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SPACE HIERARCHY RESULTS FOR RANDOMIZED MODELS
"... Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems ..."
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Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following. Let s(n) be any spaceconstructible function that is Ω(log n) and such that s(an) = O(s(n)) for all constants a, and let s ′ (n) be any function that is ω(s(n)). There exists a language computable by twosided error randomized machines using s ′ (n) space and one bit of advice that is not computable by twosided error randomized machines using s(n) space and min(s(n), n) bits of advice. There exists a language computable by zerosided error randomized machines in space s ′ (n) with one bit of advice that is not computable by onesided error randomized machines using s(n) space and min(s(n), n) bits of advice. The condition that s(an) = O(s(n)) is a technical condition satisfied by typical space bounds that are at most linear. We also obtain weaker results that apply to generic semantic models of computation. 1.
Uniform Inclusions in Nondeterministic Logspace
 Randomized Algorithms
, 1998
"... We show that the complexity class LogFew is contained in NL " SPL. Previously, this was known only to hold in the nonuniform setting. Key Words: Nondeterministic Logspace Computation, Nonuniform Complexity, Derandomization, fflbiased Sample Space. 1 Introduction In [RA97], a probabilistic ..."
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We show that the complexity class LogFew is contained in NL " SPL. Previously, this was known only to hold in the nonuniform setting. Key Words: Nondeterministic Logspace Computation, Nonuniform Complexity, Derandomization, fflbiased Sample Space. 1 Introduction In [RA97], a probabilistic construction was used to show that the complexity classes NL/poly and UL/poly coincide. That is, in the context of nonuniform complexity, nonuniform logspace is no more powerful than unambiguous logspace. It was observed in [AR98] that the equality NL=UL holds also in the uniform setting, under a plausible hypothesis concerning pseudorandom number generators. However, it remains an important open question whether NL=UL can be established without resorting to unproved assumptions. The results and techniques of [RA97] were extended in [AR98] in a number of ways. One extension involves the class LogFew, defined in [BDHM92]. (Formal definitions appear below.) No inclusion relation was known between...
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.
ReachFewL = ReachUL
, 2011
"... We show that two complexity classes introduced about two decades ago are equal. ReachUL is the class of problems decided by nondeterministic logspace machines which on every input have at most one computation path from the start configuration to any other configuration. ReachFewL, a natural general ..."
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We show that two complexity classes introduced about two decades ago are equal. ReachUL is the class of problems decided by nondeterministic logspace machines which on every input have at most one computation path from the start configuration to any other configuration. ReachFewL, a natural generalization of ReachUL, is the class of problems decided by nondeterministic logspace machines which on every input have at most polynomially many computation paths from the start configuration to any other configuration. We show that ReachFewL = ReachUL.
Space Complexity of the Directed Reacha bility Problem over SurfaceEmbedded Graphs
"... Abstract. The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachability problem over dire ..."
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Abstract. The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachability problem over directed graphs are: (1) improving Savitch’s O(log2 n) space bound, (2) designing a polynomialtime algorithm with O(n1−) space bound, and (3) designing an unambiguous nondeterministic logspace (UL) algorithm. These are wellknown open questions in complexity theory, and solving any one of them will be a major breakthrough. We will discuss some of the recent progress reported on these questions for certain subclasses of surfaceembedded directed graphs.
Unambiguity in Logspace – a Survey
, 2001
"... The restriction of nondeterminism to unambiguous computations has been used as an attempt to clarify the relationship between deterministic and nondeterministic models of computation. In this paper we will consider three different, increasingly more restrictive notions of unambiguity. While all of t ..."
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The restriction of nondeterminism to unambiguous computations has been used as an attempt to clarify the relationship between deterministic and nondeterministic models of computation. In this paper we will consider three different, increasingly more restrictive notions of unambiguity. While all of them coincide for polynomial time bounded computation, this does not seem to happen for logarithmic space bounded computation. Moreover, the least restrictive class UL seems to coincide with NL (at least under a widely believed assumption) while the more restrictive classes StUL and RUL have a deterministic algorithm that runs in time log2 n log log n.Soin this paper we will give an overview over these classes and their relationship with deterministic and nondeterministic logspace. 1
ON THE SPACE COMPLEXITY OF DIRECTED GRAPH REACHABILITY
, 2007
"... Graph reachability problems are fundamental to the study of complexity classes. It is well known that the reachability problem for general directed graphs is known to be complete for nondeterministic logspace (NL). Various restrictions of graph reachability are known to capture low level complexit ..."
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Graph reachability problems are fundamental to the study of complexity classes. It is well known that the reachability problem for general directed graphs is known to be complete for nondeterministic logspace (NL). Various restrictions of graph reachability are known to capture low level complexity classes. Recently, in a breakthrough result, it was shown that reachability problem for undirected graphs is complete for deterministic logspace (L). In this thesis we investigate the space complexity of reachability problem for directed planar graphs (denoted by PlanarReach). A graph is planar if it can be drawn on a plane without any crossing edges. Planar graphs are an important and natural subclass of general graphs and are well studied in graph theory. However, the space complexity of reachability problem for directed planar graphs is not yet well understood. It is not known whether PlanarReach is complete for NL. On the other hand researchers did not know any upper bound better than NL for this problem. We make progress on identifying the space complexity of PlanarReach. We
STCON in Directed UniquePath Graphs
"... ABSTRACT. We study the problem of spaceefficient polynomialtime algorithms for directed stconnectivity (STCON). Given a directed graph G, and a pair of vertices s,t, the STCON problem is to decide if thereexistsapathfrom s to t in G. For general graphs, thebestpolynomialtime algorithm forSTCONuse ..."
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ABSTRACT. We study the problem of spaceefficient polynomialtime algorithms for directed stconnectivity (STCON). Given a directed graph G, and a pair of vertices s,t, the STCON problem is to decide if thereexistsapathfrom s to t in G. For general graphs, thebestpolynomialtime algorithm forSTCONusesspacethatisonlyslightlysublinear. However,forspecialclassesofdirectedgraphs, polynomialtime polylogarithmicspace algorithms are known for STCON. In this paper, we continuethisthreadofresearchandstudyaclassofgraphscalleduniquepathgraphswithrespecttosource s, where there is at most one simple path from s to any vertex in the graph. For these graphs, we give a polynomialtime algorithm that uses Õ(n ε) space for any constant ε ∈ (0,1]. We also give a polynomialtime, Õ(n ε)space algorithm to recognize uniquepath graphs. Uniquepath graphs are related to configuration graphs of unambiguous logspace computations, but they can have some directed cycles. Our results may be viewed along the continuum of sublinearspace polynomialtime algorithms for STCON in different classes of directed graphs from slightly sublinearspace algorithms forgeneral graphs toO(logn) space algorithms fortrees. 1
www.stacsconf.org SPACE HIERARCHY RESULTS FOR RANDOMIZED MODELS
, 2008
"... Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems ..."
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Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following. Let s(n) be any spaceconstructible function that is Ω(log n) and such that s(an) = O(s(n)) for all constants a, and let s ′ (n) be any function that is ω(s(n)). There exists a language computable by twosided error randomized machines using s ′ (n) space and one bit of advice that is not computable by twosided error randomized machines using s(n) space and min(s(n), n) bits of advice. There exists a language computable by zerosided error randomized machines in space s ′ (n) with one bit of advice that is not computable by onesided error randomized machines using s(n) space and min(s(n), n) bits of advice. The condition that s(an) = O(s(n)) is a technical condition satisfied by typical space bounds that are at most linear. We also obtain weaker results that apply to generic semantic models of computation. 1.