Width hierarchy for k-OBDD of small width

Computer Science – Theory and Applications

Khadieva

2017

Abstract. We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to "width" complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called "reordering"), which allows to build Boolean function g from Boolean Function f , such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is 2 Ω(n/ log n) and present quantum OBDD of width O(n 2 ). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using "reordering" method we extend a hierarchy for k-OBDD of polynomial size, for k = o(n/ log 3 n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and probabilistic k-OBDDs of superpolynomial and subexponential size.

Section: Introductionsupporting

confidence: 52%

Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams

Computer Science – Theory and Applications

Khadieva

2017

“…We take this result from [6] and prove similar results for nondeterministic and probabilistic k-OBDDs in the paper. In Sections 3 and 4 we consider Boolean function SAF k,w (X) (Shuffled Address Function) which is presented in [14]. We prove that SAF k,w (X) cannot be represented by constant width k-OBDDs for k = o(n/ log n) and sublinear width k-OBDDs for k = o(n 1−α / log n), 0 < α < 0.49.…”

Section: Preliminaries and Resultsmentioning

confidence: 99%

“…Lower bound for Boolean Function SAF k,w . We consider Boolean function SAF k,w , which was defined in [14]. Function SAF k,w .…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

2016

Lobachevskii J Math

Self Cite

Abstract. The paper examines hierarchies for nondeterministic and deterministic ordered read-k-times Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n 1/2 / log 3/2 n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read k-times Branching programs for k = o( √ log n/ log log n) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003.We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/ log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs.Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/ log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs.

“…Proof (Sketch). The shuffled address function from [41] is used. The results also can be shown using the modification of the pointer jumping function from [19,57] that is called matrix XOR pointer jumping function [3].…”

Section: Theorem 3 (Khadievmentioning

confidence: 99%

Classical and Quantum Computations with Restricted Memory

Ablayev

Adventures Between Lower Bounds and Higher Altitudes

et al. 2018

Automata and branching programs are known models of computation with restricted memory. These models of computation were in focus of a large number of researchers during the last decades. Streaming algorithms are a modern model of computation with restricted memory. In this paper, we present recent results on the comparative computational power of quantum and classical models of branching programs and streaming algorithms. In addition to comparative complexity results, we present a quantum branching program for computing a practically important quantum function (quantum hash function) and prove optimality of this algorithm.