Results 1  10
of
13
Grid Graph Reachability Problems
 IN ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2006
"... We study the complexity of restricted versions of stconnectivity, which is the standard complete problem for NL. Grid graphs are a useful tool in this regard, since • reachability on grid graphs is logspaceequivalent to reachability in general planar digraphs, and • reachability on certain classes o ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
We study the complexity of restricted versions of stconnectivity, which is the standard complete problem for NL. Grid graphs are a useful tool in this regard, since • reachability on grid graphs is logspaceequivalent to reachability in general planar digraphs, and • reachability on certain classes of grid graphs gives natural examples of problems that are hard for NC 1 under AC 0 reductions but are not known to be hard for L; they thus give insight into the structure of L. In addition to explicating the structure of L, another of our goals is to expand the class of digraphs for which connectivity can be solved in logspace, by building on the work of Jakoby et al. [11], who showed that reachability in seriesparallel digraphs is solvable in L. Our main results are: • Many of the natural restrictions on gridgraph reachability (GGR) are equivalent under AC 0
Oneinputface MPCVP is Hard for L, but in LogDCFL
"... A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the oneinputface monotone planar circuit value problem (MPCVP) is in NC 2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that ev ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the oneinputface monotone planar circuit value problem (MPCVP) is in NC 2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that evaluating monotone upward stratified circuits, a restricted version of the oneinputface MPCVP, is in LogDCFL. In this paper, we prove that the unrestricted oneinputface MPCVP is also in LogDCFL. We also show this problem to be Lhard under quantifier free projections.
LTL path checking is efficiently parallelizable
 Proc. 36th Int. Conf. Autom. Lang. Program., Part II, Rhodes (Susanne Albers, Alberto MarchettiSpaccamela, Yossi Matias, Sotiris E. Nikoletseas and Wolfgang Thomas, eds.), LNCS 5556, 235– 246, 2009. Leslie Lamport. ‘Sometime’ is sometimes ‘not never’, Pr
"... Abstract. We present an AC 1 (logDCFL) algorithm for checking LTL formulas over finite paths, thus establishing that the problem can be efficiently parallelized. Our construction provides a foundation for the parallelization of various applications in monitoring, testing, and verification. Linearti ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We present an AC 1 (logDCFL) algorithm for checking LTL formulas over finite paths, thus establishing that the problem can be efficiently parallelized. Our construction provides a foundation for the parallelization of various applications in monitoring, testing, and verification. Lineartime temporal logic (LTL) is the standard specification language to describe properties of reactive computation paths. The problem of checking whether a given finite path satisfies an LTL formula plays a key role in monitoring and runtime verification [12,10,6,1,4], where individual paths are checked either online, during the execution of the system, or offline, for example based on an error report. Similarly, path checking occurs in testing [2] and in several static verification techniques, notably in MonteCarlobased probabilistic verification, where large numbers of randomly generated sample paths are analyzed [22]. Somewhat surprisingly, given the widespread use of LTL, the complexity of the path checking problem is still open [18]. The established upper bound is P: The algorithms in the literature traverse the path sequentially (cf. [10,18,12]);
Classification of Planar Upward Embedding
"... We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in R 3 and introduces new classes of planar upward drawable graphs, where some of them even ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in R 3 and introduces new classes of planar upward drawable graphs, where some of them even allow cycles. Our approach leads to a classification of planar upward embeddability. In particular, we show the coincidence of the classes of planar upward drawable graphs on the sphere and on the standing cylinder. These classes coincide with the classes of planar upward drawable graphs with a homogeneous field on a cylinder and with a radial field in the plane. A cyclic field in the plane introduces the new class RUP of upward drawable graphs, which can be embedded on a rolling cylinder. We establish strict inclusions for planar upward drawability on the plane, the sphere, the rolling cylinder, and the torus, even for acyclic graphs. Finally, upward drawability remains NPhard for the standing cylinder and the torus; for the cylinder this was left as an open problem by Limaye et al.
The Duals of Upward Planar Graphs on Cylinders
"... We consider directed planar graphs with an upward planar drawing on the rolling and standing cylinders. These classes extend the upward planar graphs in the plane. Here, we address the dual graphs. Our main result is a combinatorial characterization of these sets of upward planar graphs. It basical ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We consider directed planar graphs with an upward planar drawing on the rolling and standing cylinders. These classes extend the upward planar graphs in the plane. Here, we address the dual graphs. Our main result is a combinatorial characterization of these sets of upward planar graphs. It basically shows that the roles of the standing and the rolling cylinders are interchanged for their duals.
Rolling Upward Planarity Testing of Strongly Connected Graphs
"... A graph is upward planar if it can be drawn without edge crossings such that all edges point upward. Upward planar graphs have been studied on the plane, the standing and rolling cylinders. For all these surfaces, the respective decision problem N Phard in general. Efficient algorithms exist if the ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
A graph is upward planar if it can be drawn without edge crossings such that all edges point upward. Upward planar graphs have been studied on the plane, the standing and rolling cylinders. For all these surfaces, the respective decision problem N Phard in general. Efficient algorithms exist if the graph contains a single source and a single sink, but only for the plane and standing cylinder Here we show that there is a lineartime algorithm to test whether a strongly connected graph is upward planar on the rolling cylinder. For our algorithm, we introduce dual and directed SPQRtrees as extensions of SPQRtrees.
Efficient parallel path checking for lineartime temporal logic with past and bounds
 Logical Methods in Computer Science
"... Vol. 8(4:10)2012, pp. 1–24 www.lmcsonline.org ..."
(Show Context)
The Pcomplete Circuit Value Problem CVP, when restricted
"... to monotone planar circuits MPCVP, is known to be in NC 3, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which ..."
Abstract
 Add to MetaCart
to monotone planar circuits MPCVP, is known to be in NC 3, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC 1 (LogDCFL), while monotone circuits with oneinputface planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC 1 (LogDCFL). We reexamine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar nonmonotone circuits with polylogarithmic negationheight can be evaluated in NC.
IMPROVED UPPER BOUNDS IN NC FOR MONOTONE PLANAR CIRCUIT VALUE AND SOME RESTRICTIONS AND GENERALIZATIONS
"... and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward ..."
Abstract
 Add to MetaCart
and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we reexamine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC 1 (LogDCFL), while monotone circuits with oneinputface planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC 1 (LogDCFL). We reexamine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar nonmonotone circuits with polylogarithmic negationheight can be evaluated in NC.
Oneinputface MPCVP is Hard for L, but in
"... Abstract. A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the oneinputface monotone planar circuit value problem (MPCVP) is in NC2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the oneinputface monotone planar circuit value problem (MPCVP) is in NC2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that evaluating monotone upward stratified circuits, a restricted version of the oneinputface MPCVP, is in LogDCFL. In this paper, we prove that the unrestricted oneinputface MPCVP is also in LogDCFL. We also show this problem to be Lhard under quantifier free projections. Key Words: L, LogDCFL, monotone planar circuits. 1