Generalized representation of integers

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.

After K. Bostro/spl uml/m and T. Felbinger observed that lossless quantum data compression does not exist unless decoders know the lengths of codewords, they introduced a classical noiseless channel to inform the decoder of a quantum source about the lengths of codewords. In this paper we analyze their codes and present: 1) a sufficient and necessary condition for the existence of such codes for given lists of lengths of codes; 2) a characterization of the optimal compression rate for their codes. However our main contribution is a more efficient way to use the classical channel. We propose a more general coding scheme. It turned out that the optimal compression can always be achieved by a code obtained by this scheme. A von Neumann entropy lower bound to rates of our codes and a necessary and sufficient condition to achieve the bound are obtained. The gap between this lower bound and the compression rates is also well analyzed. For a special family of quantum sources we provide a sharper lower bound in terms of Shannon entropy. Finally, we propose some problems for further research.

Slightly improved lower bounds for homogeneous formulas of bounded depth and bounded individual degree

We describe a numbering scheme for versions with alternatives that has a useful lexicographical ordering. The version hierarchy is a tree. By inspection of the version numbers, we can easily determine whether one version is an ancestor of another. If so, we can determine the version sequence between these two versions. If not, we can determine the most recent common ancestor to these two versions (i.e., the least upper bound, lub). Sorting the version numbers lexicographically results in a version being followed by all descendants and preceded by all its ancestors. We use a representation of nonnegative integers that is self delimiting and whose lexicographical ordering matches the ordering by value. >

Due to the fact that we lack a general construction method of (N, w, λ, λ) multiple-length (ML) optical orthogonal codes (OOCs), a general construction method with high efficiency for ML OOCs with arbitrary cross-correlation λ is presented. The main idea of the method is to construct a variable-length mapping sequence with unparallel λ positions, with which to map short-length OOCs into long-length OOCs to realize the general construction of ML OOCs with correlation value λ with high efficiency by the Johnson bound. The cardinality of multiple-length mapping sequences is derived. Based on the cardinality, the fundamentals of constructing the aforementioned multiple-length mapping sequences are presented and proved. The construction method of ML OOCs with arbitrary cross-correlation λ is given. Simulation results show that the method can construct ML OOCs with arbitrary λ the applications of ML OOCs show that the method is practical for constructing ML OOCs to support multiple service.

This paper proposes two variants of improved Compression by Substring Enumeration (CSE) with a finite alphabet. In previous studies on CSE, an encoder utilizes inequalities which evaluate the number of occurrences of a substring and a minimal forbidden word (MFW) to be encoded. Also, its codeword length is proportional to the difference between the upper and lower bounds deduced from the inequalities, but the lower bound is not tight. Therefore, in this paper, we derive a new tight lower bound and consequently propose a new CSE algorithm using the new inequality. We also propose a new encoding order of substrings and MFWs which are sorted by row and column marginal totals of the proper substrings and MFWs, instead of lexicographical order used in previous studies. We then propose a new CSE algorithm which is the first proposed CSE algorithm using the new encoding order. Experimental results show that compression ratios of all files of the Calgary corpus in the proposed algorithms are better than those of a previous study on CSE with a finite alphabet. Moreover, compression ratios under the second proposed CSE get better than or equal to that under a well-known compressor for 11 files amongst 14 files in the corpus.

Single-track circuit codes (STTCs) are circuit codes with codewords of length n such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track. These codes simultaneously generalize single-track Gray codes and ordinary circuit codes. They are useful in angular quantization applications in which error detecting and/or correcting capabilities are needed. A parameter k, called the spread of the code, measures the strength of this error control capability. We consider the existence of STCCs for small lengths n/spl les/17 and spreads k/spl les/6, constructing some optimal and many good examples. We then give a general construction method for STCCs which makes use of ordinary circuit codes. We use this construction to construct examples of codes with 360 and 1000 codewords which are of practical importance. We also use the construction to prove a general result on the existence of STCCs for general spreads.

We prove two lower bounds in the statistical query (SQ) learning model. The first lower bound is on weak-learning. We prove that for a concept class of SQ-dimension d, a running time of @W(d/logd) is needed. The SQ-dimension of a concept class is defined to be the maximum number of concepts that are ''uniformly correlated'', in that each of their pair has nearly the same correlation. This lower bound matches the upper bound in Blum et al. (Weakly Learning DNF and Characterizing Statistical Query Learning using Fourier Analysis, STOC 1994, pp. 253-262), up to a logarithmic factor. We prove this lower bound against an ''honest SQ-oracle'', which gives a stronger result than the ones against the more frequently used ''adversarial SQ-oracles''. The second lower bound is more general. It gives a continuous trade-off between the ''advantage'' of an algorithm in learning the target function and the number of queries it needs to make, where the advantage of an algorithm is the probability it succeeds in predicting a label minus the probability it does not. Both lower bounds extend and/or strengthen previous results, and solve an open problem left in previous papers. An earlier version of this paper [K. Yang, New lower bounds for statistical query learning, in: The Proceedings of the 15th Annual Conference on Computational Learning Theory, COLT 2002, Sydney, Australia, July 8-10, Lecture Notes in Computer Science, vol. 2375, 2002, pp. 229-243.] appeared in the proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002).

The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. In particular, the complexities of several fundamental problems in constrained block coding are precisely classified in terms of the existing complexity-theoretic structure. One type of problem studied is that of designing encoder and decoder circuits using minimum or approximately minimum hardware; for our purposes, an "input" to this problem is (i) a deterministic, irreducible finite-state transition diagram (DIF) defining a set of constrained binary sequences, and (ii) a desired rate p:q. Several of these minimum-encoder and minimum-decoder problems are shown to be NP-hard, and more interestingly some are shown to be complete in the second and third levels of the polynomial hierarchy. Another fundamental problem is that of computing the maximum rate of a block code; that is, given a DIF and a codeword length q, find the maximum p such that a rate p:q block code exists for the constraint defined by the DIF. This problem is shown to be NP/sup #P/-complete. Although it is not known whether NP/sup #P/ contains problems of super-polynomial complexity, it lies "higher" in the complexity-class structure than NP in the sense that it is possible, given current knowledge, that NP/sup #P/ contains problems of super-polynomial complexity even if P=NP. Another question studied is whether maximum rate block codes can always be implemented by encoders and decoders of polynomial size. The answer to this question is shown to be closely related to whether the class #P lies "lower" in the complexity-class structure than currently believed-a proof of either answer would have major implications in complexity theory.

Initial investigations of the re-synchronisation properties of fifth augmentation level, variable length, self-synchronising T-codes are described. It has been found from simulation studies that within the fifth augmentation level family there are considerable differences in the resynchronisation delay statistics of individual code sets, with some code sets consistently outperforming others with the same codeword length distribution. Furthermore, the code sets with the shortest average re-synchronisation delays are generally those with the shortest average code-word length (i.e. the most efficient). The coding efficiency of these code sets has been found to be within 1% of the efficiency of a Huffman code optimised for the particular information source.

This issue of SICOMP contains six specially selected papers from STOC 2020, the Fifty-second Annual ACM Symposium on the Theory of Computing, which was held June 22--26, 2020, initially planned at Chicago, Illinois, but due to COVID-19 was an online conference in the end. The papers here were chosen to represent the range and quality of the STOC program. These papers have been revised and extended by their authors and subjected to the standard thorough reviewing process of SICOMP. The program committee for STOC 2020 consisted of an executive committee made up of Nima Anari, Boaz Barak, Sébastien Bubeck, Mark Bun, Arkadev Chattopadhyay, Chandra Chekuri, Julia Chuzhoy, Marek Cygan, Ilias Diakonikolas, Yevgeniy Dodis, Sebastian Forster, Ankit Garg, Nika Haghtalab, Prahladh Harsha, Justin Holmgren, Piotr Indyk, Rahul Jain, Sanjeev Khanna, Dakshita Khurana, Pravesh Kothari, Robert Krauthgamer, Marvin Künnemann, Tengyu Ma, Rafael Oliveira, Merav Parter, Sofya Raskhodnikova, Robert Robere, Dana Ron, Noga Ron-Zewi, Thatchaphol Saranurak, Balasubramanian Sivan, Christian Sohler, Madhur Tulsiani, Omri Weinstein, Christian Wulff-Nilsen, and Henry Yuen. The program chair was Julia Chuzhoy. Included in this issue are the following papers: ``Explicit Near-Ramanujan Graphs of Every Degree" by Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes shows a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan. ``Reducing Path TSP to TSP" by Vera Traub, Jens Vygen, and Rico Zenklusen presents a black-box reduction from the path version of the traveling salesman problem (Path TSP) to the classical tour version (TSP). ``Nearly Optimal Static Las Vegas Succinct Dictionary" by Huacheng Yu obtains a randomized dictionary data structure using ${OPT}+{poly}\lg n+O(\lg^{(\ell)} U)$ bits of space with expected constant query time for the static dictionary problem. ``Separating the Communication Complexity of Truthful and Nontruthful Algorithms for Combinatorial Auctions" by Sepehr Assadi, Hrishikesh Khandeparkar, Raghuvansh Raj Saxena, and S. Matthew Weinberg provides the first separation in the approximation guarantee achievable by truthful and nontruthful combinatorial auctions with polynomial communication. ``Strong Average-Case Circuit Lower Bounds from Nontrivial Derandomization" by Lijie Chen and Hanlin Ren establish a connection between nondeterministic algorithms estimating the acceptance probability of a given circuit and average-case lower bounds for nondeterministic time classes. ``Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs" by Siddharth Bhandari and Sayantan Chakraborty presents a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k>3\Delta$ and returns a random proper $k$-coloring of $G$. We thank the authors, the STOC 2020 program committee, the STOC 2020 external reviewers, and the SICOMP referees for all of their hard work. Arkadev Chattopadhyay, Marek Cygan, Noga Ron-Zewi, Christian Wulff-Nilsen - Guest editors

The natural relaxation for the group Steiner tree problem, as well as for its generalization, the directed Steiner tree problem, is a flow‐based linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the group Steiner tree problem, we show that the integrality ratio is $\Omega(\log^2 k)$, where k denotes the number of groups; this holds even for input graphs that are hierarchically well‐separated trees, introduced by Bartal [in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, 1996, pp. 184–193], in which case this lower bound is tight. This also applies for the directed Steiner tree problem. In terms of the number n of vertices, our results for the directed Steiner problem imply an $\Omega(\frac{\log^2 n}{(\log \log n)^2})$ integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is known to be superlogarithmic but subpolynomial. Our results and techniques have been used by Halperin and Krauthgamer [in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585–594] to show comparable inapproximability results, assuming that NP has no quasi‐polynomial Las Vegas algorithms. We also show algorithmically that the integrality ratio for the group Steiner tree problem is much better for certain families of instances, which helps pinpoint the types of instances (parametrized by optimal solutions to their flow‐based relaxations) that appear to be most difficult to approximate.

We present a reformulation of the 2n+o(n) lower bound of Bent and John [Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 213--216] for the number of comparisons needed for selecting the median of n elements. Our reformulation uses a weight function. Apart from giving a more intuitive proof for the lower bound, the new formulation opens up possibilities for improving it. We use the new formulation to show that any pair-forming median finding algorithm, i.e., a median finding algorithm that starts by comparing $\lfloor n/2\rfloor$ disjoint pairs of elements must perform, in the worst case, at least 2.01 n + o(n) comparisons. This provides strong evidence that selecting the median requires at least cn+o(n) comparisons for some c> 2.

Results of a realisation of the topological transitions hypothesis are presented. The basic difficulties in the construction of quantum topological transition theory are connected with a necessity to introduce a new non-local interaction defined on a space of topological states. So the general method of construction and study of topological transitions classical models is formulated as a necessary step towards a corresponding quantum description. Their local properties, including an asymptotic behaviour in the neighbourhood of the transition, are studied and applications to problems of gravitation and cosmology are given. The method used is shown to lead to a scalar-tensor theory of topological transitions. Different variants of this theory and its main features are discussed.

Row-column designs are useful for the experimental situations in which there are two cross classified sources of heterogeneity in the experimental material. Often it is desired to compare two or more factors in row-column set up where only two units can be accommodated in a single column. In this article, a general method of construction has been developed to generate row-column designs for factorial experiments with two rows which permit orthogonal estimation of all main effects and specific two factor interactions as per the choice of experimenters.