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49
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 31 (8 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds
 Journal of Computer and System Sciences
, 1998
"... We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting t ..."
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Cited by 30 (6 self)
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We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting the number of accepting paths of a nondeterministic logspace machine can be done in NL/poly, if the number of paths is small. This clarifies the complexity of the class LogFew (defined and studied in [BDHM91]). Using derandomization techniques, we then improve this to show that this counting problem is in NL. Determining if our other theorems hold in the uniform setting remains an The material in this paper appeared in preliminary form in papers in the Proceedings of the IEEE Conference on Computational Complexity, 1998, and in the Proceedings of the Workshop on Randomized Algorithms, Brno, 1998. y Supported in part by NSF grants CCR9509603 and CCR9734918. z Supported in part by the ...
Determinant: Combinatorics, Algorithms, and Complexity
, 1997
"... We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, a ..."
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Cited by 29 (7 self)
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We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to e#cient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.
Directed planar reachability is in unambiguous logspace
 In Proceedings of IEEE Conference on Computational Complexity CCC
, 2007
"... We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1. ..."
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Cited by 24 (6 self)
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We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.
The directed planar reachability problem
 In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science
, 2005
"... Abstract. We investigate the stconnectivity 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 logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the ..."
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Cited by 23 (9 self)
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Abstract. We investigate the stconnectivity 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 logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previouslystudied subclass of planar graphs known as grid graphs. We show that the directed planar stconnectivity problem reduces to the reachability problem for directed grid graphs. A special case of the gridgraph 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
On Arithmetic Branching Programs
 IN PROC. OF THE 13TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 1998
"... The model of arithmetic branching programs is an algebraic model of computation generalizing the model of modular branching programs. We show that, up to a polynomial factor in size, arithmetic branching programs are equivalent to complements of dependency programs, a model introduced by Pudl&apo ..."
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Cited by 16 (0 self)
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The model of arithmetic branching programs is an algebraic model of computation generalizing the model of modular branching programs. We show that, up to a polynomial factor in size, arithmetic branching programs are equivalent to complements of dependency programs, a model introduced by Pudl'ak and Sgall [20]. Using this equivalence we prove that dependency programs are closed under conjunction over every field, answering an open problem of [20]. Furthermore, we show that span programs, an algebraic model of computation introduced by Karchmer and Wigderson [16], are at least as strong as arithmetic programs; every arithmetic program can be simulated by a span program of size not more than twice the size of the arithmetic program. Using the above results we give a new proof that NL/poly ` \PhiL/poly, first proved by Wigderson [25]. Our simulation of NL/poly is more efficient, and it holds for logspace counting classes over every field.
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 ..."
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Cited by 14 (9 self)
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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
Reachability in K3,3free graphs and K5free graphs is in unambiguous logspace
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2009
"... We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm de ..."
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Cited by 13 (2 self)
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We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The nonplanar components are replaced by planar components that maintain the reachabilty properties. For K5free graphs we also need a decomposition into fourconnected components. A careful analysis finally gives a polynomial size planar graph which can be computed in logspace. We show the same upper bound for computing distances in K3,3free and K5free directed graphs and for computing longest paths in K3,3free and K5free directed acyclic graphs.
THE ISOMORPHISM PROBLEM FOR PLANAR 3CONNECTED GRAPHS IS IN UNAMBIGUOUS LOGSPACE
, 2008
"... The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace ..."
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Cited by 13 (5 self)
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The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace, in fact to UL ∩ coUL. As a consequence of our method we get that the isomorphism problem for oriented graphs is in NL. We also show that the problems are hard for L.
Planar graph isomorphism is in logspace
 In IEEE Conference on Computational Complexity
, 2009
"... Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1 ..."
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Cited by 13 (3 self)
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Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1