show Context)
Citation Context ...ng Valiant [1976] introduced the class UP, the unambiguous version of NP, which proved to be a very useful restriction to study, mainly because of its connection to certain kind of one-way functions [=-=Grollman and Selman 1988-=-]. In the logarithmic space setting, the class UL was first defined and studied by Buntrock et al. [1991] and Àlvarez and Jenner [1993]. Since then, UL and related low-space unambiguous classes have b... |

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

64 |
On the power of the compass (or, why mazes are easier to search than graphs
- Blum, Kozen
- 1978
(Show Context)
Citation Context ... st-connectivity on such graphs with constant width captures the complexity of the AC 0 hierarchy. Long before Reingold [22] showed that the undirected st-connectivity problem is in L, Blum and Kozen =-=[9]-=- gave a deterministic log-space algorithm for undirected grid graphs. Recently, the complexity of various restrictions of grid graph reachability have been studied [2, 1]. Specifically, Allender et al... |

51 | Counting quantifiers, successor relations, and logarithmic space
- Etessami
- 1997
(Show Context)
Citation Context ... which we denote by PLANARREACH in this paper. The complexity of this problem is not yet settled satisfactorily. The best known upper bound in terms of space complexity is NL. Though it is hard for L =-=[Ete97]-=-, it is not known whether it is complete for NL. Recently there has been progress in understanding the complexity of PLANARREACH. In particular, Allender, Datta & Roy [ADR05] show that PLANARREACH log... |

49 |
A very hard logspace counting class
- Alvarez, Jenner
- 1993
(Show Context)
Citation Context ...mainly because of its connection to certain kind of one-way functions [16]. In the logarithmic space setting, the class UL was first defined and studied by Buntrock et al. [10] and Àlvarez and Jenner =-=[6]-=-. Since then, UL and related low-space unambiguous classes have been of interest to researchers [10, 6, 19, 3, 23, 2]. The class UL is particularly interesting because there is increasing evidence tha... |

38 | Making nondeterminism unambiguous
- Reinhardt, Allender
(Show Context)
Citation Context ...1, AJ93, Lan97, AL98, RA00, ADR05]. Arguably the most interesting result regarding UL, due to Reinhardt & Allender, is that the nonuniform version of UL contains the whole of NL; that is NL ⊆ UL/poly =-=[RA00]-=-. In addition, Allender, Reinhardt, & Zhou showed that, under the hardness assumption that deterministic linear space has functions that can not be computed by circuits of size 2 ɛn , the construction... |

28 | D.S.: Geometric thickness of complete graphs
- Dillencourt, Eppstein, et al.
- 2000
(Show Context)
Citation Context ... however, thickness-two suffices to capture all of NL. We’ll actually show that completeness holds for an even more restrictive notion of thickness called geometric thickness [Hutchinson et al. 1995; =-=Dillencourt et al. 2000-=-]. The geometric thickness of a graph G is defined as the minimal number k such that we can assign planar point locations to the vertices of G, represent each edge as a line segment, and assign each e... |

27 |
A.: On representations of some thicknesstwo graphs
- Hutchinson, Shermer, et al.
- 1999
(Show Context)
Citation Context ...kness-one. Surprisingly, however, thickness-two suffices to capture all of NL. We’ll actually show that completeness holds for an even more restrictive notion of thickness called geometric thickness [=-=Hutchinson et al. 1995-=-; Dillencourt et al. 2000]. The geometric thickness of a graph G is defined as the minimal number k such that we can assign planar point locations to the vertices of G, represent each edge as a line s... |

25 | The complexity of planarity testing
- Allender, Mahajan
(Show Context)
Citation Context ...tices s and t, determine if there exists a directed path from s to t in G. We will not be concerned with details about the representation of planar graphs. We note that the work of Allender & Mahajan =-=[AM04]-=-, and subsequently Reingold [Rei05], implies a deterministic logarithmic space algorithm that decides whether or not a given graph is planar, and if it is, outputs a planar embedding. We will use the ... |

24 | Searching constant width mazes captures the AC 0 hierarchy
- Barrington, Lu, et al.
- 1998
(Show Context)
Citation Context ...hability problem for a strict subclass of planar graphs called grid graphs. We denote the reachability problem for grid graphs as GGR. From this result and the fact that GGR reduces to its complement =-=[BLMS98]-=-, it follows that PLANARREACH reduces to its complement problem of unreachability in planar graphs. Allender et al. [ADR05] also give a direct log-space reduction from PLANARREACH to its complement. I... |

23 | Unambiguity and fewness for logarithmic space
- Buntrock, Jenner, et al.
- 1991
(Show Context)
Citation Context ...seful restriction to study, mainly because of its connection to certain kind of one-way functions [16]. In the logarithmic space setting, the class UL was first defined and studied by Buntrock et al. =-=[10]-=- and Àlvarez and Jenner [6]. Since then, UL and related low-space unambiguous classes have been of interest to researchers [10, 6, 19, 3, 23, 2]. The class UL is particularly interesting because there... |

22 | Isolation, matching and counting uniform and nonuniform upper bounds
- ALLENDER, REINHARDT, et al.
- 1999
(Show Context)
Citation Context ...er the hardness assumption that deterministic linear space has functions that can not be computed by circuits of size 2 ɛn , the constructions given in [RA00] can be derandomized to show that NL = UL =-=[ARZ99]-=-. These results give strong indication that NL equals UL. Our result gives further evidence that this equality might hold. Since PLANARREACH reduces to reachability in grid graphs, for our upper bound... |

19 | The directed planar reachability problem
- Allender, Datta, et al.
- 2005
(Show Context)
Citation Context .... Though it is hard for L [Ete97], it is not known whether it is complete for NL. Recently there has been progress in understanding the complexity of PLANARREACH. In particular, Allender, Datta & Roy =-=[ADR05]-=- show that PLANARREACH log-space reduces to the reachability problem for a strict subclass of planar graphs called grid graphs. We denote the reachability problem for grid graphs as GGR. From this res... |

14 | An unambiguous class possessing a complete set
- Lange
- 1997
(Show Context)
Citation Context ... by Buntrock et al. [1991] and Àlvarez and Jenner [1993]. Since then, UL and related low-space unambiguous classes have been of interest to researchers [Buntrock et al. 1991; Àlvarez and Jenner 1993; =-=Lange 1997-=-; Allender and Lange 1998; Reinhardt and Allender 2000; Allender et al. 2005]. The class UL is particularly interesting because there is increasing evidence that, in fact, the whole of nondeterministi... |

12 | Grid graph reachability problems
- Allender, Barrington, et al.
- 2005
(Show Context)
Citation Context ... Kozen [1978] gave a deterministic log-space algorithm for undirected grid graphs. Recently, the complexity of various restrictions of grid graph reachability have been studied [Allender et al. 2005; =-=Allender et al. 2006-=-]. Specifically, Allender et al. [2005] show that the layered grid graph reachability problem is in UL by prescribing a weight function that makes such graphs min-unique. A layered grid graph is a gri... |

9 | The isomorphism problem for planar 3-connected graphs is in unambiguous logspace - Thierauf, Wagner - 2008 |

7 |
RUSPACE(log n) ⊆ DSPACE(log 2 n/ log log n
- Allender, Lange
- 1998
(Show Context)
Citation Context ... et al. [1991] and Àlvarez and Jenner [1993]. Since then, UL and related low-space unambiguous classes have been of interest to researchers [Buntrock et al. 1991; Àlvarez and Jenner 1993; Lange 1997; =-=Allender and Lange 1998-=-; Reinhardt and Allender 2000; Allender et al. 2005]. The class UL is particularly interesting because there is increasing evidence that, in fact, the whole of nondeterministic logarithmic space might... |

4 |
Tanmoy Chakraborty, Samir Datta & Sambuddha Roy (2009). Planar and Grid Graph Reachability Problems. Theory Comput
- Allender, Barrington
(Show Context)
Citation Context ... problem is in L, Blum and Kozen [9] gave a deterministic log-space algorithm for undirected grid graphs. Recently, the complexity of various restrictions of grid graph reachability have been studied =-=[2, 1]-=-. Specifically, Allender et al. [2] show that the layered grid graph reachability problem is in UL by prescribing a weight function that makes such graphs min-unique. A layered grid graph is a grid gr... |

3 | RUSPACE(log n) ⊆ DSPACE(log 2 n/ log log n). Theory of Computing Systems - Allender, Lange - 1998 |

1 |
determinants, permanents, and (unique) perfect matchings
- Datta, Kulkarni, et al.
(Show Context)
Citation Context ...somorphism problem for planar 3-connected graphs can be decided in UL ∩ coUL. Limaye et al. [20] have shown that the longest path problem for planar DAGs also is solvable in UL. Finally, Datta et al. =-=[11]-=- and Datta et al. [12] use similar weighting techniques to establish improved upper bounds for bipartite planar matching problems. 13s14 Acknowledgments. The third author is very grateful to Eric Alle... |

1 | Deterministically isolating a matching in bipartite planar graphs - Datta, Kulkarni, et al. |