Least upper bounds on OBDD sizes

Heap, M.A.; Mercer, M.R.

doi:10.1109/12.286311

Cited by 17 publications

(11 citation statements)

References 3 publications

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“…We first bound the number of different subtrees with at most k leaves in t. Afterwards we will estimate the size of the remaining top tree. The same strategy is used for instance in [22,34] to derive a worst-case upper bound on the size of binary decision diagrams. Claim 1.…”

Section: Size Of the Minimal Dagmentioning

confidence: 99%

Constructing small tree grammars and small circuits for formulas

Ganardi

Hucke

Je³

et al. 2017

Journal of Computer and System Sciences

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It is shown that every tree of size n over a fixed set of σ different ranked symbols can be decomposed (in linear time as well as in logspace) into O n log σ n = O n log σ log n many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O n log σ n , which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al. [6], for which only the weaker upper bound of O n log 0.19 σ n (which was very recently improved to O n·log log σ n log σ n [23]) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m ≤ n different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size O n·log m log n and depth O(log n). This refines a classical result of Brent from 1974, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n). A short version of this paper appeared as [24]. * The fourth and fifth author are supported by the DFG research project QUANT-KOMP (Lo 748/10-1). †

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Section: Size Of the Minimal Dagmentioning

confidence: 99%

Constructing small tree grammars and small circuits for formulas

Ganardi

Hucke

Je³

et al. 2017

Journal of Computer and System Sciences

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“…We can then deduce that r i ≤ 2 i and r i ≤ 2 2 n−i , which leads to the well known property (see [2] or [5]):…”

Section: Canonical Reduced Graphs Of Boolean Functionsmentioning

confidence: 93%

On maximal QROBDD's of Boolean functions

Michon

Yunès

Valarcher

2005

RAIRO-Theor. Inf. Appl.

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We investigate the structure of "worst-case" quasi reduced ordered decision diagrams and Boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" Boolean functions as functions whose QROBDD are "worstcase" ones. So we exhibit the relation between hard functions and the Storage Access function (also known as Multiplexer).

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“…Instead, we will describe an OBDD WP(f ) from [17,13,4] that meets the upperbound OBDD(P f ) ≤ (2 + o (1))2 m /m. This OBDD also has the benefit of having optimal length m. Based on this result, even if f is an arbitrary Boolean database, the BddCpir protocol has communication complexity Θ(k · log 2 n) and server's online computational complexity Θ(n/ log n).…”

Section: Class 1: Arbitrary Databasesmentioning

confidence: 99%

“…Now, let f : {0, 1} m → {0, 1} for some ≥ 1. By the already mentioned upperbound of [17,13,4], clearly BDD(f ) ≤ · (2 + o(1))2 m /m by just evaluating BDDs in parallel. Thus, there exists a sublinear-computation CPIR protocol for say any ≤ m/3.…”

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First CPIR Protocol with Data-Dependent Computation

Lipmaa

2010

Lecture Notes in Computer Science

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We design a new (n, 1)-CPIR protocol BddCpir for -bit strings as a combination of a noncryptographic (BDD-based) data structure and a more basic cryptographic primitive (communication-efficient (2, 1)-CPIR). BddCpir is the first CPIR protocol where server's online computation depends substantially on the concrete database. We then show that (a) for reasonably small values of , BddCpir is guaranteed to have simultaneously log-squared communication and sublinear online computation, and (b) BddCpir can handle huge but sparse matrices, common in data-mining applications, significantly more efficiently compared to all previous protocols. The security of BddCpir can be based on the well-known Decisional Composite Residuosity assumption.

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Least upper bounds on OBDD sizes

Cited by 17 publications

References 3 publications

Constructing small tree grammars and small circuits for formulas

Constructing small tree grammars and small circuits for formulas

On maximal QROBDD's of Boolean functions

First CPIR Protocol with Data-Dependent Computation

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