Strengthened Lazy Heaps: Surpassing the Lower Bounds for Binary Heaps

Let $n$ denote the number of elements currently in a data structure. An in-place heap is stored in the first $n$ locations of an array, uses $O(1)$ extra space, and supports the operations: minimum, insert, and extract-min. We introduce an in-place heap, for which minimum and insert take $O(1)$ worst-case time, and extract-min takes $O(\lg{} n)$ worst-case time and involves at most $\lg{} n + O(1)$ element comparisons. The achieved bounds are optimal to within additive constant terms for the number of element comparisons. In particular, these bounds for both insert and extract-min -and the time bound for insert- surpass the corresponding lower bounds known for binary heaps, though our data structure is similar. In a binary heap, when viewed as a nearly complete binary tree, every node other than the root obeys the heap property, i.e. the element at a node is not smaller than that at its parent. To surpass the lower bound for extract-min, we reinforce a stronger property at the bottom levels of the heap that the element at any right child is not smaller than that at its left sibling. To surpass the lower bound for insert, we buffer insertions and allow $O(\lg^2{} n)$ nodes to violate heap order in relation to their parents

Memory-Adjustable Navigation Piles with Applications to Sorting and Convex Hulls

We consider space-bounded computations on a random-access machine (RAM) where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the input is $N$ elements, the length of the output is limited by the computation, and the capacity of the workspace is $O(S)$ bits for some predetermined parameter $S$. We present a state-of-the-art priority queue---called an adjustable navigation pile---for this restricted RAM model. Under some reasonable assumptions, our priority queue supports $\mathit{minimum}$ and $\mathit{insert}$ in $O(1)$ worst-case time and $\mathit{extract}$ in $O(N/S + \lg{} S)$ worst-case time for any $S \geq \lg{} N$. We show how to use this data structure to sort $N$ elements and to compute the convex hull of $N$ points in the two-dimensional Euclidean space in $O(N^2/S + N \lg{} S)$ worst-case time for any $S \geq \lg{} N$. Following a known lower bound for the space-time product of any branching program for finding unique elements, both our sorting and convex-hull algorithms are optimal. The adjustable navigation pile has turned out to be useful when designing other space-efficient algorithms, and we expect that it will find its way to yet other applications.Comment: 21 page

UBVRI photopolarimetry of the long period eclipsing AM Herculis binary V1309

We report simultaneous UBVRI photo-polarimetric observations of the long period (7.98 h) AM Her binary V1309 Ori. The length and shape of the eclipse ingress and egress varies from night to night. We suggest this is due to the variation in the brightness of the accretion stream. By comparing the phases of circular polarization zero-crossovers with previous observations, we confirm that V1309 Ori is well synchronized, and find an upper limit of 0.002 percent for the difference between the spin and orbital periods. We model the polarimetry data using a model consisting of two cyclotron emission regions at almost diametrically opposite locations, and centered at colatitude 35 (deg) and 145 (deg) on the surface of the white dwarf. We also present archive X-ray observations which show that the negatively polarised accretion region is X-ray bright.Comment: 11 pages, 12 figures (2 colour), Fig1 and Fig 4 are in lower resolution than in original paper, accepted for publication in Monthly Notices of the Royal Astronomical Societ

The Open Graph Archive: A Community-Driven Effort

In order to evaluate, compare, and tune graph algorithms, experiments on well designed benchmark sets have to be performed. Together with the goal of reproducibility of experimental results, this creates a demand for a public archive to gather and store graph instances. Such an archive would ideally allow annotation of instances or sets of graphs with additional information like graph properties and references to the respective experiments and results. Here we examine the requirements, and introduce a new community project with the aim of producing an easily accessible library of graphs. Through successful community involvement, it is expected that the archive will contain a representative selection of both real-world and generated graph instances, covering significant application areas as well as interesting classes of graphs.Comment: 10 page

QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average

In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithms that incur approximately n log n - 1.26n +o(n) comparisons on the average. A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n. Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs: when sorting integers it is slower by only 15% to STL-Introsort

XMM-Newton observation of the long-period polar V1309 Ori: The case for pure blobby accretion

Using XMM-Newton we have obtained the first X-ray observation covering a complete orbit of the longest period polar, V1309 Ori. The X-ray light curve is dominated by a short, bright phase interval with EPIC pn count rates reaching up to 15 cts/sec per 30 sec resolution bin. The bright phase emission is well described by a single blackbody component with kT_bb = (45 +- 3) eV. The absence of a bremsstrahlung component at photon energies above 1 keV yields a flux ratio F_bb/F_br > 6700. This represents the most extreme case of a soft X-ray excess yet observed in an AM Herculis star. The bright, soft X-ray emission is subdivided into a series of individual flare events supporting the hypothesis that the soft X-ray excess in V1309 is caused by accretion of dense blobs. In addition to the bright phase emission, a faint, hard X-ray component is visible throughout the binary orbit with an almost constant count rate of 0.01 cts/sec. Spectral modelling indicates that this emission originates from a complex multi-temperature plasma. At least three components of an optically thin plasma with temperatures kT= 0.065, 0.7, and 2.9 keV are required to fit the observed flux distribution. The faint phase emission is occulted during the optical eclipse. Eclipse ingress lasts about 15--20 min and is substantially prolonged beyond nominal ingress of the white dwarf. This and the comparatively low plasma temperature provide strong evidence that the faint-phase emission is not thermal bremsstrahlung from a post-shock accretion column above the white dwarf. A large fraction of the softer faint-phase emission could be explained by scattering of photons from the blackbody component in the infalling material above the accretion region. The remaining hard X-ray flux could be produced in the coupling region, so far unseen in other polars.Comment: 10 pages, 5 figures, A&A publishe

An extensive photometric study of the recently discovered intermediate polar V515 And (XSS J00564+4548)

We report results of photometry of the intermediate polar V515 And. The observations were obtained over 33 nights in 2008 and 2009. The total duration of the observations was 233 h. We clearly detected two oscillations with periods of 465.48493\pm0.00007$ and 488.61822\pm0.00009 s, which may be the white dwarf spin period and the orbital sideband. The semi-amplitudes of the oscillations are 25 and 20 mmag, accordingly. The oscillation with a period of 465.48493 s has a stable smooth asymmetric pulse profile whereas the pulse profile of the oscillation with a period of 488.61822 s reveals significant changes from a quasi-sinusoidal shape to a shape somewhat resembling a light-curve of an eclipsing binary. Two detected oscillations imply an orbital period of 2.73 h. V515 And is one of the most rapidly spinning intermediate polars with orbital periods less than 3 h and may be not in spin equilibrium. This can be proved by future observations. For this purpose we obtained oscillation ephemerises with a formal shelf life of about 100 yr. (a 1 sigma confidence level).Comment: 10 pages, 8 figures, will be published in MNRA

The secondary star and distance of the polar V1309 Ori

The first phase-resolved JHK light curves of the eclipsing polar (AM Herculis binary) V1309 Ori are presented and interpreted. We separate the contributions from the secondary star and from other sources with the aim of determining a photometric distance. Simple model calculations show that the accretion stream and the cyclotron source on the accreting white dwarf are minor contributors to the infrared light, allowing an accurate determination of spectral type and absolute flux of the secondary star. The unilluminated backside of the secondary star as seen in eclipse has spectral type dM0 to dM0+. Its dereddened magnitude is K = 13.58 at orbital phase phi = 0 (eclipse). Using the calibrated surface brightness of M-stars and the published mass of the secondary, M2 = 0.46 Msun, we obtain a distance d = 600 +/- 25 pc which scales as M2^(1/2). The radius of the Roche-lobe filling secondary exceeds the main-sequence radius of an M0 star by 21 +11/-6 %. The debated origin of the infrared light of V1309 Ori has been settled in favor of the secondary star as the main contributor and an accurate distance has been derived that will place estimates of the luminosity and synchronization time scale on a more secure basis.Comment: 5 pages, 3 figures, accepted for publication in Astronomy and Astrophysic