Paint cost spectrum of perfect k-ary trees

In this paper we introduce a self-adjusting k-ary search tree scheme to implement the abstract data type DICTIONARY. Sleator and Tarjan introduced splay trees and the splay heuristic in 1983 [ST83]. They proved that the amortized time efficiency of splay trees is within a constant factor of the efficiency of both balanced binary trees (such as AVL trees) and static optimal binary trees. Sleator and Tarjan's splay heuristic is defined only for binary search trees. In this paper, we consider a self-adjustment heuristic for k-ary search trees. We present a heuristic called k-splaying and prove that the amortized number of node READs per operation in k-ary trees maintained using this heuristic is O(log2n). (Note: All constants in our time bounds are independent of both k and n). This is within a factor of O(log2k) of the amortized number of node READs required for a B-tree operation. A k-ary tree maintained using the k-splay heuristic can be thought of as a self-adjusting B-tree. It differs from a B-tree in that leaves may be at different depths and the use of space is optimal. We also prove that the time efficiency of k-splay trees is comparable to that of static optimal k-ary trees. If sequence s in a static optimal tree takes time t, then sequence s in any k-splay tree will take time O(tlog2k+n2). These two results are k-ary analogues of two of Sleator and Tarjan's results for splay trees. As part of our static optimality proof, we prove that for every static tree (including any static optimal tree) there is a balanced static tree which takes at most twice as much time on any sequence of search operations. This lemma allows us to improve our static optimality bound to O(tlog2k+nlogkn), and similarly improve Sleator and Tarjan's static optimality result.

An N-ary tree (N≥2) is a connected graph that does not contain cycles and has up to N children for any node. It can be used efficiently to represent data in well-structured hierarchical clusters and to process the data through the parent–child relationships. Several branches of a tree can be handled concurrently, the data hierarchy is described explicitly, and recursion can easily be applied. Thus this model is very appropriate for parallel high-performance computations in areas such as data processing (e.g. sort and search), priority queue management, combinatorial searches and so forth. N-ary trees have been profoundly studied (primarily for N=2) and are supported by software libraries. FPGAs have large embedded dual-port memories with programmable data width for different ports, advanced logic capabilities, and a large potential for parallelism and these features enable N-ary trees with data operations associated with their nodes to be represented more compactly and processed more efficiently in FPGAs than in software. A number of recent research efforts are dedicated to high-performance computations in electronic circuits and systems without the direct use of processing elements, which undoubtedly introduce many constraints (e.g. pre-defined operand sizes, fixed instruction sets, limited concurrency and parallelism). This chapter presents recent advances in this area and is composed of four basic parts: (1) an overview of N-ary trees, their applications, and potential varieties; (2) a discussion of common techniques for implementing and processing N-ary trees in hardware, including their representation in memory, models of computations and algorithms; (3) a description of hierarchical finite-state machines (HFSMs) with extended capabilities (with datapath, in particular) that enable N-ary trees to be processed in hardware and provide support for parallelism, hierarchy and recursion; (4) examples, practical applications, experiments and comparisons of HFSMs. The last part shows that the circuits that have been implemented are faster than the alternatives, and this conclusion is confirmed by examples and experiments in several application areas.

In federated learning, workers train local models with their private data sets and only upload local gradients to the remote aggregator. Data privacy is well preserved and parallelism is achieved. In large-scale deep learning tasks, however, frequent interactions between workers and the aggregator to transmit parameters can cause tremendous degradation of system performance in terms of communication costs, the needed number of iterations, the latency of each iteration and the accuracy of the trained model because of system “churns” (i.e., devices frequently joining and leaving the network). Existing research leverages different network topologies to improve the performance of federated learning. In this paper, we propose a novel hybrid network topology design that integrates ring (R) and n-ary tree (T) to provide flexible and adaptive convergecast in federated learning. Specifically, multiple participated peers within one-hop are formed as a local ring to adapt to device dynamics (i.e., “churns”) and carry out local cooperation shuffling; an n-ary convergecast tree is formed from local rings to the aggregator to assure the communication efficiency. Theoretical analysis shows the superiority of the proposed hybrid (R+T) convergecast design in terms of system latency as compared to existing topologies. Prototype-based simulation on CloudLab shows that the hybrid (R+T) design is able to reduce the rounds of iterations while achieving the best model accuracy under system “churns” as compared to the state of the art.

EFFECTS OF DIAGRAMS ON STRATEGY CHOICE IN PROBABILITY PROBLEM SOLVING Chenmu Xing The role of diagrammatic representations and visual reasoning in mathematics problem solving has been extensively studied. Prior research on visual reasoning and problem solving has provided evidence that the format of a diagram can modulate solvers’ interpretations of the structure and concept of the represented problem information, and influence their problem solving outcomes. In this dissertation, two studies investigated how different types of diagrams influence solvers’ choice of solution strategy and their success rate in solving probability word problems. Participants’ solution strategies suggested that problem solvers tended to construct solutions that reflect the structure of a provided diagram, resulting in different representations of the mathematical structure of the problem. For the present set of problems, a binary tree or a binary table tends to steer solvers to use a sequential-sampling strategy, which defines simple or conditional probabilities for each selection stage and calculates the intersection of these probabilities as the final probability value, using the multiplication rule of probability. This strategy choice is structurally matched with the diagrammatic structure of a binary tree or a binary table, which represents unequally-likely outcomes at the event level. In contrast, an N-byN (outcome) table steers solvers to use of an outcome-search strategy, which involves searching for the total number of target outcomes and all the possible outcomes at the equally-likely outcome level, and calculates the part-over-the-whole value as the final probability, using the classical definition of probability. This strategy is strongly cued by the N-by-N (outcome) table, because the table structure represents all equally-likely outcomes for a probability problem, and organizes the information so that the target outcomes can be seen as a subset embedded in the whole outcome space. When an N-ary (outcome) tree was provided, choices were split between the two solutions, because the N-ary tree structure not only cues searching for equally-likely outcomes but also organizes the problem information in a sequential-sampling, stage-by-stage way. Furthermore, different diagrams seem to be associated with different patterns of characteristic errors. For example, solving a combinations problem with an N-by-N table tended to elicit erroneous solutions involving miscounting those self-repeated combinations represented by the table’s diagonal cells as valid outcomes. Typical errors associated with the use of a binary tree involved incorrect value definitions of the conditional probability of the outcome of a selection. And the N-ary tree may lead to less successful coordination of all the target outcomes for the studied problems, because the target outcomes were dispersed in the outcome space depicted by the tree, thus not salient. The findings support arguments (e.g., Tversky, Morrison, & Betrancourt, 2002) that in order to promote problem solving success, a diagrammatic representation must be carefully selected or designed so that its structure and content can be well-matched to the problem structure and content. And for computational efficiency, information should be spatially organized so that it can be processed readily and accurately. In addition to the implications for effective diagram design for problem solving activities, the findings also offer important insights for probability education. It is suggested that a variety of diagram types be utilized in the educational activities for novice learners of probability, because they tend to highlight different probability concepts and structures even for the same probability topic.

Innovations are necessary to ride the inevitable tide of change. Virtualization is used in cloud computing to efficiently use the available resources. The increasing demand of efficient computing resources is leading to adaption of Service Oriented Architecture (SOA). Cloud provides computing resources in various means, including Software as a Service (SaaS), Platform as a Service (PaaS) and Infrastructure as a Service (IaaS). Cloud Computing is still at its infant stage and a very new technology, To make the cloud resource efficient for better profits as service, task scheduling plays a vital role. This research paper tackles this issue of task distribution over the cloud resources by the use of k-way virtual machine tree based task scheduling. Heterogeneous Cloud Computing Environment was created for meeting the real-time environments needs. CloudSim toolkit is used to simulate the results, the results show that the proposed algorithm gives better results in terms of response time of the cloud leading to a better QoS.

Permutation representation of k-ary trees

Extremal problems on k-ary trees with respect to the cover cost and reverse cover cost

This paper discusses the Ll -restricted k-ary trees. We represent a k-ary tree by the level numbers of its leaves. Thus every k-ary tree with n leaves corresponds to a sequence of n positive integers. We first give the necessary and sufficient conditions for a sequence representing a k-ary tree, and then present an algorithm for generating all the Ll -restricted feasible sequences lexicographically as a list.

We count the number of vertices with given outdegree in plane trees and k-ary trees, and get the following results: the total number of vertices of outdegree i among all plane trees with n edges is $${2n-i-1 \atopwithdelims ()n-1}$$ ; the total number of vertices of degree i among all plane trees with n edges is twice this number; and the total number of vertices of outdegree i among all k-ary trees with n edges is $${k\atopwithdelims ()i}{kn\atopwithdelims ()n-i}$$ . For all these results we give bijective proofs.

The paper demonstrates that N-ary trees (N>;2) can efficiently be used to model and process data in hardware. It is done through: 1) representation of data by N-ary trees; 2) compact coding of N-ary trees in memory; 3) common methods for data processing based on the model of a hierarchical finite state machine (HFSM). The proposed techniques have the following advantages: 1) similarity of processing N-ary trees with different characteristics such as the size of data M, the value N, and the depth d of trees; 2) fixed number of processing steps from the root to leaves for the given depth d; 3) the ease of reconfiguration (customization) of HFSM for different values of N, d, and M; 4) potential parallel processing of nodes' children. The results of experiments confirm effectiveness of the proposed techniques and their applicability for solving practical problems.

Enumerating k-way trees

A dynamic authenticated data structure based on k-ary trees is here proposed to improve the performance of certificate revocation in vehicular ad-hoc networks. Such a structure allows taking advantage of a duplex construction of the new standard SHA-3. In particular, efficient algorithms for search, insertion, deletion and restructuring the used k-ary trees are presented. This is a work in progress, and in the near future an implementation of a proof-of-concept prototype for smartphones will be available.

Confidence Based Learning (CBL) is a newer method of teaching and learning system that not only identify the knowledge of a typical learner but also takes care of the confidence level in the knowledge for its practical use in professional world. Unlike, other e-learning systems, the CBL first assesses a learner and identifies the knowledge level and skill gaps and the limitation of confidence level. The authors in this manuscript propose a k-ary tree structure to represent tasks and Atomic Competencies (ACs) related to an Instructional Objective (IO). Structure of Learning Object (LO) and 2-dimensional assessment technique for CBL is available, but it lacks in proper trace of progressions. A mining technique using LRS for improvised identification of limitations of a learner is also suggested in the research. In this research proposal, the authors explain how a k-ary tree may be used for implementation of the tasks and ACs so that proper diagnosis can be performed, and a customized learning content can be provided. The authors proposed an optimization technique using Huffman/Dynamic Huffman Tree to quantify the progression towards mastery. The research article emphasizes of the progressive model towards the learners’ mastery. This not only provide a learning map to the learner but also help the course designer to trace the respective learners’ performance.

Shifts and loopless generation of k-ary trees

We consider the problem of solving equations over k-ary trees. Here an equation is a pair of labeled α-ary trees, where α is a function associating an arity to each label. A solution to an equation is a morphism from α-ary trees to k-ary trees that maps the left and right hand side of the equation to the same k-ary tree.