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16
Exponential Algorithmic Speedup by a Quantum Walk
"... We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fouri ..."
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Cited by 102 (4 self)
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We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk e#ciently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
Randomization and Derandomization in SpaceBounded Computation
 In Proceedings of the 11th Annual IEEE Conference on Computational Complexity
, 1996
"... This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators tha ..."
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Cited by 36 (0 self)
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This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic spacebounded computations, and the application of such generators to obtain deterministic simulations.
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 25 (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].
A Sublinear Space, Polynomial Time Algorithm for Directed st Connectivity
 IN PROCEEDINGS, STRUCTURE IN COMPLEXITY THEORY, SEVENTH ANNUAL CONFERENCE
, 1992
"... Directed st connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the first known deterministic sublinear space, polynomial time algorithm for directed st connectivity. For nvertex graphs, our algorithm can use as little as ..."
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Cited by 19 (4 self)
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Directed st connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the first known deterministic sublinear space, polynomial time algorithm for directed st connectivity. For nvertex graphs, our algorithm can use as little as n=2 \Theta( p log n) space while still running in polynomial time.
TimeSpace Lower Bounds for Directed st Connectivity on JAG Models (Extended Abstract)
, 1993
"... Directed st connectivity is the problem of detecting whether there is a path from a distinguished vertex s to a distinguished vertex t in a directed graph. We prove timespace lower bounds of ST = \Omega\Gamma n 2 = log n) and S 1=2 T = \Omega\Gamma mn 1=2 ) for Cook and Rackoff's JAG mode ..."
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Cited by 11 (2 self)
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Directed st connectivity is the problem of detecting whether there is a path from a distinguished vertex s to a distinguished vertex t in a directed graph. We prove timespace lower bounds of ST = \Omega\Gamma n 2 = log n) and S 1=2 T = \Omega\Gamma mn 1=2 ) for Cook and Rackoff's JAG model [8], where n is the number of vertices and m the number of edges in the input graph, and S is the space and T the time used by the JAG. We also prove a timespace lower bound of S 1=3 T = \Omega\Gamma m 2=3 n 2=3 ) on the more powerful nodenamed JAG model of Poon [13]. These bounds approach the known upper bound of T = O(m) when S = \Theta(n log n).
Tight lower bounds for stconnectivity on the NNJAG model
 SIAM J. on Computing
, 1999
"... Abstract. Directed stconnectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Compu ..."
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Cited by 8 (1 self)
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Abstract. Directed stconnectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, 1993, pp. 218–227]. Let n be the number of nodes in the input graph and S and T be the space and time used by the NNJAG, respectively. We show that, for any δ>0, if an NNJAG uses space S ∈ O(n1−δ), then T ∈ 2Ω(log2 (n/S)) ; otherwise n log n) / log log n) S
TimeSpace Lower Bounds for Undirected and Directed STConnectivity on JAG
, 1993
"... Directed and undirected stconnectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graph ..."
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Cited by 5 (2 self)
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Directed and undirected stconnectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graphs), these upper bounds are not simultaneously achievable. This uses new entropy techniques to prove tight bounds on a game involving a helper and a player that models a computation having precomputed information about the input stored in its bounded space. The second result proves that a JAG requires a timespace tradeoff of T \Theta S 1 2 2\Omega i mn 1 2 j to compute directed stconnectivity. The third result proves a timespace tradeoff of T \Theta S 1 3 2\Omega i m 2 3 n 2 3 j on a version of the...
Monotone circuits for connectivity have depth (log ) 2(1
 SIAM Journal on Computing
, 1998
"... We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been p ..."
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Cited by 4 (0 self)
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We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been published in SIAM Journal on Computing is hence subject to copyright restrictions. It is for personal use only. 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
Improved Depth Lower Bounds for Small Distance Connectivity
, 1995
"... We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fanin circuits, requires depth \Omega\Gammapth log k), improving on a depth lower b ..."
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Cited by 3 (0 self)
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We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fanin circuits, requires depth \Omega\Gammapth log k), improving on a depth lower bound of \Omega\Gamma/16 k) when k = log O(1) n given in [2, 8]. In addition we show that there is a constant c such that for k log n, any depth d unbounded fanin circuit for this problem requires size at least n ck ffl d where ffl d = OE \Gamma2d =3 and OE is the golden mean. This latter result improves on an n \Omega\Gamma711 (d+3) k) bound from [2, 8] where log (i) is the ifold composition of log with itself. The key to our technique is a new form of `switching lemma' which combines some of the features of iteratively shortening terms due to Furst, Saxe, and Sipser [13] and Ajtai [1] with the kinds of switching lemma arguments introduced by Yao [18], Hastad [14], and Cai [9]...