Results 21  30
of
41
Quantum Branching Programs and SpaceBounded Nonuniform Quantum Complexity
, 2005
"... In this paper, the space complexity of nonuniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and nonuniform quantum Turing machines, simulations between these two models are presented which allow to trans ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
In this paper, the space complexity of nonuniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and nonuniform quantum Turing machines, simulations between these two models are presented which allow to transfer upper and lower bound results. Exploiting additional insights about the connection between the running time and the precision of amplitudes, it is shown that nonuniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Furthermore, quantum ordered binary decision diagrams (QOBDDs) are considered, which are restricted QBPs that can be regarded as a nonuniform analog of oneway quantum finite automata. Upper and lower bounds are proved that allow a classification of the computational power of QOBDDs in comparison to usual deterministic and randomized variants of the model. Finally, an extension of QBPs is proposed where the performed unitary operation may depend on the result of a previous measurement. A simulation of randomized BPs by this generalized QBP model as well as exponential lower bounds for its ordered variant are presented.
BDDs  Design, Analysis, Complexity, and Applications
 Discrete Applied Mathematics 138
, 2004
"... BDDs (binary decision diagrams) and their variants are the most frequently used representation types or data structures for boolean functions. Research on BDD variants has turned out to be one of the areas where the symbiosis between theoretical investigations in algorithm design and analysis, compl ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
BDDs (binary decision diagrams) and their variants are the most frequently used representation types or data structures for boolean functions. Research on BDD variants has turned out to be one of the areas where the symbiosis between theoretical investigations in algorithm design and analysis, complexity theory, and applications has led to progress in theory and in applications. Here the different roots of the interest in BDDs are described, the main BDD variants and their algorithmic properties are presented, the representation size of selected functions is investigated, lower bound techniques are discussed and applications to algorithmic graph problems and hardware verification problems are presented.
Some Topics in Parallel Computation and Branching Programs
, 1995
"... There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition, several restrictions of branching programs, for example ordered binary decision diagrams, have proven to be successful data structures in several VLSI design and verification applications. We construct a branching program of o(n log 3 n) nodes for computing any threshold function on n variables and a branching program of o(n log 4 n) nodes for determining the sum of n variables modulo a fixed divisor. These are improvements over constructions of size 2(n 3=2 ) due to Lupanov [Lup65]. The second p...
Binary Decision Diagrams
"... Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerfu ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerful lower bound techniques, various types of restricted decision diagrams are investigated. They lead to new lower bound techniques and some of them allow efficient algorithms for a list of operations on boolean functions. Indeed, restricted decision diagrams like ordered binary decision diagrams (OBDDs) are the most common data structure for boolean functions with many applications in verification, model checking, CAD tools, and graph problems. From a complexity theoretical point of view also randomized and nondeterministic decision diagrams are of interest.
Approximation of Boolean Functions by Combinatorial Rectangles
 Electr. Coll. on Comp. Compl
, 2000
"... This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of Boolean functions in this model is very sensitive to the allowed error: There is an explicitly defined sequence of functions f n : {0, 1} n # {0, 1} such that f n has rectangle approximations with a constant number of rectangles and onesided error 1/3+o(1) or twosided error 1/4+2 #(n) , but, on the other hand, f n requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant. Rectangle partitions and rectangle approximations with the same partition of the input variables for all rectangles have been thoroughly investigated in communication complexity theory. The complexity measures where each r...
Computing Jacobi symbols modulo sparse integers and polynomials and some applications
 J. Algorithms
"... ..."
(Show Context)
Amplifying Circuit Lower Bounds Against Polynomial Time With Applications
 In IEEE Conference on Computational Complexity
"... We give a selfreduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. • Amplifying SizeDepth Lower Bounds. If CircEval has Boolean circuits of n k size and n1−δ depth for some k and δ, then for every ε> 0, there is a δ ′> 0 such that CircEval has circui ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We give a selfreduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. • Amplifying SizeDepth Lower Bounds. If CircEval has Boolean circuits of n k size and n1−δ depth for some k and δ, then for every ε> 0, there is a δ ′> 0 such that CircEval has circuits of n1+ε size and n1−δ ′ depth. Moreover, the resulting circuits require only Õ(nε) bits of nonuniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound). • Lower Bounds for Quantified Boolean Formulas. Let c, d> 1 and e < 1 satisfy c < (1 − e + d)/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n c], or the Circuit Evaluation problem cannot be solved with circuits of n d size and n e depth. This implies unconditional polynomialtime uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ctime uniform NC circuits, for all c < 2. 1
TimeSpace TradeOffs For Undirected STConnectivity on a JAG
"... The following is a second proof of (basically) the same undirected stconnectivity result using recursive flyswatters as given in my thesis and in STOC93 [Ed93a, EdPHD]. The input graph and the reduction techniques in the two proofs are similar. The main difference is that JAG result is reduced to ..."
Abstract
 Add to MetaCart
The following is a second proof of (basically) the same undirected stconnectivity result using recursive flyswatters as given in my thesis and in STOC93 [Ed93a, EdPHD]. The input graph and the reduction techniques in the two proofs are similar. The main difference is that JAG result is reduced to a different game. In this paper, the game consists of a pebble walking on a line. The movements of the pebble are directed by a player and a random input. The conjecture is that the player cannot get the pebble across the line much faster than that done by a random walk. Likely, however, this is hard to prove. What can be proven is that this game becomes equivalent to the game in the original paper, if the player who is directing the pebble always knows where in the line pebble is. Therefore, the lower bound for the original game applies to this new game. Hence, the JAG lower bound proved in this paper is the same as that proven before. Two advantages of this new proof are that it is a litt...