Models of adding relations to an organization structure of a complete K-ary tree

This paper proposes three models of adding relations to an organization structure which is a complete K-ary tree of height H: (i) a model of adding an edge between two nodes with the same depth N, (ii) a model of adding edges between every pair of nodes with the same depth N and (iii) a model of adding edges between every pair of siblings with the same depth N. For each of the three models, an optimal depth N* is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes. (c) 2005 Elsevier B.V. All rights reserved.

Optimal radio labellings of complete m-ary trees

A radio labelling of a connected graph G is a mapping f : V (G) → {0, 1, 2, ...} such that | f (u) - f (v) | ≥ diam (G) - d (u, v) + 1 for each pair of distinct vertices u, v ∈ V (G), where diam (G) is the diameter of G and d (u, v) the distance between u and v. The span of f is defined as max_u, _{v ∈ V (G)} | f (u) - f (v) |, and the radio number of G is the minimum span of a radio labelling of G. A complete m-ary tree (m ≥ 2) is a rooted tree such that each vertex of degree greater than one has exactly m children and all degree-one vertices are of equal distance (height) to the root. In this paper we determine the radio number of the complete m-ary tree for any m ≥ 2 with any height and construct explicitly an optimal radio labelling.

Comparison of K-ary N-cube and de Bruijn Overlays in QoS-Constrained Multicast Applications

Research on the construction of logical overlay networks has gained significance in recent times. This is partly due to work on peer-to-peer (P2P) systems for locating and retrieving distributed data objects, and also scalable content distribution using end-system multicast techniques. However, there are emerging applications that require the real-time transport of data from various sources to potentially many thousands of subscribers, each having their own quality-of-service (QoS) constraints. This paper primarily focuses on the properties of two popular topologies found in interconnection networks, namely k-ary n-cubes and de Bruijn graphs. The regular structure of these graph topologies makes them easier to analyze and determine possible routes for real-time data than complete or irregular graphs. We show how these overlay topologies compare in their ability to deliver data according to the QoS constraints of many subscribers, each receiving data from specific publishing hosts. Comparisons are drawn on the ability of each topology to route data in the presence of dynamic system effects, due to end-hosts joining and departing the system. Finally, experimental results show the service guarantees and physical link stress resulting from efficient multicast trees constructed over both kinds of overlay networks.

Bounds on isoperimetric values of trees

Let G = (V, E) be a finite, simple and undirected graph. For S subset of V, let delta(S, G) = {(u, v) is an element of E : u is an element of S and v is an element of V - S} be the edge boundary of S. Given an integer i, 1 <= i <= vertical bar V vertical bar, let the edge isoperimetric value of G at i be defined as b(e)(i, G) = min(S subset of V:vertical bar S vertical bar=i)vertical bar delta(S, G)vertical bar. The edge isoperimetric peak of G is defined as b(e)(G) = max(1 <= j <=vertical bar V vertical bar)b(e)(j, G). Let b(v)(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi: 10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T-d(2)), c(1)d <= b(e) (T-d(2)) <= d and c(2)d <= b(v)(T-d(2)) <= d where c(1), c(2) are constants. For a complete t-ary tree of depth d (denoted as T-d(t)) and d >= c log t where c is a constant, we show that c(1)root td <= b(e)(T-d(t)) <= td and c(2)d/root t <= b(v) (T-d(t)) <= d where c(1), c(2) are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T = (V, E, r) be a finite, connected and rooted tree - the root being the vertex r. Define a weight function w : V -> N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index eta(T) be defined as the number of distinct weights in the tree, i.e eta(T) vertical bar{w(u) : u is an element of V}vertical bar. For a positive integer k, let l(k) = vertical bar{i is an element of N : 1 <= i <= vertical bar V vertical bar, b(e)(i, G) <= k}vertical bar. We show that l(k) <= 2(2 eta+k k)

On the ordering, enumeration and ranking of k-ary trees /

"UILU-ENG 77 1706."

Generating k-ary trees lexicographically /

"UILU-ENG 77 1758."

A game-theoretic network formation model

We study the dynamics of a game-theoretic network formation model that yields large-scale small-world networks. So far, mostly stochastic frameworks have been utilized to explain the emergence of these networks. On the other hand, it is natural to seek for game-theoretic network formation models in which links are formed due to strategic behaviors of individuals, rather than based on probabilities. Inspired by Even-Dar and Kearns (2007), we consider a more realistic model in which the cost of establishing each link is dynamically determined during the course of the game. Moreover, players are allowed to put transfer payments on the formation of links. Also, they must pay a maintenance cost to sustain their direct links during the game. We show that there is a small diameter of at most 4 in the general set of equilibrium networks in our model. Unlike earlier model, not only the existence of equilibrium networks is guaranteed in our model, but also these networks coincide with the outcomes of pairwise Nash equilibrium in network formation. Furthermore, we provide a network formation simulation that generates small-world networks. We also analyze the impact of locating players in a hierarchical structure by constructing a strategic model, where a complete b-ary tree is the seed network.

M-ary trees for combinatorial asset management decision problems

A novel m-ary tree based approach is presented to solve asset management decisions which are combinatorial in nature. The approach introduces a new dynamic constraint based control mechanism which is capable of excluding infeasible solutions from the solution space. The approach also provides a solution to the challenges with ordering of assets decisions.

Secondary attribute retrieval using tree data structures

Several techniques are known for searching an ordered collection of data. The techniques and analyses of retrieval methods based on primary attributes are straightforward. Retrieval using secondary attributes depends on several factors. For secondary attribute retrieval, the linear structures—inverted lists, multilists, doubly linked lists—and the recently proposed nonlinear tree structures—multiple attribute tree (MAT), K-d tree (kdT)—have their individual merits. It is shown in this paper that, of the two tree structures, MAT possesses several features of a systematic data structure for external file organisation which make it superior to kdT. Analytic estimates for the complexity of node searchers, in MAT and kdT for several types of queries, are developed and compared.

A comparative study of multiple attribute tree and inverted file structures for large bibliographic files

A variety of data structures such as inverted file, multi-lists, quad tree, k-d tree, range tree, polygon tree, quintary tree, multidimensional tries, segment tree, doubly chained tree, the grid file, d-fold tree. super B-tree, Multiple Attribute Tree (MAT), etc. have been studied for multidimensional searching and related problems. Physical data base organization, which is an important application of multidimensional searching, is traditionally and mostly handled by employing inverted file. This study proposes MAT data structure for bibliographic file systems, by illustrating the superiority of MAT data structure over inverted file. Both the methods are compared in terms of preprocessing, storage and query costs. Worst-case complexity analysis of both the methods, for a partial match query, is carried out in two cases: (a) when directory resides in main memory, (b) when directory resides in secondary memory. In both cases, MAT data structure is shown to be more efficient than the inverted file method. Arguments are given to illustrate the superiority of MAT data structure in an average case also. An efficient adaptation of MAT data structure, that exploits the special features of MAT structure and bibliographic files, is proposed for bibliographic file systems. In this adaptation, suitable techniques for fixing and ranking of the attributes for MAT data structure are proposed. Conclusions and proposals for future research are presented.

Data mining approach identifies research priorities and data requirements for resolving the red algal tree of life

Background. The assembly of the tree of life has seen significant progress in recent years but algae and protists have been largely overlooked in this effort. Many groups of algae and protists have ancient roots and it is unclear how much data will be required to resolve their phylogenetic relationships for incorporation in the tree of life. The red algae, a group of primary photosynthetic eukaryotes of more than a billion years old, provide the earliest fossil evidence for eukaryotic multicellularity and sexual reproduction. Despite this evolutionary significance, their phylogenetic relationships are understudied. This study aims to infer a comprehensive red algal tree of life at the family level from a supermatrix containing data mined from GenBank. We aim to locate remaining regions of low support in the topology, evaluate their causes and estimate the amount of data required to resolve them. Results. Phylogenetic analysis of a supermatrix of 14 loci and 98 red algal families yielded the most complete red algal tree of life to date. Visualization of statistical support showed the presence of five poorly supported regions. Causes for low support were identified with statistics about the age of the region, data availability and node density, showing that poor support has different origins in different parts of the tree. Parametric simulation experiments yielded optimistic estimates of how much data will be needed to resolve the poorly supported regions (ca. 103 to ca. 104 nucleotides for the different regions). Nonparametric simulations gave a markedly more pessimistic image, some regions requiring more than 2.8 105 nucleotides or not achieving the desired level of support at all. The discrepancies between parametric and nonparametric simulations are discussed in light of our dataset and known attributes of both approaches. Conclusions. Our study takes the red algae one step closer to meaningful inclusion in the tree of life. In addition to the recovery of stable relationships, the recognition of five regions in need of further study is a significant outcome of this work. Based on our analyses of current availability and future requirements of data, we make clear recommendations for forthcoming research.

Development of a small-plant bioassay to assess banana grown from tissue culture for consistent infection by Fusarium oxysporum f. sp. cubense

Two reliable small-plant bioassays were developed using tissue-cultured banana, resulting in consistent symptom expression and infection by Fusarium oxysporum f. sp. cubense (Foc). One bioassay was based on providing a constant watertable within a closed pot and the second used free-draining pots. Culture medium for spore generation influenced infectivity of Foc. Inoculation of potted banana by drenching potting mix with a conidial suspension, consisting mostly of microconidia, few macroconidia and no chlamydospores, generated from one-quarter-strength potato dextrose agar + streptomycin sulfate, resulted in inconsistent infection. When a conidial suspension that consisted of all three spore types, microconidia, macroconidia and chlamydospores, prepared from spores generated on carnation leaf agar was used, all plants became infected, indicating that the spore type present in conidial suspensions may contribute to inconsistency of infection. Inconsistency of infection was not due to loss of virulence of the pathogen in culture. Millet grain precolonised by Foc as a source of inoculum resulted in consistent infection between replicate plants. Sorghum was not a suitable grain for preparation of inoculum as it was observed to discolour roots and has the potential to stunt root growth, possibly due to the release of phytotoxins. For the modified closed-pot system, a pasteurised potting mix consisting of equal parts of bedding sand, perlite and vermiculite plus 1 g/L Triabon slow release fertiliser was suitable for plant growth and promoted capillary movement of water through the potting mix profile. A suitable potting mix for the free-draining pot system was also developed.

A unified construction of space-time codes with optimal rate-diversity tradeoff

The problem of constructing space-time (ST) block codes over a fixed, desired signal constellation is considered. In this situation, there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code. The transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal-to-noise ratio (SNR). In the setting of a quasi-static channel model, let n(t) denote the number of transmit antennas and T the block interval. For any n(t) <= T, a unified construction of (n(t) x T) ST codes is provided here, for a class of signal constellations that includes the familiar pulse-amplitude (PAM), quadrature-amplitude (QAM), and 2(K)-ary phase-shift-keying (PSK) modulations as special cases. The construction is optimal as measured by the rate-diversity tradeoff and can achieve any given integer point on the rate-diversity tradeoff curve. An estimate of the coding gain realized is given. Other results presented here include i) an extension of the optimal unified construction to the multiple fading block case, ii) a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes, iii) the providing of a linear dispersion form for the underlying binary block codes, iv) a Gray-mapped version of the unified construction, and v) a generalization of construction of the S-ary case corresponding to constellations of size S-K. Items ii) and iii) are aimed at simplifying the decoding of this class of ST codes.

Automatic 3d free form shape matching using the graduated assignment algorithm.

Liu, Yonghuai. Automatic 3d free form shape matching using the graduated assignment algorithm. Pattern Recognition, vol. 38, no. 10, pp. 1615-1631, 2005.