Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting.We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p−1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.     

Highly nonlinear balanced Boolean functions with good local and global avalanche characteristics

Here we deal with an interesting subset of n-variable balanced Boolean functions which satisfy strict avalanche criteria. These functions achieve the sum-of-square indicator value (a measure for global avalanche criteria) strictly less than 2²ⁿ⁺¹ and nonlinearity strictly greater than 2ⁿ⁻¹−2^⌊n/2⌋. These parameters are currently best known. Moreover, these functions do not possess any nonzero linear structure. The technique involves a well-known simple construction coupled with very good initial functions obtained by computer search, which were not known earlier.     

An improved FPTAS for Restricted Shortest Path

Given a graph with a cost and a delay on each edge, Restricted Shortest Path (RSP) aims to find a min-cost s-t path subject to an end-to-end delay constraint. The problem is NP-hard. In this note we present an FPTAS with an improved running time of O(mn/ε) for acyclic graphs, where m and n denote the number of edges and nodes in the graph. Our algorithm uses a scaling and rounding technique similar to that of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42]. The novelty of our algorithm lies in its “adaptivity”. During each iteration of our algorithm the approximation parameters are fine-tuned according to the quality of the current solution so that the running time is kept low while progress is guaranteed at each iteration. Our result improves those of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42], Phillips [Proc. 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 776-785], and Raz and Lorenz [Technical Report, 1999].     

Upper bound for algebraic immunity on a subclass of Maiorana McFarland class of bent functions

Studying algebraic immunity of Boolean functions is recently a very important research topic in cryptography. It is recently proved by Courtois and Meier that for any Boolean function of n-variable the maximum algebraic immunity is . We found a large subclass of Maiorana McFarland bent functions on n-variable with a proven low level of algebraic immunity . To the best of our knowledge we provide for the first time a new upper bound for algebraic immunity for a nontrivial class of Boolean functions. We also discuss that this result has some fascinating implications.     

On the lower bounds of the second order nonlinearities of some Boolean functions

The rth order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code R(r,n). In this paper we deduce the lower bounds of the second order nonlinearities of the following two types of Boolean functions:

1.

with d=2^2r+2^r+1 and , where n=6r.

2.

, where x,y∈F_2^t,n=2t,n?6 and i is an integer such that 1?i<t,gcd(2^t-1,2ⁱ+1)=1.

For some λ, the functions of the first type are bent functions, whereas Boolean functions of the second type are all bent functions, i.e., they possess the maximum first order nonlinearity. It is demonstrated that in some cases our bounds are better than the previously obtained bounds.     

Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Ω(n²) lower bound and the O(n3+?) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered “cross-sections” of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+?) with lower bound Ω(n²).In this paper, we determine the combinatorial complexities of these four problems for some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n²) or O(n2+?) and lower bounds of Ω(n²).     

Attribute estimation and testing quasi-symmetry

A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every quasi-symmetric function and, except with an error probability at most δ>0, rejects every function that differs from every quasi-symmetric function on at least a fraction ε>0 of the inputs. For a function of n arguments, the test probes the function at O((n/ε)log(n/δ)) inputs. Our quasi-symmetry test acquires information concerning the arguments on which the function actually depends. To do this, it employs a generalization of the property testing paradigm that we call attribute estimation. Like property testing, attribute estimation uses random sampling to obtain results that have only “one-sided” errors and that are close to accurate with high probability.     

A new lower bound on the monotone network complexity of Boolean sums

Summary Neciporuk [3], Lamagna/Savage [1] and Tarjan [6] determined the monotone network complexity of a set of Boolean sums if each two sums have at most one variable in common. By this result they could define explicitely a set of n Boolean sums which depend on n variables and whose monotone complexity is of order n ^3/2. In the main theorem of this paper we prove a more general lower bound on the monotone network complexity of Boolean sums. Our lower bound is for many Boolean sums the first nontrivial lower bound. On the other side we can prove that the best lower bound which the main theorem yields is the n ^3/2-bound cited above. For the proof we use the technical trick of assuming that certain functions are given for free.     

Improved artificial bee colony algorithm for global optimization

The artificial bee colony algorithm is a relatively new optimization technique. This paper presents an improved artificial bee colony (IABC) algorithm for global optimization. Inspired by differential evolution (DE) and introducing a parameter M, we propose two improved solution search equations, namely “ABC/best/1” and “ABC/rand/1”. Then, in order to take advantage of them and avoid the shortages of them, we use a selective probability p to control the frequency of introducing “ABC/rand/1” and “ABC/best/1” and get a new search mechanism. In addition, to enhance the global convergence speed, when producing the initial population, both the chaotic systems and the opposition-based learning method are employed. Experiments are conducted on a suite of unimodal/multimodal benchmark functions. The results demonstrate the good performance of the IABC algorithm in solving complex numerical optimization problems when compared with thirteen recent algorithms.     

Randomized vs. deterministic decision tree complexity for read-once Boolean functions

We consider the deterministic and the randomized decision tree complexities for Boolean functions, denotedDC(f) andRC(f), respectively. A major open problem is how smallRC(f) can be with respect toDC(f). It is well known thatRC(f)DC(f) ^0.5 for every Boolean functionf (called 0.5-exponent). On the other hand, some Boolean functionf is known to haveRC(f) = (DC(f))^0.753...) (or 0.753...-exponent). It is not known whether there is a Boolean function with exponent smaller than 0.753... Likewise, no lower bound for arbitrary Boolean functions with exponent greater than 0.5 is known.Our result is a 0.51 lower bound on the exponent for everyread-once function. Read-once means that each input variable appears exactly once in the Boolean formula representing the function. To obtain this result we generalize an existing lower bound technique and combine it with restriction arguments. This result provides a lower bound ofn ^0.51 on the number of positions that have to be evaluated by any randomized - pruning algorithm computing the value of any two-person zero-sum game tree withn final positions.     

Fast non-blocking atomic commit: an inherent trade-off

This paper investigates the time-complexity of the non-blocking atomic commit (NBAC) problem in a synchronous distributed model where t out of n processes may fail by crashing. We exhibit for t?3 an inherent trade-off between the fast abort property of NBAC, i.e., aborting a transaction as soon as possible if some process votes “no”, and the fast commit property, i.e., committing a transaction as soon as possible when all processes vote “yes” and no process crashes. We also give two algorithms: the first satisfies fast commit and a weak variant of fast abort, whereas the second satisfies fast abort and a weak variant of fast commit.     

Results on rotation symmetric polynomials over GF(p)

In this paper, we generalize the recent counting results about rotation symmetric Boolean functions to the rotation symmetric polynomials over finite fields GF(p). By using Möbius function, we obtain some formulas for more general n, the number of variables. Some known formula in Boolean case are simplified.     

Construction of 1-resilient Boolean functions with optimum algebraic immunity

In this paper, for an integer n≥10, two classes of n-variable Boolean functions with optimum algebraic immunity (AI) are constructed, and their nonlinearities are also determined. Based on non-degenerate linear transforms to the proposed functions, we can obtain 1-resilient n-variable Boolean functions with optimum AI and high nonlinearity if n?1 is never equal to any power of 2.     

On another Boolean matrix

In his paper “On a Boolean matrix”, Nechiporuk gave an explicit example of a set of n homogeneous monotone Boolean functions of the first degree in n variables that require Ω(n^3/2) two-input gates in any monotone Boolean network computing them. In this note we show how this can be extended to Ω(n^5/3) two-input gates.     

Block sensitivity of minterm-transitive functions

Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures.In a recent work of this type, Sun (2007) [9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f)=Ω(N^1/3). He also showed that there exists such a function for which bs(f)=O(N^3/7lnN). His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) [3].We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f)=Ω(N^3/7). Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which bs(f)=O(N^3/7ln^1/7N).     

Optimal search for rationals

We present a Θ(log₂M)-time algorithm that determines an unknown rational number x in by asking at most 2log₂M+O(1) queries of the form “Is x?y?”.     

A note on the query complexity of the Condorcet winner problem

Given an unknown tournament over {1,…,n}, we show that the query complexity of the question “Is there a vertex with outdegree n−1?” (known as a Condorcet winner in social choice theory) is exactly 2n−⌊log(n)⌋−2. This stands in stark contrast to the evasiveness of this property in general digraphs.     

A demonstration of level-anchored ratio scaling for prediction of grip strength

Level-anchored ratio scaling, such as the Borg CR10 scale^® and the Borg CR100 scale^®, uses verbal anchors in congruence with numbers to give ratio data together with natural levels of intensity. This presupposes that the anchors possess natural positions in the subjective dynamic range and also “numerical” inter-relations. In an experiment, subjects had to produce a force of handgrip corresponding to their conception of “Strong”, followed by a “Maximal” performance. By using the previously found relationship between “Strong” and “Maximal” of 1:2 together with knowledge of the exponent in the power S-R-function (R = c × Sⁿ) for grip strength, n = 1.8, predictions of individual maximal performances were obtained. The predicted values correlated 0.76 with, and deviated only 3% (ns) from, actual maximal performances of grip strength. This result –as previously also found for aerobic capacity–gives a strong support for the use of verbal anchors, so common in category scaling, also in “ratio scaling” and that the Borg CR-scales fulfill the requirements for ratio scales. For estimation of muscular strength, such as grip strength, this present study points to the value of using submaximal determinations as a compliment to maximal performances (e.g., to obtain measures of functional capacity). The results also support the increasingly common use of the CR-methodology in other ergonomic settings concerning suitable design of tools and equipment.     

Malign Distributions for Average Case Circuit Complexity

In contrast to machine models like Turing machines or random access machines, circuits are a static computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, size and depth are natural and simple measures for circuits and provide a worst-case analysis. We consider a new model in which an internal gate is evaluated as soon as its result has been determined by a partial assignment of its inputs. This way, a dynamic notion of delay is obtained which gives rise to an average case measure for the time complexity of circuits. In a previous paper we have obtained tight upper and lower bounds for the average case complexity of several basic Boolean functions. This paper examines the asymptotic average case complexity for the set of alln-ary Boolean functions. In contrast to worst case analysis a simple counting argument does not work. We prove that with respect to the uniform probability distribution almost all Boolean functions require at leastn−log n−log log nexpected time. On the other hand, there is a significantly large subset of functions that can be computed with a constant average delay. Finally, for an arbitrary Boolean function we compare its worst case and average case complexity. It is shown that for each function that requires circuit depthd, i.e. of worst-case complexityd, the expected time complexity will be at leastd−log n−log dwith respect to an explicitly defined probability distribution. In addition, a nontrivial upper bound on the complexity of such a distribution will be obtained.