Arbitrary Tree Research Articles

Admixture, Population Structure, and F-Statistics.

Many questions about human genetic history can be addressed by examining the patterns of shared genetic variation between sets of populations. A useful methodological framework for this purpose isF-statistics that measure shared genetic drift between sets of two, three, and four populations and can be used to test simple and complex hypotheses about admixture between populations. This article provides context from phylogenetic and population genetic theory. I review how F-statistics can be interpreted as branch lengths or paths and derive new interpretations, using coalescent theory. I further show that the admixture tests can be interpreted as testing general properties of phylogenies, allowing extension of some ideas applications to arbitrary phylogenetic trees. The new results are used to investigate the behavior of the statistics under different models of population structure and show how population substructure complicates inference. The results lead to simplified estimators in many cases, and I recommend to replace F3 with the average number of pairwise differences for estimating population divergence.

Detecting Character Dependencies in Stochastic Models of Evolution.

Stochastic models of biological evolution generally assume that different characters (runs of the stochastic process) are independent and identically distributed. In this article we determine the asymptotic complexity of detecting dependence for some fairly general models of evolution, but simple models of dependence. A key difference from much of the previous work is that our algorithms work without knowledge of the tree topology. Specifically, we consider various stochastic models of evolution ranging from the common ones used by biologists (such as Cavender-Farris-Neyman and Jukes-Cantor models) to very general ones where evolution of different characters can be governed by different transition matrices on each edge of the evolutionary tree (phylogeny). We also consider several models of dependence between two characters. In the most specific model, on each edge of the phylogeny the joint distribution of the dependent characters undergoes a perturbation of a fixed magnitude, in a fixed direction from what it would be if the characters were evolving independently. More general dependence models don't require such a strong "signal." Instead they only require that on each edge, the perturbation of the joint distribution has a significant component in a specific direction. Our main results are nearly tight bounds on the induced or operator norm of the transition matrices that would allow us to detect dependence efficiently for most models of evolution and dependence that we consider. We make essential use of a new concentration result for multistate random variables of a Markov random field on arbitrary trivalent trees: We show that the random variable counting the number of leaves in any particular state has variance that is subquadratic in the number of leaves.

Rendezvous in networks in spite of delay faults

Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adver- sarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not feasible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost O(nl), where l is the smaller label, working in arbitrary trees, and we show that \Omega(l) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L.

Design of sampled analog wavelet processor architecture for cochlear implant application

This paper presents the design of sampled analog wavelet processor architecture for cochlear implant application using discrete Haar wavelet transform. Basic element of sampled analog wavelet processor architecture is operational amplifier based switched capacitor integrator circuit. This architecture distributes input signal frequencies into different predefined frequency band and based on cochlear sensitivity region for different frequencies by using arbitrary tree structure. The presented architecture is beneficial in terms of processing time of input signal, area and complexity by avoiding quantization and encoding process part of data converter circuits. Simulation results for this architecture is carried out with CMOS 90 nm technology.

Non-Haar MRA on local fields of positive characteristic

We propose a simple method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field F(s) of positive characteristic p. To construct this mask we use two new ideas. First, we consider local field as vector space over the finite field GF(ps). Second, we construct scaling function by arbitrary tree that has ps vertices. By fixed prime number p there exist ps(ps−2) such trees.

Embedding an Arbitrary Tree in a Graceful Tree

A function f is called a graceful labeling of a graph G with m edges if f is an injective function from V(G) to \(\{0,1,2,\ldots ,m\}\) such that when every edge uv is assigned the edge label \(|f(u)-f(v)|\), then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular Graceful Tree Conjecture states that every tree is graceful. The Graceful Tree Conjecture remains open for over four decades. Although there are a few general results and techniques on the construction of graceful trees, settling the conjecture seems to be very hard. In this paper, we have introduced a new and different method of constructing graceful trees from a given arbitrary tree. More precisely, we show that every tree can be embedded in a graceful tree with at most km edges, \(k<\lceil \frac{m}{4} \rceil \), where m is the number of edges of the given arbitrary tree. Further, we pose a related open problem toward settling the Graceful Tree Conjecture.

Towards Optimal Embedding of an Arbitrary Tree in a Graceful Tree

A function f is called a graceful labeling of a graph G with m edges, if f is an injective function from V(G) to {0,1,2,⋯,m} such that when every edge uv is assigned the edge label |f(u)−f(v)|, then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular and notorious Graceful Tree Conjecture: “All trees are graceful”, remains unsettled over four decades inspite of many rigorous attempts and investigations. Inspired by the interesting result of Acharya et al. [B.D. Acharya, S.B. Rao, S. Arumugam, Embedding and NP-Complete problems for Graceful Graphs, Labelings of Discrete Structures and Applications, in: B.D. Acharya, S. Arumugam, Alexander Rosa (Eds.), Narosa Publishing House, New Delhi, 2008, pp. 57–62], in this paper we show that every tree can be embedded in a graceful tree. Practically, we observe that any tree with m edges can be embedded in a graceful tree with less than 2m edges and we discuss a related open problem towards settling the Graceful Tree Conjecture through the embedding approach.

On p-adic Ising–Vannimenus model on an arbitrary order Cayley tree

In this paper, we continue an investigation of the p-adic Ising–Vannimenus model on the Cayley tree of an arbitrary order k(k ⩾ 2). We prove the existence of p-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional p-adic system of equations. We are also able to show the uniqueness of translation-invariant p-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising–Vannimenus model depending on the order k of the Cayley tree and the prime p. Note that the methods used in the paper are not valid in the real setting, since all of them are based on p-adic analysis and p-adic probability measures.

Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of $\epsilon$, with the guarantee that for each $\epsilon$ simultaneously, the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the $\ell_2$ distortion is $O(\sqrt{\log n})$. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies constant $\ell_q$-distortion for every fixed $q<\infty$.

The Power of Linear Programming for General-Valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrized by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimization problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. For example, if $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to 3-SAT. More generally, if $\Gamma$ contains only $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to CSP($\Gamma$). If $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,1\}$-valued functions, VCSP($\Gamma$) corresponds to Min-3-SAT, in which the goal is to minimize the number of unsatisfied clauses in a 3-CNF instance. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree submodular on arbitrary trees.

On $m$-Closed Graphs

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gröbner basis with respect to the lexicographic order induced by $x_1 &gt; \ldots &gt; x_n &gt; y_1&gt; \ldots &gt; y_n$. In this paper, we generalize this notion and study the so called $m$-closed graphs. We find equivalent condition to $3$-closed property of an arbitrary tree $T$. Using it, we classify a class of $3$-closed trees. The primary decomposition of this class of graphs is also studied.

N-ary implicit blends with topology control

Constructive implicit surfaces are attractive for modeling and animation because they seamlessly handle shapes with complex and dynamic topology. However, the way they merge shapes is difficult to control. This paper introduces a solution: an improved blend operator that provides control over how topology changes are handled. It is based on a correction applied to the standard blending operator: the sum. Building on summation preserves the n-ary nature of the blend, providing the simplicity of arbitrary (e.g. flat) construction trees and segmentation invariance. The correction is based on projection to a reference case in the variation-space defined by the field and the norm of its gradient. It provides a single parameter, allowing for tuning behavior to achieve effects ranging from avoiding topological combination, through merging only during overlap, to merging at a distance. Dynamic adjustment of the parameter allows for context-dependent effects. Applications range from skeleton-based modeling, where shapes keep the topology of their skeleton, to objects that change topology during animation, with controllable merging. We illustrate the latter with Manga-style hair, where merging depends on the angle between hair wisps.

Tree Edit Distance and Maximum Agreement Subtree

This paper presents an interesting relation between the Maximum Agreement Subtree (MAST) problem and the Tree Edit Distance (TED) problem, both of which have been intensively studied in the literature. To be specific, we show that, for an arbitrary tree edit distance metric that is a derivative of the Taï tree edit distance metric, there exists a MAST-like pattern extraction problem, named Mostly Adjusted Sub-Forest (MASF) problem, such that computing a distance between trees x and y is equivalent to finding an optimal pattern shared between x and y. The MASF problem is different from the MAST problem in: (1) A pattern of the MASF problem may be a forest instead of a tree; (2) Instead of requiring exact match of labels, a pattern of a MASF problem is multi-labeled, and our flexible matching rule only requires that the label set of a vertex of a pattern includes the label of the corresponding vertex in a target tree; (3) To control ambiguity of matching caused by the flexible rule, the objective function includes a penalty function. As an application of this generic framework, we equate the Lu* tree edit distance metric with a pattern extraction problem, named the Mostly Adjusted Agreement Sub-Tree (MAAST) problem. The MAAST problem aims to find optimal agreement subtrees under the flexible matching rule and is solved in O(n2dlog⁡d)-time, where n and d are the size and the minimum degree of the input trees. The MAST problem requires O(n2.5)-time computation.

On Network Functional Compression

In this thesis, we consider different aspects of the functional compression problem. In functional compression, the computation of a function (or, some functions) of sources is desired at the receiver(s). The rate region of this problem has been considered in the literature under certain restrictive assumptions. In Chapter 2 of this Thesis, we consider this problem for an arbitrary tree network and asymptotically lossless computations. In particular, for one-stage tree networks, we compute a rate-region and for an arbitrary tree network, we derive a rate lower bound based on the graph entropy. We introduce a new condition on colorings of source random variables' characteristic graphs called the coloring connectivity condition (C.C.C.). We show that unlike the condition mentioned in Doshi et al., this condition is necessary and sufficient for any achievable coding scheme based on colorings. We also show that, unlike entropy, graph entropy does not satisfy the chain rule. For one stage trees with correlated sources, and general trees with independent sources, we propose a modularized coding scheme based on graph colorings to perform arbitrarily closely to the derived rate lower bound. We show that in a general tree network case with independent sources, to achieve the rate lower bound, intermediate nodes should perform some computations. However, for a family of functions and random variables called chain rule proper sets, it is sufficient to have intermediate nodes act like relays to perform arbitrarily closely to the rate lower bound. In Chapter 3 of this Thesis, we consider a multi-functional version of this problem with side information, where the receiver wants to compute several functions with different side information random variables and zero distortion. Our results are applicable to the case with several receivers computing different desired functions. We define a new concept named multi-functional graph entropy which is an extension of graph entropy defined by K6rner. We show that the minimum achievable rate for this problem is equal to conditional multi-functional graph entropy of the source random variable given the side information. We also propose a coding scheme based on graph colorings to achieve this rate. In these proposed coding schemes, one needs to compute the minimum entropy coloring (a coloring random variable which minimizes the entropy) of a characteristic graph. In general, finding this coloring is an NP-hard problem. However, in Chapter 4, we show that depending on the characteristic graph's structure, there are some interesting cases where finding the minimum entropy coloring is not NP-hard, but tractable and practical. In one of these cases, we show that, by having a non-zero joint probability condition on random variables' distributions, for any desired function, finding the minimum entropy coloring can be solved in polynomial time. In another case, we show that if the desired function is a quantization function, this problem is also tractable. We also consider this problem in a general…

Color ordering in QCD

We derive color decompositions of arbitrary tree and one-loop QCD amplitudes into color-ordered objects called primitive amplitudes. Furthermore, we derive general fermion flip and reversion identities spanning the null space among the primitive amplitudes and use them to prove that all color-ordered tree amplitudes of massless QCD can be written as linear combinations of color-ordered tree amplitudes of $\mathcal{N}=4$ super Yang-Mills theory.

Graceful labeling for mushroom trees

One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261–290, 1973) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star.

Competitive Phylogenetics

This exercise demonstrates the principle of parsimony in constructing cladograms. Although it is designed using mammalian cranial characters, the activity could be adapted for characters from any group of organisms. Students score categorical traits on skulls and record the data in a spreadsheet. Using the Mesquite software package, students generate arbitrary cladograms and measure tree length. They then move taxa around to reduce tree length. The exercise can become competitive when students report out on tree lengths and try to achieve shorter trees than their peers. The resulting cladograms can be compared with a published mammalian phylogeny. The exercise illustrates phylogenetics, the principle of parsimony, and hypothesis testing using morphological data.

DECOR: Distributed construction of load balanced routing trees for many to one sensor networks

Many sensor networks suffer from the energy hole problem which is a special case of load imbalance caused by the funnelling effect of many sensor nodes transmitting their data to a single, central sink. In order to mitigate the problem, a balanced routing tree is often required and this can be constructed with either a centralised or distributed algorithm. Distributed solutions are typically less effective but are significantly cheaper than centralised solutions in terms of communication overhead and they scale better for the same reason.In this paper we propose a novel distributed algorithm for the construction of a load balanced routing tree. Our proposed solution, Degree Constrained Routing, is unique in that it aims to maximise global balance during construction rather that relying on rebalancing an arbitrary tree or only maximising local balance. The underlying principle is that if all nodes adopt the same number of children as each other while the routing tree grows, then the final tree will be globally balanced. Simulation results show that our algorithm can produce trees with improved balance which results in lifetimes increased by up to 80% compared to the next best distributed algorithm.

THEORY OF DIGITAL FILTER BANKS REALIZED VIA MULTIVARIATE EMPIRICAL MODE DECOMPOSITION

Undecimated and decimated multivariate empirical mode decomposition filter banks (MEMDFBs) are introduced in order to incorporate MEMD equipped with downsampling into any arbitrary tree structure and provide flexibility in the choice of frequency bands. Undecimated MEMDFBs show the same results as those of original MEMD for an octave tree structure. Since the exact cut-off frequencies of MEMD are not known (i.e. due to data-driven decomposition), employing just simple downsampling in MEMD might cause aliasing. However, decimated MEMDFBs in this paper achieve perfect reconstruction with aliasing cancelled for any arbitrary tree. Applications of decimated/undecimated MEMDFBs for speech/audio and image signals are also included. Since decimated MEMDFBs can be applied into any arbitrary tree structure, this extends into MEMD packets. Arbitrary tree structures in decimated MEMDFBs also lead to more diverse choices in frequency bands for various multivariate applications requiring decimations.

Time versus space trade-offs for rendezvous in trees

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. The main result of this paper is a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with $$k$$ memory bits, we show that optimal rendezvous time is $$\Theta (n+n^2/k)$$ in $$n$$ -node trees. More precisely, if $$k \ge c\log n$$ , for some constant $$c$$ , we design agents accomplishing rendezvous in arbitrary trees of size $$n$$ (unknown to the agents) in time $$O(n+n^2/k)$$ , starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time $$o(n+n^2/k)$$ , even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a $$n$$ -node line.