Finding 𝐶₃-strong pseudoprimes

We consider the problem of testing graph cluster structure: given access to a graph G = (V, E), can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e. being a good single cluster) and the graph having a sparse cut (i.e. being a union of at least two clusters). A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing k-clusterability in time O(n^1/2 poly(k)). Their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points. This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph G. In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant. Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes. We complement our algorithm with a matching lower bound on the query complexity of testing clusterability. Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools. As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time.

The regularity lemma of Szemeredi turned out to be the most powerful tool for studying the testability of graph properties in the dense graph model. In fact, as we argue in this paper, this lemma can be used in order to prove (essentially) all the previous results in this area. More precisely, a barrier for obtaining an efficient testing algorithm for a graph property P was having an efficient regularity lemma for graphs satisfying P. The problem is that for many natural graph properties (e.g. triangle freeness) it is known that a graph can satisfy P and still only have regular partitions of tower-type size. This means that there was no viable path for obtaining reasonable bounds on the query complexity of testing such properties. In this paper we consider the property of being induced C_4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for testing this property. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox. We thus obtain the first example of an efficient testing algorithm that cannot be derived from an efficient version of the regularity lemma.

Let ψ m \psi _m be the smallest strong pseudoprime to the first m m prime bases. This value is known for 1 ≤ m ≤ 11 1 \leq m \leq 11 . We extend this by finding ψ 12 \psi _{12} and ψ 13 \psi _{13} . We also present an algorithm to find all integers n ≤ B n\le B that are strong pseudoprimes to the first m m prime bases; with reasonable heuristic assumptions we can show that it takes at most B 2 / 3 + o ( 1 ) B^{2/3+o(1)} time.

Efficient march test for 3-coupling faults in random access memories

We consider the on-line scheduling problem of jobs with precedence constraints on m parallel identical machines. Each job has a time processing requirement, and may depend on other jobs (has to be processed after them). A job arrives only after its predecessors have been completed. The cost of an algorithm is the time that the last job is completed. We show lower bounds on the competitive ratio of on-line algorithms for this problem in several versions. We prove a lower bound of 2 − 1/m on the competitive ratio of any deterministic algorithm (with or without preemption) and a lower bound of 2 − 2/(m + 1) on the competitive ratio of any randomized algorithm (with or without preemption). The lower bounds for the cases that preemption is allowed require arbitrarily long sequences. If we use only sequences of length O(m 2), we can show a lower bound of 2 − 2/(m + 1) on the competitive ratio of deterministic algorithms with preemption, and a lower bound of 2 − O(1/m) on the competitive ratio of any randomized algorithm with preemption. All the lower bounds hold even for sequences of unit jobs only. The best algorithm that is known for this problem is the well known List Scheduling algorithm of Graham. The algorithm is deterministic and does not use preemption. The competitive ratio of this algorithm is 2 − 1/m. Our randomized lower bounds are very close to this bound (a difference of 0(1/m)) and our deterministic lower bounds match this bound.

J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p p -pseudoprimes and elementary Abelian p p -pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p p -pseudoprime to two bases only, where p = 2 p=2 or 3 3 . In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2 2 -pseudoprimes to the same bases. In Section 3, we give examples of Sylow p p -pseudoprimes to the first several prime bases for the first several primes p p . We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow p p -pseudoprime to the same bases for all p ∈ { 2 , 3 , 5 , 7 , 11 , 13 } p\in \{2,3,5,7,11,13\} . In Section 4, we define n n to be a k k - fold Carmichael Sylow pseudoprime , if it is a Sylow p p -pseudoprime to all bases prime to n n for all the first k k smallest odd prime factors p p of n − 1 n-1 . We find and tabulate all three 3 3 -fold Carmichael Sylow pseudoprimes > 10 16 >10^{16} . In Section 5, we define a positive odd composite n n to be a Sylow uniform pseudoprime to bases b 1 , … , b k b_1,\ldots ,b_k , or a Syl-upsp ( b 1 , … , b k ) (b_1,\ldots ,b_k) for short, if it is a Syl p _p -psp ( b 1 , … , b k ) (b_1,\ldots ,b_k) for all the first ω ( n − 1 ) − 1 \omega (n-1)-1 small prime factors p p of n − 1 n-1 , where ω ( n − 1 ) \omega (n-1) is the number of distinct prime factors of n − 1 n-1 . We find and tabulate all the 17 Syl-upsp ( 2 , 3 , 5 ) (2,3,5) ’s > 10 16 >10^{16} and some Syl-upsp ( 2 , 3 , 5 , 7 , 11 ) (2,3,5,7,11) ’s > 10 24 >10^{24} . Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6.

Spin Transfer Torque Magnetic Random Access Memory (STT-MRAM) is a promising alternative technology for on-chip memories due to several advantages such as high density, non-volatility, scalability, high endurance and CMOS compatibility. Due to the emerging fabrication processes of magnetic layers in the fabrication of STTMRAM devices, they are more susceptible to manufacturing defects and have different failure mechanisms compared to conventional CMOS memories. This mandates specific fault modeling and development of proper test algorithms for high coverage testing of these memories in order to ensure correct functionality in the field. In this paper, a detailed defect injection is performed based on the magnetic devices and layout characteristics of STT-MRAM and unique fault models are constructed for these memories. Based on the derived fault models and behaviors, efficient test algorithms are developed to fully cover these faults.

Testing and diagnosis techniques play a key role during the advance of semiconductor memory technologies. The challenge of failure detection has created intensive investigation on efficient testing and diagnosis algorithm for better fault coverage and yield improvement. The test and diagnosis complexity are 12N for bit-oriented diagnosis algorithm, N is the number of addresses is proposed for fault detection and diagnosis. Using the proposed march-based algorithm (march-12N), 100% of the faults under the fault model are covered and partial distinguished. They also can locate the faulty cells and identify their types. The complete fault and diagnosis procedures for state coupling faults, idempotent coupling faults and inversion coupling faults are written in this paper. Therefore, all the coupling faults that occur in SRAM memories are verified and proved the valid results. Furthermore, the realistic 12N test and diagnosis algorithm has shown the improvement of diagnostic resolution and test time.

An efficient parallel algorithm for testing whether a graph G is k-vertex connected is presented. The algorithm runs in $O(k^2 \log n)$ time and uses $(n + k^2 )kC(n,m)$ processors on a CROW PRAM, where n and m are the number of vertices and edges of G, and $C(n,m)$ is the number of processors required to compute the connected components of G in logarithmic time. For fixed k, the algorithm runs in logarithmic time and uses $nC(n,m)$ processors. To develop our algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem. Given a graph G, and two specified vertices s and t, find k vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most $k-1$ vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in $O(k^2 \log n)$ time and uses $kC(n,m)$ processors. The way to modify the algorithm to find k-edge disjoint paths, if they exist, is shown. This yields an efficient parallel algorithm for testing w...

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast Δ-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al. [6], we prove that the randomized complexity of Δ-coloring a tree with maximum degree Δ is θ(logΔ log n), for any Δ ≥ 55, whereas its deterministic complexity is θ(logΔn) for any Δ ≥ 3. This also establishes a large separation between the deterministic complexity of Δ-coloring and (Δ+1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(logΔn) rounds can be transformed to run in O(log* n -- log*Δ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(logΔn) time deterministically. (This gives an alternate proof that deterministically Δ-coloring a tree with small Δ takes Ω(logΔn) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n}. This shows that a deterministic Ω(logΔn) lower bound for any problem (Δ-coloring a tree, for example) implies a randomized Ω(logΔ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) terms in the complexities of the best MIS and (Δ+1)-coloring algorithms without also improving the 2O(√log n})-round Panconesi-Srinivasan algorithms.

We extend the line of research initiated by Fortnow and Klivans [6] that studies the relationship between efficient learning algorithms and circuit lower bounds. In [6], it was shown that if a Boolean circuit class \(\mathcal{C}\) has an efficient deterministic exact learning algorithm, (i.e. an algorithm that uses membership and equivalence queries) then \(\mathsf{EXP}^{\mathsf{NP}} \not \subseteq \mathsf{P/poly}[\mathcal{C}]\). Recently, in [14] EXP NP was replaced by DTIME(n ω(1)). Yet for the models of randomized exact learning or Valiant’s PAC learning, the best result so far is a lower bound against BPEXP (the exponential-time analogue of BPP). In this paper, we derive stronger lower bounds as well as some other consequences from randomized exact learning and PAC learning algorithms, answering an open question posed in [6] and [14]. In particular, we show that

SummaryIn this paper lower bounds are obtained for the intensity of the sonic bang resulting from a slender aircraft flying straight and level at supersonic speeds. The lower bounds are obtained for an aircraft subject to various constraints on its total lift, volume, area distribution and centre-of-pressure position. The work described proves the results quoted in an earlier paper, corrects the results given in that paper for the general configuration, and extends the results to cover more constraints, mainly the centre-of-pressure position.

Disturb failures are considered the most predominant failure mode in flash memories. Disturb faults are highly dependant on the core memory cell structure, manufacturing technology, and array organization and operation. In this paper, we develop appropriate fault primitives for all possible disturb faults in flash memories. Further, we analyze the origins of such disturbs and propose a method that uses cell structure and array organization information to identify the relevant disturbs to create reduced list of faults. As an example, the method was used to create a minimized faults list for NOR and NAND flash memory arrays. Moreover, we show how the reduced fault list developed can be used to devise more efficient test algorithms.

In conjunction with the 2012 Yosemite hantavirus outbreak, the number of sera our facility tested for hantavirus antibodies increased. We tracked test results and used the data set to determine if a more efficient testing algorithm was possible. Sera were screened using laboratory-developed pan-hantavirus IgG and IgM enzyme immunoassays (EIAs), with an index of >1.10 defined as positive. Sera that were IgM positive by screening (screen IgM(+)) were tested for Sin Nombre virus (SNV)-specific IgM using a laboratory-developed EIA; screen IgM(+) IgG(+) sera were also tested for SNV IgG using a laboratory-developed immunoblot assay. SNV antibody-positive samples were sent to state public health laboratories (PHL) or the CDC for confirmation. Of 3,946 sera tested from July through December 2012, 205 were screen IgM(+) IgG negative (IgG(-)); 7/205 were SNV IgM(+), but only 1/5 sent to PHL/CDC was confirmed as SNV IgM(+). Of 61 screen IgM(+) IgG(+) sera, 16 were SNV antibody positive; 13/16 sera (from 11 patients) went to PHL/CDC, where SNV infection was confirmed for all patients. Of 12 confirmed patients, 7 had been exposed at Yosemite. A modified algorithm defining screen indices of ≥2.00 as positive identified 11/12 confirmed cases while reducing the number of sera requiring SNV-specific antibody testing by 65%; the patient missed was not tested until 3 months after the onset of symptoms. Hantavirus antibody testing at our facility identified 12 SNV-infected patients, including 7 exposed at Yosemite. Some screen IgM(+) IgG(-) SNV IgM(+) results were false positives, emphasizing the value of PHL/CDC confirmatory testing. We identified a modified algorithm requiring analysis of fewer specimens for SNV-specific antibodies without loss of sensitivity.

In this letter, we obtain lower and upper bounds on the minimum distance $d_{\min }$ of low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value $a_{\max }$ , exhaustively. We also derive a lower bound $L_{ne}$ on the weight of non-elementary code words. By performing the search with $a_{\max } = L_{ne}$ , we either obtain an elementary code word with the smallest weight $d_{\min }$ , or establish the lower bound of $L_{ne}$ on $d_{\min }$ . For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on $d_{\min }$ . We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $d_{\min }$ . Finding $d_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.