Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

The interaction of ultrathin films of Ni and Pd with W(110): an XPS study

The interaction of ultrathin films of Ni and Pd with W(110) has been examined using X-ray photoelectron spectroscopy (XPS) and the effects of annealing temperature and adsorbate coverage (film thickness) are investigated. The XPS data show that the atoms in a monolayer of Pd or Ni supported on W(110) are electronically perturbed with respect to the surface atoms of Pd(100) and Ni(100). The magnitude of the electronic perturbations is larger for Pd than for Ni adatoms. Our results indicate that the difference in Pd(3d_5/2) XPS binding energies between a pseudomorphic monolayer of Pd on W(110) and the surface atoms of Pd(100) correlates with the variations observed for the desorption temperature of CO (i.e., the strength of the Pd---CO bond) on these surfaces. A similar correlation is seen for the Ni(2p_3/2) XPS binding energies of Ni/W(110) and Ni(100) and the CO desorption temperatures from the surfaces. The shifts in XPS binding energies and CO desorption temperatures can be explained in terms of: (1) variations that occur in the Ni---Ni and Pd---Pd interactions when Ni and Pd adopt the lattice parameters of W(110) in a pseudomorphic adlayer; and (2) transfer of electron density from the metal overlayer to the W(110) substrate upon adsorption. Measurements of the Pd(3d_5/2) XP binding energy of Pd/W(110) as a function of film thickness indicate that the Pd---W interaction affects the electronic properties of several layers of Pd atoms.     

Normality of numbers generated by the values of entire functions

We show that the number generated by the q-ary integer part of an entire function of logarithmic order, where the function is evaluated over the natural numbers and the primes, respectively, is normal in base q. This is an extension of related results for polynomials over the real numbers established by Nakai and Shiokawa.     

Accurate confidence intervals in regression analyses of non-normal data

A linear model in which random errors are distributed independently and identically according to an arbitrary continuous distribution is assumed. Second- and third-order accurate confidence intervals for regression parameters are constructed from Charlier differential series expansions of approximately pivotal quantities around Student’s t distribution. Simulation verifies that small sample performance of the intervals surpasses that of conventional asymptotic intervals and equals or surpasses that of bootstrap percentile-t and bootstrap percentile-|t| intervals under mild to marked departure from normality.     

Solving the pallet loading problem

This paper presents new bounds, heuristics, and an exact algorithm for the Pallet Loading Problem (PLP). PLP maximizes the number of boxes placed on a rectangular pallet. All boxes have identical rectangular dimensions and, when placed, must be located completely within the pallet. Boxes may be rotated 90° so long as they are placed with edges parallel to the pallet’s edges. The set of all PLP instances with an area ratio (pallet area divided by box area) less than 101 boxes can be represented by 3,080,730 equivalent classes. Our G5-heuristic finds optimal solutions to 3,073,724 of these 3,080,730 classes and in the remaining 7006 classes only differs from the best known bound by one box. We develop three other heuristics that solve another 54 instances. Finally, we solve the 6952 remaining classes with our exact HVZ algorithm. Only a subset of these classes has been solved previously.     

Projective DNF formulae and their revision

Combinatorial and learnability results are proven for projective disjunctive normal forms, a class of DNF expressions introduced by Valiant.     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.