Complete Binary Tree Research Articles

How to Make Knockout Tournaments More Popular?

Given a mapping from a set of players to the leaves of a complete binary tree (called a seeding), a knockout tournament is conducted as follows: every round, every two players with a common parent compete against each other, and the winner is promoted to the common parent; then, the leaves are deleted. When only one player remains, it is declared the winner. This is a popular competition format in sports, elections, and decision-making. Over the past decade, it has been studied intensively from both theoretical and practical points of view. Most frequently, the objective is to seed the tournament in a way that ``assists'' (or even guarantees) some particular player to win the competition. We introduce a new objective, which is very sensible from the perspective of the directors of the competition: maximize the profit or popularity of the tournament. Specifically, we associate a ``score'' with every possible match, and aim to seed the tournament to maximize the sum of the scores of the matches that take place. We focus on the case where we assume a total order on the players' strengths, and provide a wide spectrum of results on the computational complexity of the problem.

Optimization of layout for embedding half hypercube into conventional tree architectures

SummaryEmbedding a graph into another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. This has the potential to enhance the physical layout of network‐on‐chip (NoC) devices as well as to investigate their virtualization possibilities. Layout is one of the many indicators of graph embedding. An optimal layout in NoC design can result in a decreased wiring area and cost, as well as the reduction in communication delay between parallel processing components. In this work, the guest graph is the half hypercube, which possess efficient routing, fault tolerance, and Hamiltonian features. The edge isoperimetric problems is solved for the half hypercube using a novel technique called the isochronal ordering. The host graphs considered in our work are complete binary trees, ‐rooted complete binary trees, and other few predominantly investigated tree based architectures.

A modification of the random cutting model

AbstractWe propose a modification to the random destruction of graphs: given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon (J. Austral. Math. Soc.11, 1970) and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.

Induced Subgraphs and Path Decompositions

A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.

Joint Classification of Hyperspectral and LiDAR Data Using Binary-Tree Transformer Network

The joint utilization of multi-source data is of great significance in geospatial observation applications, such as urban planning, disaster assessment, and military applications. However, this approach is confronted with challenges including inconsistent data structures, irrelevant physical properties, scarce training data, insufficient utilization of information and an imperfect feature fusion method. Therefore, this paper proposes a novel binary-tree Transformer network (BTRF-Net), which is used to fuse heterogeneous information and utilize complementarity among multi-source remote sensing data to enhance the joint classification performance of hyperspectral image (HSI) and light detection and ranging (LiDAR) data. Firstly, a hyperspectral network (HSI-Net) is employed to extract spectral and spatial features of hyperspectral images, while the elevation information of LiDAR data is extracted using the LiDAR network (LiDAR-Net). Secondly, a multi-source transformer complementor (MSTC) is designed that utilizes the complementarity and cooperation among multi-modal feature information in remote sensing images to better capture their correlation. The multi-head complementarity attention mechanism (MHCA) within this complementor can effectively capture global features and local texture information of images, hence achieving full feature fusion. Then, to fully obtain feature information of multi-source remote sensing images, this paper designs a complete binary tree structure, binary feature search tree (BFST), which fuses multi-modal features at different network levels to obtain multiple image features with stronger representation abilities, effectively enhancing the stability and robustness of the network. Finally, several groups of experiments are designed to compare and analyze the proposed BTRF-Net with traditional methods and several advanced deep learning networks using two datasets: Houston and Trento. The results show that the proposed network outperforms other state-of-the-art methods even with small training samples.

Approximating Pathwidth for Graphs of Small Treewidth

We describe a polynomial-time algorithm which, given a graphGwith treewidtht, approximates the pathwidth ofGto within a ratio of\(O(t\sqrt {\log t})\). This is the first algorithm to achieve anf(t)-approximation for some functionf.Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at leastth+2 has treewidth at leasttor contains a subdivision of a complete binary tree of heighth+1. The boundth+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (withc=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constantcsuch that every graph with pathwidth Ω(kc) has treewidth at leastkor contains a subdivision of a complete binary tree of heightk.Our main technical algorithm takes a graphGand some (not necessarily optimal) tree decomposition ofGof widtht′ in the input, and it computes in polynomial time an integerh, a certificate thatGhas pathwidth at leasth, and a path decomposition ofGof width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of heighth. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC’05) for treewidth.

Tree-Depth and the Formula Complexity of Subgraph Isomorphism

For a fixed “pattern” graph , the colored -subgraph isomorphism problem (denoted by ) asks, given an -vertex graph and a coloring , whether contains a properly colored copy of . The complexity of this problem is tied to parameterized versions of and , among other questions. An overarching goal is to understand the complexity of , under different computational models, in terms of natural invariants of the pattern graph . In this paper, we establish a close relationship between the formula complexity of and an invariant known as tree-depth (denoted by). is known to be solvable by monotone formulas of size . Our main result is an lower bound for formulas that are monotone or have sublogarithmic depth. This complements a lower bound of Li, Razborov, and Rossman [SIAM J. Comput., 46 (2017), pp. 936–971] relating tree-width and circuit size. As a corollary, it implies a stronger homomorphism preservation theorem for first-order logic on finite structures [B. Rossman, An improved homomorphism preservation theorem from lower bounds in circuit complexity, in 8th Innovations in Theoretical Computer Science Conference, LIPIcs. Leibniz Int. Proc. Inform. 67, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Germany, 2017, 27]. The technical core of this result is an lower bound in the special case where is a complete binary tree of height , which we establish using the pathset framework introduced in B. Rossman [SIAM J. Comput., 47 (2018), pp. 1986–2028]. (The lower bound for general patterns follows via a recent excluded-minor characterization of tree-depth [W. Czerwiński, W. Nadara, and M. Pilipczuk, SIAM J. Discrete Math., 35 (2021), pp. 934–947; K. Kawarabayashi and B. Rossman, A polynomial excluded-minor approximation of treedepth, in Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 234–246]. Additional results of this paper extend the pathset framework and improve upon both the best known upper and lower bounds on the average-case formula size of when is a path.

Embedding hierarchical folded cubes into linear arrays and complete binary trees with minimum wirelength

Embedding hierarchical folded cubes into linear arrays and complete binary trees with minimum wirelength

Permutations encoding the local shape of level curves of real polynomials via generic projections

The non-convexity of a smooth and compact connected component of a real algebraic plane curve can be measured by a combinatorial object called the Poincare-Reeb tree associated to the curve and to a direction of projection. In this paper we show that if the chosen projection avoids the bitangents and the inflectional tangencies to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincare-Reeb tree becomes a complete binary tree and its vertices become endowed with a total order relation. Such a projection direction is called generic. We prove that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the choice of the direction of projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, that we call snakes. The snakes offer an effective description of the local geometry and topology, well-suited for further computations.

“Pass the Buck” on a Complete Binary Tree

Summary We employ the stochastic abacus to compute winning probabilities at each level of the game “Pass the Buck” on a complete binary tree with the starting vertex being the root of the tree. The derivation is also generalized to play on a complete -ary trees.

A Fast Parallelizable Algorithm for Constructing Balanced Binary Search Trees

We suggest a new non-recursive algorithm for constructing a binary search tree given an array of numbers. The algorithm has $O(N)$ time and $O(1)$ memory complexity if the given array of $N$ numbers is sorted. The resulting tree is of minimal height and can be transformed to a complete binary search tree (retaining minimal height) with $O(\log N)$ time and $O(1)$ memory. The algorithm allows simple and effective parallelization.

A Keyword-Grouping Inverted Index Based Multi-Keyword Ranked Search Scheme Over Encrypted Cloud Data

With the comprehensive development of cloud computing technology, more and more enterprises and individuals tend to outsource computing, data, and other resources to the cloud service providers to save the data management cost. Since the plaintext data outsourcing in the cloud could leak users’ private information, it is highly recommended to encrypt them before outsourcing. However, it is a challenge to perform searches over encrypted cloud data. In this paper, we adopt the keyword grouping idea into the traditional inverted index and propose a keyword-grouping inverted index (KGI-index). Based on the index, we propose a privacy-preserving KGI-index based multi-keywords ranked search scheme (KMRS). To improve the search efficiency, we adopt two strategies including grouping high relevant keywords and using the complete binary tree structure to optimize the index. The security analysis and experimental result show that the proposed scheme is a privacy-preserving and efficient multi-keyword ranked search scheme over encrypted cloud data.

Hopf dreams and diagonal harmonics

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday–Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on ν $\nu$ -trees related to ν $\nu$ -Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics.

Trees with minimum number of infima closed sets

Let T be a rooted tree, and V(T) its set of vertices. A subset X of V(T) is called an infima closed set of T if for any two vertices u,v∈X, the first common ancestor of u and v is also in X. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with n vertices is also provided.

Dynamic sampling from a discrete probability distribution with a known distribution of rates

In this paper, we consider several efficient data structures for the problem of sampling from a dynamically changing discrete probability distribution, where some prior information is known on the distribution of the rates, in particular the maximum and minimum rate, and where the number of possible outcomes N is large. We consider three basic data structures, the Acceptance–Rejection method, the Complete Binary Tree and the Alias method. These can be used as building blocks in a multi-level data structure, where at each of the levels, one of the basic data structures can be used, with the top level selecting a group of events, and the bottom level selecting an element from a group. Depending on assumptions on the distribution of the rates of outcomes, different combinations of the basic structures can be used. We prove that for particular data structures the expected time of sampling and update is constant when the rate distribution follows certain conditions. We show that for any distribution, combining a tree structure with the Acceptance–Rejection method, we have an expected time of sampling and update of Oleft( log log {r_{max}}/{r_{min}}right) is possible, where r_{max} is the maximum rate and r_{min} the minimum rate. We also discuss an implementation of a Two Levels Acceptance–Rejection data structure, that allows expected constant time for sampling, and amortized constant time for updates, assuming that r_{max} and r_{min} are known and the number of events is sufficiently large. We also present an experimental verification, highlighting the limits given by the constraints of a real-life setting.

Non-crossing monotone paths and binary trees in edge-ordered complete geometric graphs

An edge-ordered graph is a graph with a total ordering of its edges. A path \(P=v_1v_2\ldots v_k\) in an edge-ordered graph is called increasing if \((v_iv_{i+1}) < (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\); and it is called decreasing if \((v_iv_{i+1}) > (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\). We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing.Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let \(\overline{\alpha}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length \(\overline{\alpha}(G)\). Let \(\overline{\tau}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of \(\overline{\tau}(G)\) edges. In this paper we show that \(\overline \alpha(K_n) = \Omega(\log\log n)\), \(\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)\) and \(\overline \tau(K_n) = O(\sqrt{n \log n})\).

Optimal Layout of Recursive Circulant Graphs

The Recursive Circulant has several attractive topological properties. Though the embedding of parallel architectures such as Hypercubes and Mesh into Recursive Circulant have been studied, the embedding of Recursive Circulant into other architectures has not been taken up so far. In this paper, we compute the wirelength of embedding even into paths (MinLA), 1-rooted complete binary trees, regular caterpillars and ladders.

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

A parallel construction method based on linear hierarchical bounding box without stack

The intersect operation of geometry is a common method of collision detection. To achieve a fast intersect operation, a hierarchical bounding box must be quickly constructed. This paper proposes an effective space division method for constructing high-quality linear hierarchical bounding boxes through ray tracing. This method can generate more regular axial bounding boxes into a complete binary tree. This structure can be completely parallel on the GPU, and effective parallelization can ensure the fastest tree-building time. The algorithm in this paper uses a non-stack ray tracing method to increase the frame rate of the 3D model rendering in the graphics processing unit and reduce the construction time of the bounding box. The experimental analysis of construction time, frame rate, average number of intersections and other performance indicators, the experimental results show that the use of stack-free midpoint splitting method and surface heuristic splitting method to ensure rapid construction time and efficient ray traversal performance, improve the object’s light traversal efficiency.

KCB-BC-SSE: A Keyword Complete Binary Tree Searchable Symmetric Encryption Scheme using Blockchain

Internet of things (IoT) devices generate a massive amount of data every second in our lives. While cloud storage brings convenience to the users, it has the disadvantages of low scalability and weak reliability. Blockchain-based distributed storage eliminates data loss due to the tamper-resistant characteristic. Nonetheless, blockchain storage raises some confidentiality and retrieval efficiency issues of data. Searchable Symmetric Encryption (SSE) is an effective search scheme based on ciphertexts, which ensures data security. However, with the increase of data on the blockchain, the data retrieval on the blockchain will be difficult. In this paper, a Complete Binary Tree Searchable Symmetric Encryption using Blockchain (KCB-BC-SSE) scheme is proposed. According to the number of searchers, the scheme is divided into single searcher and multiple searchers where the search efficiency is improved in theory. And the simulation on ethereum shows the feasibility of our scheme. The security analysis illustrates our scheme achieves security under the chosen keyword attack.