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30
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 27 (6 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.
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 23 (7 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 22 (4 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 ...
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 18 (7 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
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 15 (4 self)
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We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 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'ak ..."
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Cited by 13 (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.
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 8 (1 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
Arithmetic Complexity, Kleene Closure, and Formal Power Series
, 1999
"... The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC¹ and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity c ..."
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Cited by 6 (2 self)
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The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC¹ and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC¹ and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: There is a language in AC 0 for which inversion and root extraction are GapLcomplete, and there is a finite set for which inversion and root extraction are GapNC¹complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC¹, such as GapAC^0. We initiate work in this direction by classifyi...
On some Recognizable Picturelanguages
 Proceedings of the 23th Conference on Mathematical Foundations of Computer Science, number 1450 in Lecture Notes in Computer Science
, 1998
"... . We show that the language of pictures over fa; bg, where all occurring b's are connected is recognizable, which solves an open problem in [Mat98]. We generalize the used construction to show that monocausal deterministically recognizable picture languages are recognizable, which is surprisingly no ..."
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Cited by 4 (2 self)
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. We show that the language of pictures over fa; bg, where all occurring b's are connected is recognizable, which solves an open problem in [Mat98]. We generalize the used construction to show that monocausal deterministically recognizable picture languages are recognizable, which is surprisingly nontrivial. Furthermore we show that the language of pictures over fa; bg, where the number of a's is equal to the number of b's is nonuniformly recognizable. 1 Introduction In [GRST94] pictures are defined as twodimensional rectangular arrays of symbols of a given alphabet. A set (language) of pictures is called recognizable if it is recognized by a finite tiling system. It was shown in [GRST94] that a picture language is recognizable iff it is definable in existential monadic secondorder logic. In [Wil97] it was shown that starfree picture expressions are strictly weaker than firstorder logic. A comparison to other regular and contextfree formalisms to describe picture languages can be...
On the Power of Unambiguity in Logspace
, 2010
"... We report progress on the NL vs UL problem. We show unconditionally that the complexity class ReachFewL ⊆ UL. This improves on the earlier known upper bound ReachFewL ⊆ FewL. We investigate the complexity of minuniqueness a central notion in studying the NL vs UL problem. – We show that minuniq ..."
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Cited by 4 (3 self)
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We report progress on the NL vs UL problem. We show unconditionally that the complexity class ReachFewL ⊆ UL. This improves on the earlier known upper bound ReachFewL ⊆ FewL. We investigate the complexity of minuniqueness a central notion in studying the NL vs UL problem. – We show that minuniqueness is necessary and sufficient for showing NL = UL. – We revisit the class OptL[log n] and show that ShortestPathLength computing the length of the shortest path in a DAG, is complete for OptL[log n]. – We introduce UOptL[log n], an unambiguous version of OptL[log n], and show that (a) NL = UL if and only if OptL[log n] = UOptL[log n], (b) LogFew ≤ UOptL[log n] ≤ SPL. We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in UL.