## The directed planar reachability problem (2005)

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Venue: | In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science |

Citations: | 20 - 8 self |

### BibTeX

@INPROCEEDINGS{Allender05thedirected,

author = {Eric Allender and Samir Datta and Sambuddha Roy},

title = {The directed planar reachability problem},

booktitle = {In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science},

year = {2005},

pages = {238--249},

publisher = {Springer}

}

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### Abstract

Abstract. We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs. A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1

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Citation Context ...itions. A closed orientable surface is one that can be obtained by adding handles to a sphere in 3-space. The genus of the resulting surface is equal to the number of handles added; see also the text =-=[GT87]-=-. Given a graph G, the genus of the graph is the genus of the (closed orientable) surface of least genus on which the graph can be embedded. Given a graph G embedded on a closed orientable surface, an... |

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Citation Context ...raph is 1. We introduce here some terminology and definitions relating to graphs on surfaces. It will be sufficient to give informal definitions of various notions; the interested reader can refer to =-=[MT01]-=- for more rigorous definitions. A closed orientable surface is one that can be obtained by adding handles to a sphere in 3-space. The genus of the resulting surface is equal to the number of handles a... |

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Citation Context ... directed graphs (STCONN) is complete for nondeterministic logspace (NL); the restriction of this problem to undirected graphs, called USTCONN, has recently been shown to be complete for logspace (L) =-=[Rei05]-=-; thus this problem has the same complexity as the s-t-connectivity problem for graphs of outdegree 1 (and even for graphs of indegree and outdegree at most 1 [Ete97]). Grid graphs are an important re... |

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Citation Context ...r grid graphs have proved easier to work with than the corresponding problems for general graphs. For instance, the reachability problem for undirected grid graphs was shown to lie in Lsin the 1970’s =-=[BK78]-=-, although more than a quarter-century would pass before Reingold proved the corresponding theorem for general undirected graphs. Barrington also defined what we will refer to as the layered grid grap... |

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Citation Context ...n to be complete for logspace (L) [Rei05]; thus this problem has the same complexity as the s-t-connectivity problem for graphs of outdegree 1 (and even for graphs of indegree and outdegree at most 1 =-=[Ete97]-=-). Grid graphs are an important restricted class of graphs for which the reachability problem has significant connections to complexity classes. (The vertices in a grid graph are a subset of IN × IN, ... |

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Citation Context ...Lemma 6.1. Let G be a graph of genus g>0, and let T be a spanning tree of G.Then there is an edge e ∈ E(G) such that T ∪{e} contains a surface-nonseparating cycle. Proof. The proof follows ideas from =-=[Tho90]-=- which introduces the “3-path condition”: Definition 6.2. Let K be a family of cycles of G as follows. We say that K satisfies the 3-path condition if it has the following property. If x, y are vertic... |

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Citation Context ...ty class UL. This must be viewed as a slight improvement, since it is shown in [ARZ99] that NL = UL if there is any problem in DSPACE(n) that requires circuits of exponential size, and it is shown in =-=[RA00]-=- that NL/poly = UL/poly (unconditionally). We actually show that LGGR lies in UL∩coUL, although (in contrast to all of the other reachability problems we consider) it remains open if LGGR reduces to i... |

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Citation Context ...ure under complement, but it is also useful in presenting our reduction to grid-graph reachability. Let G be a directed graph. Testing if G is planar reduces to the undirected s-tconnectivity problem =-=[AM04]-=- and thus can be done in logarithmic space [Rei05]. Furthermore, if a graph is planar then a planar combinatorial embedding (i.e., a cyclic ordering of the edges adjacent to each vertex) can be comput... |

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Citation Context ... results we will present in this paper) it is sufficient to restrict attention to grid graphs where the start vertex s lies in the first column, and the terminal vertex t lies in the final column. In =-=[BLMS98]-=-, Barrington et al showed that the reachability problem in (directed or undirected) grid graphs of width k captures the complexity of depth k AC 0 . Barrington also considered general grid graphs with... |

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Citation Context ... 13] shows that GGR reduces to its complement. Our final technical contribution is to show that LGGR lies in the complexity class UL. This must be viewed as a slight improvement, since it is shown in =-=[ARZ99]-=- that NL = UL if there is any problem in DSPACE(n) that requires circuits of exponential size, and it is shown in [RA00] that NL/poly = UL/poly (unconditionally). We actually show that LGGR lies in UL... |

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Citation Context ...lity problems we consider) it remains open if LGGR reduces to its complement. (Note also that it remains open if UL = coUL.) Some other examples of reachability problems in UL were presented by Lange =-=[Lan97]-=-; these problems are obviously in UL (in the sense that the positive instances consist of certain graphs that contain only one path from s to t), and the main contribution of [Lan97] is to present a c... |

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Citation Context ... given an embedding Π of a graph G onto a surface of genus 1. (Unlike the planar case, it doessnot appear to be known if testing if a graph has genus g>0 can be accomplished in logspace, even for g =1=-=[MV00]-=-.) Given such an embedding, using [AM04], we can check in logspace if the minimal genus of the graph is 1. We introduce here some terminology and definitions relating to graphs on surfaces. It will be... |

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Citation Context ...direction) between LGGR and the problems considered in [Lan97]. Series-parallel graphs are an important and well-studied subclass of planar directed graphs. Jakoby, Liskiewicz, and Reischuk showed in =-=[JLR01]-=- that s-t-connectivity in series-parallel graphs can be computed in logspace (and in fact is complete for L). They also show the much stronger result that counting the number of paths between s and t ... |

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Citation Context ...es an alternate construction of a ≤log m reduction of PLANAR.STCONN to its complement. 5 More Closure Properties Different types of logspace reductions were introduced and studied by Ladner and Lynch =-=[LL76]-=-, who showed that logspace Turing and truth-table reducibilities coincided (A≤ log T B iff A≤log tt B). They also introduced a more restrictive version of logspacecomputable truth-table reducibility, ... |

2 |
Grid graph reachability problems. Talk presented at Dagstuhl Seminar on Complexity of Boolean Functions, Seminar number 02121
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(Show Context)
Citation Context ...) grid graphs of width k captures the complexity of depth k AC 0 . Barrington also considered general grid graphs without the width restriction, calling this the Grid Graph Reachability problem (GGR) =-=[Bar02]-=-. The construction of [BLMS98, Lemma 13] shows that GGR reduces to its complement via uniform projections. (The problems STCONN and USTCONN also reduce to their complements via uniform projections, as... |

2 |
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(Show Context)
Citation Context ...aphs with s and t on the outer face to the complement of the s-t-connectivity problem for directed planar graphs. It has recently come to our attention that a very similar theorem and proof appear in =-=[HRS93]-=-[Theorem 3]. First, however, we define some terms more carefully. As observed in the preceding subsection, it is easy in logspace to enumerate all of the vertices that appear on a given face of a plan... |