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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.
Reachability Problems: An Update
"... Abstract. There has been a great deal of progress in the fifteen years that have elapsed since Wigderson published his survey on the complexity of the graph connectivity problem [Wig92]. Most significantly, Reingold solved the longstanding question of the complexity of the st connectivity problem i ..."
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Cited by 3 (0 self)
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Abstract. There has been a great deal of progress in the fifteen years that have elapsed since Wigderson published his survey on the complexity of the graph connectivity problem [Wig92]. Most significantly, Reingold solved the longstanding question of the complexity of the st connectivity problem in undirected graphs, showing that this is complete for logspace (L) [Rei05]. This survey talk will focus on some of the remaining open questions dealing with graph reachability problems. Particular attention will be paid to these topics: – Reachability in planar directed graphs (and more generally, in graphs of low genus) [ADR05,BTV07]. – Reachability in different classes of grid graphs [ABC + 06]. – Reachability in mangroves [AL98]. The problem of finding a path from one vertex to another in a graph is the first problem that was identified as being complete for a natural subclass of P; it was shown to be complete for nondeterministic logspace (NL) by Jones [Jon75]. Restricted versions of this problem were subsequently shown to be complete for other natural complexity
NLprintable sets and Nondeterministic Kolmogorov Complexity
, 2003
"... This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity. ..."
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Cited by 2 (0 self)
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This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity.
On the Complexity of Free Monoid Morphisms
, 1999
"... . We locate the complexities of evaluating, of inverting, and of testing membership in the image of, morphisms h : \Sigma ! \Delta . By and large, we show these problems complete for classes within NL. Then we develop new properties of finite codes and of finite sets of words, which yield imag ..."
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. We locate the complexities of evaluating, of inverting, and of testing membership in the image of, morphisms h : \Sigma ! \Delta . By and large, we show these problems complete for classes within NL. Then we develop new properties of finite codes and of finite sets of words, which yield image membership subproblems that are closely tied to the unambiguous space classes found between L and NL. 1 Introduction Free monoid morphisms h : \Sigma ! \Delta , for finite alphabets \Sigma and \Delta, are an important concept in the theory of formal languages (e.g. [6, 11]), and they are relevant to complexity theory. Indeed, it is well known (e.g. [7]) that NP = Closure( AC 0 m ; HOM n:e: ) ae Closure( AC 0 m ; HOM) = R:E:; where AC 0 m denotes manyone AC 0 reducibility, HOM (resp. HOM n:e: ) is the set of morphisms (resp. nonerasing morphisms), and Closure denotes the smallest class of languages containing a finite nontrivial language and closed under the relations speci...
Unambiguous Functions in Logarithmic Space
"... Abstract. We investigate different variants of unambiguity in the context of computing multivalued functions. We propose a modification to the standard computation models of Turing Machines and configuration graphs, which allows for unambiguitypreserving composition. We define a notion of reductio ..."
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Abstract. We investigate different variants of unambiguity in the context of computing multivalued functions. We propose a modification to the standard computation models of Turing Machines and configuration graphs, which allows for unambiguitypreserving composition. We define a notion of reductions (based on function composition), which allows nondeterminism but controls its level of ambiguity. In light of this framework we establish reductions between different variants of path counting problems. We obtain improvements of results related to inductive counting.
RUSPACE(log n) \subseteq DSPACE(log² n/log log n)
 THE 7TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC’96
, 1998
"... We present a deterministic algorithm running in space O , log n= log log n # solving the connectivity problem on strongly unambiguous graphs. In addition, we presentanO#log n# timebounded algorithm for this problem running on a parallel pointer machine. ..."
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We present a deterministic algorithm running in space O , log n= log log n # solving the connectivity problem on strongly unambiguous graphs. In addition, we presentanO#log n# timebounded algorithm for this problem running on a parallel pointer machine.
StUSPACE(log n) \subseteq DSPACE(log² n/ log log n)
"... We present a deterministic algorithm running in space O \Gamma log 2 n= log log n \Delta solving the connectivity problem on strongly unambiguous graphs. In addition, we present an O(logn) timebounded algorithm for this problem running on a parallel pointer machine. ..."
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We present a deterministic algorithm running in space O \Gamma log 2 n= log log n \Delta solving the connectivity problem on strongly unambiguous graphs. In addition, we present an O(logn) timebounded algorithm for this problem running on a parallel pointer machine.
COMPLEXITY THEORETIC ASPECTS OF PLANAR RESTRICTIONS AND OBLIVIOUSNESS
, 2006
"... In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized ci ..."
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In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the wellknown circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NLcomplete). We prove that this problem logspacereduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining
SpaceEfficient Algorithms for Reachability in SurfaceEmbedded Graphs
, 2010
"... We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let G(m, g) be the class of directed acyclic graphs with m = m(n) source vertices embedded on a surface (orientable or nonorientable) of genus g = g(n). We give a logspace reduction that on in ..."
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We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let G(m, g) be the class of directed acyclic graphs with m = m(n) source vertices embedded on a surface (orientable or nonorientable) of genus g = g(n). We give a logspace reduction that on input G, u, v where G ∈G(m, g) and u and v are two vertices of G, outputs G,u,v where G is directed graph, and u,v are vertices of G, so that (a) there is a directed path from u to v in G if and only if there is a directed path from u to v in G and (b) G has O(m + g) vertices. By a direct application of Savitch’s theorem on the reduced instance we get a deterministic O(log n + log 2 (m + g))space algorithm for the reachability problem for graphs in G(m, g). By setting m and g to be 2O(√log n) we get that the reachability problem for directed acyclic graphs with 2O(√log n) O( sources embedded on surfaces of genus 2 √ log n) is in L (deterministic logarithmic