Independent Ofk Research Articles

Maximizing k -Submodular Functions and Beyond

We consider the maximization problem in the value oracle model of functions defined on k -tuples of sets that are submodular in every orthant and r -wise monotone, where k ⩾ 2 and 1 ⩽ r ⩽ k . We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r ). For r = k , we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1/(1+√ k /2. In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP . Extending a result of Ando et al., we show that for any k ⩾ 3, submodularity in every orthant and pairwise monotonicity (i.e., r = 2) precisely characterize k -submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k ) for the maximization problem of k -submodular functions.

Clique Counting in MapReduce

We tackle the problem of counting the number q k of k -cliques in large-scale graphs, for any constant k ≥ 3. Clique counting is essential in a variety of applications, including social network analysis. Our algorithms make it possible to compute q k for several real-world graphs and shed light on its growth rate as a function of k . Even for small values of k , the number q k of k -cliques can be in the order of tens or hundreds of trillions. As k increases, different graph instances show different behaviors: while on some graphs q k + 1 < q k , on other benchmarks q k + 1 » q k , up to two orders of magnitude in our observations. Graphs with steep clique growth rates represent particularly tough instances in practice. Due to the computationally intensive nature of the clique counting problem, we settle for parallel solutions in the MapReduce framework, which has become in the last few years a de facto standard for batch processing of massive datasets. We give both theoretical and experimental contributions. On the theory side, we design the first exact scalable algorithm for counting (and listing) k -cliques in MapReduce. Our algorithm uses O ( m 3/2 ) total space and O ( m k /2 ) work, where m is the number of graph edges. This matches the best-known bounds for triangle listing when k = 3 and is work optimal in the worst case for any k , while keeping the communication cost independent of k . We also design sampling-based estimators that can dramatically reduce the running time and space requirements of the exact approach, while providing very accurate solutions with high probability. We then assess the effectiveness of different clique counting approaches through an extensive experimental analysis over the Amazon EC2 platform, considering both our algorithms and their state-of-the-art competitors. The experimental results clearly highlight the algorithm of choice in different scenarios and prove our exact approach to be the most effective when the number of k -cliques is large, gracefully scaling to nontrivial values of k even on clusters of small/medium size. Our approximation algorithms achieve extremely accurate estimates and large speedups, especially on the toughest instances for the exact algorithms.

Computing the inverse sort transform in linear time

The Sort Transform (ST) can significantly speed up the block sorting phase of the Burrows-Wheeler Transform (BWT) by sorting the limited order contexts. However, the best result obtained so far for the inverse ST has a time complexity O ( N log k ) and a space complexity O ( N ), where N and k are the text size and the context order of the transform, respectively. In this article, we present a novel algorithm that can compute the inverse ST for any k -order contexts in an O ( N ) time and space complexity, a linear result independent of k . The main idea behind the design of this linear algorithm is a set of cycle properties of k -order contexts that we explore for this work. These newly discovered cycle properties allow us to quickly compute the Longest Common Prefix (LCP) between any pair of adjacent k -order contexts that may belong to two different cycles, which eventually leads to the proposed linear-time solution.

Anomalous diffusion in disordered multi-channel systems

We study diffusion of a particle in a system composed ofK parallel channels, where the transition rates within the channels arequenched random variables whereas the inter-channel transition ratev is homogeneous. A variant of the strong disorder renormalization groupmethod and Monte Carlo simulations are used. Generally, we observeanomalous diffusion, where the average distance traveled by the particle,[⟨x(t)⟩]av, has a power-law timedependence [⟨x(t)⟩]av ∼ tμK(v), witha diffusion exponent 0 ≤ μK(v) ≤ 1. In the presence of left–right symmetry of the distribution of randomrates the recurrent point of the multi-channel system is independent ofK and the diffusion exponent is found to increase withK and decreasewith v. In the absence of this symmetry, the recurrent point may be shifted withK and the current can be reversed by varying the lane change ratev.

A fast bit-vector algorithm for approximate string matching based on dynamic programming

The approximate string matching problem is to find all locations at which a query of length m matches a substring of a text of length n with k -or-fewer differences. Simple and practical bit-vector algorithms have been designed for this problem, most notably the one used in agrep . These algorithms compute a bit representation of the current state-set of the k -difference automaton for the query, and asymptotically run in either O ( nm/w ) or O ( nm log σ/ w ) time where w is the word size of the machine (e.g., 32 or 64 in practice), and σ is the size of the pattern alphabet. Here we present an algorithm of comparable simplicity that requires only O ( nm/w) time by virtue of computing a bit representation of the relocatable dynamic programming matrix for the problem. Thus, the algorithm's performance is independent of k , and it is found to be more efficient than the previous results for many choices of k and small m . Moreover, because the algorithm is not dependent on k , it can be used to rapidly compute blocks of the dynamic programming matrix as in the 4-Russians algorithm of Wu et al.(1996). This gives rise to an O(kn/w) expected-time algorithm for the case where m may be arbitrarily large. In practice this new algorithm, that computes a region of the dynamic progr amming (d.p.) matrx w entries at a time using the basic algorithm as a subroutine is significantly faster than our previous 4-Russians algorithm, that computes the same region 4 or 5 entries at a time using table lookup. This performance improvement yields a code that is either superior or competitive with all existing algorithms except for some filtration algorithms that are superior when k/m is sufficiently small.

Weighted Estimates for the Helmholtz Equation and Some Applications

We characterize the radial functionsVin Rnfor which the a priori inequality ‖∇u1/2‖2+k‖uV1/2‖2⩽C‖V−1/2(Δ+k2)u‖2holds with constant independent ofk. The condition is forVto have the X-rays transform everywhere bounded. We apply these estimates to the well posedness of evolution Schrödinger equations with time dependent drift terms and to the restriction of the Fourier transform to Euclidean spheres.

Choosing Subsets with Maximum Weighted Average

Given a set ofnreal values, each with a positive weight, we wish to find the subset ofn−kvalues having maximum weighted average. This is equivalent to the following form of parametric selection: givennobjects with values decreasing linearly with time, find the time at which then−kmaximum values add to zero. We show that these problems can be solved in timeO(n) (independent ofk). A generalization in which weights are allowed to be negative is NP-complete.

Efficient vector and parallel manipulation of tensor products

We present efficient vector and parallel methods for manipulating tensor products of matrices. We consider both computing the matrix-vector product (A 1 ļÄA K )x and solving the system of linear equations (A 1 ļÄA K )x=b. The methods described are independent of K . We accompany this article with a companion algorithm which describes an implementation of a complete set of tensor product routines based on LAPACK and the Level 2 and 3 Basic Linear Algebra Subprograms (BLAS) which provide vectorization and parallelization.

Regularized semigroups, iterated Cauchy problems and equipartition of energy

The iterated Cauchy problem under consideration is $$\Pi _{k = 1}^n (d/dt - A_k )u(t) = 0(t \geqslant 0).(*)$$ Here {A1,..., An} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachAk is a (C0) semigroup generator, this semigroup is of class (C0) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachAk generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC−1f→0; hereC is bounded and injective. When theAk are commuting generator withAk-Aj injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form\(u = \sum\nolimits_{i = 1}^n {u_i } \) whereui is a solution ofu′i=Aiui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.

Scattering of elastic waves by a fracture zone containing randomly distributed cracks

We theoretically study the scattering ofP, SV andSH waves by a zonal distribution of cracks, which simulates a fault fracture zone. An investigation is conducted how the geometrical properties of the crack distribution and the frictional characteristics of the crack surface are reflected in the attenuation and dispersion of incident waves, as well as in the amplitudes of the transmitted and reflected waves from the zone. If the crack distribution within the fault zone changes temporally during the preparation process of the expected earthquake, it will be important for earthquake prediction to monitor it, utilizing the scattering-induced wave phenomena. We consider the two-dimensional problem. Aligned cracks with the same length are assumed to be randomly distributed in a zone with a finite width, on which elastic waves are assumed to be incident. The distribution of cracks is assumed to be homogeneous and sparse. The crack surface is assumed to be stress-free, or to undergo viscous friction; the latter case simulates fluid-filled cracks. The opening displacement of the crack is assumed to be negligibly small. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed. When the crack surface is stress-free, it is commonly observed for every wave mode (P, SV andSH) that the attenuation coefficientQ −1 peaks aroundka∼1, the phase velocity is almost independent ofk in the rangeka 1, wherek is the intrinsicS wavenumber anda is the half length of the crack. The effect of the friction is to shift the peak ofQ −1 and the corner of the phase velocity curve to the low wavenumber range. The high wavenumber asymptote ofQ −1 is proportional tok −1 independently of model parameters and the wave modes. If the seismological observation thatQ −1 ofS waves has a peak at around 0.5 Hz in the earth's crust is combined with our results, the upper limit of crack size within the crust is estimated about 4 km. The information regarding the transmitted and reflected waves, such as the high wavenumber limit of the amplitude of the transmitted wave etc., allows estimation of the strength of the friction.

Theh?p version of the finite element method for elliptic equations of order2m

The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown in [3, 4, 5] that the solutions of the second order and fourth order equations belong to the spacesB β 1 where the weighted Sobolev norms of thek-th derivatives are bounded byCd k?l (k?l)!,k?l, l?2 whereC andd are constants independent ofk. In this case theh?p version of the finite element method leads to an exponential rate of convergence measured in the energy norm [6, 12, 13]. Theh?p version was implemented in the code PROBE1 [18] and has been very successfully used in the industry. We will discuss in this paper the generalization of these results for problems of order2m. We will show also that the exponential rate can be achieved if the exact solution belongs to the spacesB β 1 where the weighted Sobolev norm of thek-th derivatives is bounded byCd k?l (k?l)!,k?l=m+1, C andd are independent ofk. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spacesB β m+1 . Problems of this type are related obviously to many engineering problems, such as problems of plates and shells, and are also important in connection with well-known locking problems.

How thick is a liquid-vapor interface?

The structure of the interface of an argonlike fluid in equilibrium with its vapor near the triple point is studied using Monte Carlo simulation. By referring particle coordinates to the capillary waves and calculating the transverse structure factor in that reference frame, one can determine the penetration depth of the capillary waves into the bulk as a function of the wave vector. The penetration depth of the lowest capillary mode is consistent with the interfacial thickness as determined from the intrinsic density profile, and the data for the higher modes suggest that the penetration depth might be independent ofk.

Crack branching and arrest in environmental cracking of polyethylene

At intermediateK values a region where the crack speed is constant, that is, independent ofK, is observed in detergent cracking of low density polyethylene. This region is terminated at highK by one of two processes: in thin specimens the crack arrests, and in thick specimens it branches. The mechanism of crack branching involves the crack front twisting and then the initiation of a new crack at the centre of the specimen. A model has been proposed to explain this mechanism and also the fact that crack arrest or branching occur at approximately the sameK value. After branching the cracks continue to propagate at constant speed along paths whereK| ≈ 0 and if the specimen is wide enough, can branch again.

Caustic Stress Corrosion Cracking of Mild Steel

The stress corrosion cracking (SCC) behavior of cold worked mild steel in hot, aqueous, 33 pct NaOH solutions was studied with prefatigue cracked double cantilever beam specimens. SCC kinetics were studied under freely corroding potentials (E corr ≈ −1.00 VSHE) and potentiostatic potentials of −0.76 VSHE near the active-passive transition. The pH of the liquid within the crack was determined and fractography was studied by scanning electron microscopy. Cracking was transgranular atE corr, intergranular at −0.76 VSHE, and produced no detectable change in crack liquid pH from that of the bulk solution. Crack rates were dependent upon temperature, potential, and stress intensity (K 1). The apparent activation energy in Region II, where crack growth rate was independent ofK, was ∼ 24kJ/mol for both cracking modes. This was considered to be due to mixed rate control involving activation polarization and mass transport processes. The mechanism of cracking was entirely consistent with metal dissolution at –0.76 VSHE and may involve hydrogen embrittlement and/or dissolution effects atE corr.

Skewness and asymmetry of distributions

The ‘skewness’ of a distribution, a poorly-defined term, is conventionally deemed to be invariant under linear transformations. A comparison is made of three criteria of it: the sign of odd central moments; the several relationships of the mean, the median and the mode; and asymmetry proper which is the set of ratios of the probability densities of all pairs of points equidistant above and below some arbitrary point, usually the principal mode. Some useful general relationships are discussed. The skeness of convolutions is briefly discussed. A class of distributions is identified for which the skewness of the minimum ofk competing processes is independent ofk. It has importance in the parametric study of survivorship.

Gaseous hydrogen embrittlement of high strength steels

The kinetics of sustained-load subcritical crack growth in hydrogen were determined for 18Ni(200) and 18Ni(250) maraging steels over a range of hydrogen pressures and temperatures. Crack growth in each steel was characterized by an apparent threshold stress intensity, a domain where the growth rate increased sharply with stress intensity (K) (Stage I), and a range where the growth rate was independent ofK (Stage II). The rate-limited Stage II crack growth in these steels exhibited three distinct regions of temperature dependency, with a different isothermal pressure dependence in each region. In the low temperature region, Stage II crack growth was thermally activated with δH = 18.2 ±1.7 kj/mol; (δH being independent of hydrogen pressure and yield strength). The growth rates at a givenK were proportional to the square root of hydrogen pressure. In the intermediate temperature region, Stage II growth rates increased at slower rates, passed through a maximum and then decreased with increasing temperature. Within this region, the pressure dependence for crack growth increased from 1/2-power to 2.0-power with increasing temperature. Above a transition temperature, each grade of maraging steel became essentially immune to gaseous hydrogen embrittlement for the hydrogen pressure range considered. The transition temperature was strongly affected by yield strength and hydrogen pressure. Plausible explanations for these phenomenological results are considered.

Stress-corrosion cracking of Ti-8Al-lMo-lV in aqueous environments: 1. The kinetics of subcritical crack propagation

Precracked specimens of Ti-8Al-lMo-lV were tested in salt-water, and the subcritical crack velocity was measured as a function of applied stress-intensity. The minimum applied stress intensity which produces subcritical cracking,K Iscc, varies inversely with the average chloride-ion concentration of the environment, butK Iscc does not vary significantly with temperature. Plots of subcritical crack velocity,a, against applied stress intensity,K I, reveal two types of cracking behavior. At low stress intensitiesa is approximately proportional toK I (Stage I crack propagation). At high stress intensities,a is approximately constant, independent ofK I (Stage II crack propagation). The apparent activation energy for subcritical crack propagation is 5600 cal-(g-mole)-1, anda ∞ [Cl-]1/6. It is suggested that the instantaneous subcritical crack velocity is limited by the rate of production or by the diffusion of some embrittling species.

Low-energy expansion of theS(k) matrix for the Schrödinger equation with a certain type of spherically symmetric potentials

The potentialV(γ) treated is generally of such a long-range type that, when terms which for large γ decrease at least exponentially are neglected, the limit potential equals\(V_1 (r) = \sum\limits_{n = 1}^M {c_n r^{ - e_n } } \) whereen>0 for alln. TheSL(k) matrix corresponding to theL-th partial wave can in this case be reached in two steps. First the phase shifts produced by the above-defined limit potentialV1(r) are determined. Subsequently the effect of the difference potentialV1(r)=V(r)−V1(r) can be determined. TheSL(k) matrix is now obtained in a form, which is convergent (1) when |k| 0}^\infty {\left| {V_e (r)} \right|} \). convergent. The case (2–4)V(r) =cr−1 +dr−2 +Ve(r), which is of obvious importance, will be studied explicitly. In the case that\(\mathop {\lim }\limits_{r \to 0} r^2 V_e (r) \geqslant - d\) the physical solution takes for larger the formuLRaa (k, r) ∼ sin (kr −Lπ/2 −γ log 2kr +δL(k) +ηL(k)). HereδL(k) is the phase shift produced by the difference potentialVe(r) whileηL(k) = argΓ(Γ′ + 1 +iγ) + (L −L′)π/2 is the phase shift produced by the limit potentialV1(r). We obtain $$\begin{gathered} ctg \delta _L (k) = \frac{{1 - exp[2\gamma \pi ]}}{{\sin (2L'\pi )}} - tg(L'\pi ) + \hfill \\ + \frac{{k^{ - 2L' - 1} \Gamma (2L' + 2)\Gamma (2L' + 1)}}{{\left( {2^{L'} \left| {\Gamma (L' + 1 + i\gamma )} \right|} \right)^2 }}exp[\gamma \pi ] \left( {\sum\limits_{n = 0}^\infty {a_{L_n } k^{2n} } } \right)/\left( {\sum\limits_{n = 0}^\infty {b_{L_n } k^{2n} } } \right) \hfill \\ \end{gathered} $$ where γ=c/2k andL′=(1/4+L(L+1)=d)1/2−1/2 differ from an integer or half-integer. HereaLn andbLn are independent ofk and essentially integrals over all space and whose integrands decrease at least exponentially for large values of the arguments,aLn andbLn are therefore well suited for numerical calculations.

I. On the analytical theory of the attraction of solids bounded by surfaces of a hypothetical class including the ellipsoid

The following investigation is the result of an attempt to simplify the analytical treatment of the problem of the Attraction of Ellipsoids. The application to this particular case, of certain known propositions relating to closed surfaces in general, showed that the principal theorems could easily be deduced without taking account of any other properties of the ellipsoid than those expressed by two differential equations, of which the truth is evident on inspection. In fact if we take the equation x 2 / a 2 + h + y 2 / b 2 + h + z 2 / c 2 + h = k , we see at once that the expression on the left side, considered as a function of x, y, z, h , satisfies the two partial differential equations d 2 u / dx 2 + d 2 u / dy 2 + d 2 u / dz 2 = 2 (1/ a 2 + h + 1/ b 2 + h + 1/ c 2 + h ) ( du/dx ) 2 + ( du/dy ) 2 + ( du/dz ) 2 + 4 du/dh = 0, and these equations express all that we require to know about the ellipsoid, except the fact that the surface is capable of being extended to infinity in every direction by the variation of h , without ceasing to be closed. But it appeared also that the success of the method depended only on the circumstance that the right-hand member of the first equation, and the coefficient of du/dh in the second, are constants independent of k . It was therefore possible to generalize the process by taking indeterminate functions of h for these two constants. As, however, the coefficient of du/dh could always be reduced to a constant independent of h , by taking a function of h as a parameter instead of h , we may suppose, without loss of generality, that this reduction has been effected.