External Path Length Research Articles

Upper tail analysis of bucket sort and random tries

Bucket Sort is known to run in expected linear time when the input keys are distributed independently and uniformly at random in the interval [0,1). The analysis holds even when a quadratic time algorithm is used to sort the keys in each bucket. We show how to obtain linear time guarantees on the running time of Bucket Sort that hold with very high probability. Specifically, we investigate the asymptotic behavior of the exponent in the upper tail probability of the running time of Bucket Sort. We consider large additive deviations from the expectation, of the form cn for large enough (constant) c, where n is the number of keys that are sorted.Our analysis shows a profound difference between variants of Bucket Sort that use a quadratic time algorithm within each bucket and variants that use a Θ(blog⁡b) time algorithm for sorting b keys in a bucket. When a quadratic time algorithm is used to sort the keys in a bucket, the probability that Bucket Sort takes cn more time than expected is exponential in Θ(nlog⁡n). When a Θ(blog⁡b) algorithm is used to sort the keys in a bucket, the exponent becomes Θ(n). We prove this latter theorem by showing an upper bound on the tail of a random variable defined on tries, a result which we believe is of independent interest. This result also enables us to analyze the upper tail probability of a well-studied trie parameter, the external path length, and show that the probability that it deviates from its expected value by an additive factor of cn is exponentially small in Θ(n).

Some properties of exponential trees

Exponential recursive trees and exponential PORTs are introduced in H. Mahmoud (Profile of random exponential recursive trees. Methodology and Computing in Applied Probability (accepted) 2021). In that reference, the author investigates the order and node profile of these species. Several other equally important properties remain to be explored. The aim of the present manuscript is to establish fundamental properties concerning leaves (and their profile level by level) and distances in these trees. Some results fall back on the order of a tree. For the number of leaves in both flavours, we find (under appropriate scaling for each) a limit distribution uniquely characterized by inductively constructed moments. We find an limit for the scaled total (external) path length in an exponential recursive tree (PORT) in terms of the known distribution of the scaled order given in Mahmoud [11]. These total path lengths are indicative of the depth of a randomly chosen node (external node) in an exponential recursive tree (PORT).

Changes in Root Architecture and Aboveground Traits of Red Clover Cultivars Driven by Breeding to Improve Persistence

Red clover persistence has been one of the major targets for the most breeding programs worldwide. A phenotypic characterization at above- and belowground level was performed to a set of highly persistent red clover populations. The objective was to identify phenotypic changes occurring after empirical selection for plant survival, which is the consensus criteria of persistence. Eleven red clover populations were established on mesocosms of polyvinyl chloride (PVC) tubes of 11 cm in diameter and 100 cm depth, containing as substrate a mixture (v/v) of sand, vermiculite, soil and perlite. The trial was organized in a randomized complete block design with four replicates, each replicate consisting of five mesocosms per population and one plant per mesocosm. In total, 220 mesocosms were handled. At aboveground level, growth parameters, specific leaf area (SLA) and shoot dry matter (ShootDM) were measured. At belowground level, root morphology (volume, diameter and length) and topology (altitude, external path length [EPL] and dichotomous branching index [DBI]) were measured through image analyses. Analyses of variance were performed implementing a phenotypic linear mixed model using the Restricted Maximum Likelihood method. Additionally, variance components were estimated and broad-sense heritability was calculated for each phenotypic trait. Highly persistent cultivars exhibited 30% higher ShootDM and 10% lower leaf size and SLA than the oldest low-persistent cultivar Quiñequeli. At root level, they showed 20, 50 and 50% higher crown diameter, root length density and root volume than Quiñequeli, respectively, but 20% lower DBI. Root traits exhibited medium-low values of genetic control; broad sense heritability ranged between 0.20 and 0.48. In conclusion, highly persistent red clover cultivars and experimental lines bred in Chile modified their phenotypic expression of individual plants at shoot and root levels relative to the oldest low persistent cultivar Quiñequeli. Associations among above- and belowground traits offer opportunities for designing more efficient selection strategies. For instance, the strong relationship between SLA and root traits offers tremendous potential for indirect phenotypic selection.

Leakage detection of closed vials based on two-line water-vapor TDLAS

If the water vapor in the ambient air entered vials during the freeze-drying and packaging processes, it could lead to the deterioration of pharmaceuticals. Therefore, the leakage detection of vials was vital in the pharmaceuticals industry. In the work, a sensor based on two-line tunable diode laser absorption spectroscopy (TDLAS) was developed for leakage detection of vials with the advantages of non-invasive, fast response and high-accuracy. 15 vials were measured by the senor and the results were compared with a three-lightpath TDLAS method. It revealed that the leakage coefficient k of No. 11–15 vials obtained by the sensor were much higher than that of No. 1–10 vials, and No. 11–15 vials were considered as leaked vials. It matched well with the results determined by the three-lightpath TDLAS method. And the dependence of k on different external path lengths (0/1/2/3 cm) was discussed. It was found that k increased proportionally with the increasing external path length. Furthermore, the sensor was applied for instantly punctured and totally opened ideal-closed vial, respectively. It showed that k tended to be constant when the RH in the vial was the same as that of the environment.

Dependence between path-length and size in random digital trees

AbstractWe study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

Optical feedback in dfb quantum cascade laser for mid-infrared cavity ring-down spectroscopy

A simple external optical feedback system has been applied to a distributed feedback quantum cascade laser (DFB QCL) for cavity ring-down spectroscopy (CRDS) and a clear effect of feedback was observed. A long external feedback path length of up to 4m can decrease the QCL linewidth to around 50kHz, which is of the order of the transmission linewidth of our high finesse ring-down cavity. The power spectral density of the transmission signal from high finesse cavity reveals that the noise at frequencies above 20kHz is reduced dramatically.

Laser feedback interferometry based on high density cosine-like intensity fringes with phase quasi-quadrature

A novel laser feedback interferometry based on high-order feedback is presented and realized for the first time. The interferometer uses a birefringence dual frequency laser and a tilted feedback mirror with high amplitude reflectivity to generate high density cosine-like optical fringes. These optical fringes have nanoscale resolution. Particularly, phase quasi-quadrature between the dual frequency fringes is obtained because of the phase shift caused by the changes of external optical path length. This phase characteristic can be used to distinguish the direction of movement easily. Under typical room conditions, the system's resolution is 0.51nm in 850μm range, and its 2 min displacement accuracy is 5nm.

Approximating Optimal Binary Decision Trees

We give a (ln n+1)-approximation for the decision tree (DT) problem. An instance of DT is a set of m binary tests T=(T 1,…,T m ) and a set of n items X=(X 1,…,X n ). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the total external path length of the tree is minimized. Total external path length is the sum of the depths of all the leaves in the tree. DT has a long history in computer science with applications ranging from medical diagnosis to experiment design. It also generalizes the problem of finding optimal average-case search strategies in partially ordered sets which includes several alphabetic tree problems. Our work decreases the previous best upper bound on the approximation ratio by a constant factor. We provide a new analysis of the greedy algorithm that uses a simple accounting scheme to spread the cost of a tree among pairs of items split at a particular node. We conclude by showing that our upper bound also holds for the DT problem with weighted tests.

Formula omitted]: The internal extent formula for compacted tries

It is well known (Knuth, 1997 [5, pp. 399–400]) that in a binary tree the external path length minus the internal path length is exactly 2 n − 2 , where n is the number of external nodes. We show that a generalization of the formula holds for compacted tries, replacing the role of paths with the notion of extent, and the value 2 n − 2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.

Left and right length of paths in binary trees or on a question of Knuth

We consider extended binary trees and study the common right and left depth of leaf $j$, where the leaves are labelled from left to right by $0, 1, \ldots, n$, and the common right and left external pathlength of binary trees of size $n$. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf $j$ in a random size-$n$ binary tree when $j \sim \rho n$ with $0< \rho < 1$, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-$n$ binary tree.

LTP interferometer—noise sources and performance

The LISA Technology Package (LTP) uses laser interferometry to measure the changes in relative displacement between two inertial test masses. The goals of the mission require a displacement measuring precision of 10 pm Hz−1/2 at frequencies in the 3–30 mHz band. We report on progress with a prototype LTP interferometer optical bench in which fused silica mirrors and beamsplitters are fixed to a ZERODUR® substrate using hydroxide catalysis bonding to form a rigid interferometer. The couplings to displacement noise of this interferometer of two expected noise sources—laser frequency noise and ambient temperature fluctuations—have been investigated, and an additional, unexpected, noise source has been identified. The additional noise is due to small amounts of signal at the heterodyne frequency arriving at the photodiode preamplifiers with a phase that quasistatically changes with respect to the optical signal. The phase shift is caused by differential changes in the external optical paths the beams travel before they reach the rigid interferometer. Two different external path length stabilization systems have been demonstrated and these allowed the performance of the overall system to meet the LTP displacement noise requirement.

Concentration of Size and Path Length of Tries

We show that, for a certain class of probabilistic models, the number of internal nodes $S_n$ of a trie built from $n$ independent and identically distributed keys is concentrated around its mean, in the sense that $\Var S_n=\Oh(\EE S_n)$. Keys are sequences of symbols which may be taken from varying alphabets, and the choice of the alphabet from which the $k$th symbol is taken, as well as its distribution, may depend on all the preceding symbols. In the construction of the trie we also allow for bucket sizes greater than 1. The property that characterizes our models is the following: there is a constant $C$ such that for any word $v$ that may occur as a prefix of a key the size $S^v_n$ of a trie built from the suffixes of $n$ independent keys conditioned to have the common prefix $v$ has the property $\EE S^v_n\leq Cn$. This class of models contains memoryless and Markovian source models as well as the probabilistic dynamical source models that were recently introduced and thoroughly developed by Vallee [Algorithmica 29 (2001) 262–306], in particular the continued fraction source. Furthermore we study the external path length $L_n$, which obeys $\EE L_n=\Oh(n\ln n)$ and $\Var L_n=\Oh(n\ln^2 n)$.

Topology: a novel method to describe branching patterns in Peronospora viciae colonies

Topology provides a novel means to describe branching patterns and has not been applied to fungal colonies previously. For any branched structure, various topologies are possible, and these lie between two extremes, a herringbone pattern (main axis with primary laterals) and a dichotomous pattern (highly branched system). We applied topological methods to colonies of Peronospora viciae 48 h after inoculation of Pisum sativum leaves. The methods are based on two simulations, one developed for channel networks such as found in river systems and another for biological systems. Although not a true herringbone form, the Peronospora viciae colonies have a strong herringbone element within their growth pattern. All 25 colonies analysed fell into the random distribution according to the confidence limits calculated from simulations for biological systems. These confidence limits, however, represent the percentile distribution of all simulated networks, and only those structures with a perfect herringbone or dichotomous topology fall outside the range. The tendency of P. viciae colonies towards herringbone growth is reflected by the topological indices for altitude and external pathlength (a(obs)/E(a) and pe(obs)/E(pe), where a = altitude, pe = external pathlength, obs = observed for the P. viciae colonies and E = expected values for random growth), and the slope of the regression analysis for a(obs) and pe(obs). We consider this trend as significant because it was consistent for all but one of the colonies, and implies that growth can be envisaged as an intermediate between random and herringbone topology. It is proposed that initial herringbone growth may reflect a strategy that is aimed at overcoming host resistance, achieving rapid colonisation of infected tissue and maximising the potential for nutrient acquisition. This topology would also increase the likelihood of finding a compatible mating type for reproduction between heterothallic isolates.

Comparative root system structure of post-fire Pinus halepensis Mill. and Cistus monspeliensis L saplings.

To describe root system topology of Pinus halepensis and Cistus monspeliensis saplings co-inhabiting natural post-fire sites, 55 P. halepensis and 26 C. monspeliensis saplings were extracted by the total excavation method from a burnt pine stand. Seedlings were individually labelled when emerging after fire and extracted three years later, at the sapling phase. In order to evaluate the effect of inter-specific competition of C. monspeliensis on P. halepensis root system, a stratified sampling was carried out according to density and height of the saplings. Topological parameters considered in the analysis were magnitude, total external pathlength, and altitude of the root systems. Weight and length of roots were also measured in order to estimate the specific root length, an index commonly used in morphological studies. Results clearly evidenced greater variability in root system topology of P. halepensis than C. monspeliensis saplings. Herringbone architecture (i.e., the most ordered pattern possible, with branching confined to the main axis) characterised small pine saplings, regardless of competition from C. monspeliensis, which changed to random branching in large saplings. In medium sized saplings, the root system was affected by inter-specific competition, which delayed changes in root branching. In contrast, C. monspeliensis invariably adopted randomly branched architecture, regardless of intra-specific competition. It is concluded that such different topological patterns make C. monspeliensis more competitive during the early stages of post-fire succession, because its root system is much more transport-efficient in the nutrient-rich environment. Those pines which finally branch roots by random pattern will reach higher stem height and magnitude, a factor which allows them to successfully compete with C. monspeliensis for soil nutrients and water. The morphological analysis showed a significant increase in the specific root length with competition, both in P. halepensis and C. monspeliensis saplings, which could be interpreted as a consequence of the reduction of root diameter in response to nutrient depletion.

An almost sure result for path lengths in binary search trees

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

An almost sure result for path lengths in binary search trees

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

Lopsided Trees, I: Analyses

Lopsided trees are rooted, ordered trees in which the length of an edge from a node to its i th child depends upon the value of i . These trees model a variety of problems and have therefore been extensively studied. In this paper we combine analytic and combinatorial techniques to address three open problems on such trees: 1. Given n , characterize the combinatorial structure of a lopsided tree with n leaves that has minimal external path length. 2. Express the cost of the minimal external path length tree as a function of n . 3. Calculate exactly how many nodes of depth ≤ x exist in the infinite lopsided tree.

Minimal Huffman trees

For a given set of real weights, Huffman trees minimize the weighted external path length. Over the years, several algorithms have been proposed for constructing Huffman trees that minimize additional natural cost functions such as the external path length, the variance and, more generally, the central moments. We show that all these cost functions are minimized on exactly the same class of Huffman trees, and we characterize the class: it consists of all Huffman trees of minimal level set. It follows that a Huffman tree minimizing one of the cost functions in fact minimizes all of them, and has the minimal level set; in particular, it has minimum height. We show that the unique Huffman tree produced by the simplest construction method, the bottom-merge algorithm of Schwartz, belongs to the class. Finally, we prove that several natural variants of Huffman's algorithm, that appear to be nondeterministic, in fact all lead to the single Huffman tree obtained by Schwartz's algorithm.

The Construction of Huffman Codes is a Submodular ("Convex") Optimization Problem Over a Lattice of Binary Trees

We show that the space of all binary Huffman codes for a finite alphabet defines a lattice, ordered by the imbalance of the code trees. Representing code trees as path-length sequences, we show that the imbalance ordering is closely related to a majorization ordering on real-valued sequences that correspond to discrete probability density functions. Furthermore, this tree imbalance is a partial ordering that is consistent with the total orderings given by either the external path length (sum of tree path lengths) or the entropy determined by the tree structure. On the imbalance lattice, we show the weighted path-length of a tree (the usual objective function for Huffman coding) is a submodular function, as is the corresponding function on the majorization lattice. Submodular functions are discrete analogues of convex functions. These results give perspective on Huffman coding and suggest new approaches to coding as optimization over a lattice.

Binary search trees constructed from nondistinct keys with/without specified probabilities

We investigate binary search trees formed from sequences of nondistinct keys under two models. In the first model, an input sequence is composed from elements of a given finite multiset and all possible sequences are equally-likely. In the second model, an input sequence is composed from n elements of a (possibly infinite) set, where each key has a specified probability; the n keys are independently chosen from the given set. Under both models, we shall derive general closed-form expressions for the expected values of the characteristic parameters defined on the corresponding binary search trees. These parameters include the (left, right) depth of a given key, the level of a given external node and the left (right) side or the internal (external) path length of a search tree. Furthermore, we find some nonobvious relations between these expected values. In some respects, the second model tends to the first model for large n. All results are illustrated by concrete examples sometimes showing unexpected phenomena.