OPTIMAL PARALLEL ENCODING AND DECODING ALGORITHMS FOR TREES

In integrated circuit (IC) design for advanced manufacturing processes, iterative improvement processes are an approach commonly used for improving design quality, particularly for low-energy, high-frequency circuits. Layout-dependent effects (LDEs) affect IC design and fabrication, resulting in differences in circuit functionality and performance in presimulation versus postsimulation. These differences can greatly increase the length of design iteration cycles. Foundries have attempted to accelerate the iterative design process by providing process design kit libraries to IC designers. However, these kits can neither fully mitigate the negative affect of LDEs on design for manufacturability (DFM) nor eliminate the difference between presimulation and postsimulation results. To address this problem, this study proposed a novel algorithm-based design process in which an IC designer can use layout parasitic extraction to extract the DFM parameters of all components in a circuit prior to the routing process; these parameters include netlists that describe the internal connectivity of components as well as their interconnectivity with other components. Accordingly, the IC designer can determine the effects of DFM parameters on the circuit and modify them in advance if necessary. When the LDEs generated by device placement have been confirmed to not affect circuit properties, physical verification of the routing of metal wires in the IC layout can be performed. In this verification, postsimulations only need to focus on problems regarding the wire loading and timing effects of metal routing. The proposed design process is useful for mitigating LDEs in IC design and considerably reduces the time required for iterative improvement processes. Specifically, this study provides a method of enhancing design iteration cycles and provides guidelines for analyzing factors that hinder IC device functionality and performance. In sum, various problems can be resolved by separately extracting the DFM parameters associated with LDEs and parasitic parameters associated with routing.

This paper examines the use of binary trees in the design of efficient parallel algorithms. Using binary trees, we develop efficient algorithms for several scheduling problems. The shared memory model for parallel computation is used. Our success in using binary trees for parallel computations, indicates that the binary tree is an important and useful design tool for parallel algorithms.

3-Dimensional reconstruction (3D reconstruction) generates a 3D computer model of a real object or scene from data such as images, it involves many stages and open problems. Existing methods focus on point clouds and reconstructed polygonal mesh within Manhattan-world constrains in urban scenes reconstruction. However, when dealing with structures like steel truss cable-stayed bridges with complex topology (i.e., connectivity and genus), existing methods fail to recover an appealing polygonal mesh from highly unstructured and noisy point clouds. A topology-aware 3D reconstruction method which can obtain high-level structures and low-level shapes is proposed in this paper. A convolutional neural network and point cloud network is designed to encode multi-view images and 3D point cloud into a compact code, which is then decoded into structure layouts (i.e., a hierarchical binary structural parsing tree) and 3D shapes (i.e., leaf nodes on the binary tree) by designing a recursive neural network and a distance field network respectively. These high-level structures and low-level shapes constitute a 3D digital model.

The height of a tree with n nodes, that is the number of nodes on a maximal simple path starting at the root, is of interest in computing because it represents the maximum size of the stack used in algorithms that traverse the tree. In the classical paper of de Bruijn, Knuth and Rice, there is computed the average height of planted plane trees with n nodes assuming that all n-node trees are equally likely. The first section of this paper is devoted to the computation of the cumulative distribution function of this problem; we give an asymptotic equivalent in terms of familiar functions (Theorem 1). Then we derive an explicit expression and an asymptotic equivalent for the sth moment about origin of this distribution (Theorem 2). In the last section we compute the average stack size after t units of time during postorder-traversing of a binary tree with n leaves. Thereby, in one unit of time, a node is stored in the stack or is removed from the top of the stack.KeywordsBinary TreeAverage HeightSimple PolisArithmetical FunctionAsymptotic CaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We investigate the search cost in terms of number of messages generated for routing queries in tree-based P2P structured systems including binary and k-ary tree structures with different arities and different degrees of imbalance in the tree shape. This work is motivated by the fact that k-ary balanced tree access structures can greatly reduce the number of hops for searching compared to the binary trees. We study to what extent the same fact is true when the tree-like structures for access in P2P environments are unbalanced. Another important issue related to P2P environments is how to build these structures in a self-organizing way. We propose a mechanism for constructing k-ary tree based decentralized access structure in a self-organizing way and based on local interactions only. The ability to search efficiently also on unbalanced k-ary trees opens interesting opportunities for load balancing as has been shown in earlier work on P-Grid, our approach to structured P2P systems.KeywordsCommunication CostAccess StructureSearch CostDistribute Hash TableRout Table EntryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Restricted rotation distance between binary trees

Design of data structure is valuable to the concrete realization of DNA computer.Therefore,a method of designing a binary tree based on sequential storage model in DNA computer was proposed,which utilized the biological characteristics of DNA molecules and restriction endonucleases to complete sequential storage structure and basic operations of the binary tree.All the biological technology mentioned here could be practically implemented in the laboratory.To prove the feasibility of this method,an actual binary tree with detailed nucleotide encodings and an instance of algorithm were given.The process of an algorithm implemented on this binary tree was demonstrated.

Computer-aided design (CAD) tools play a critical role in optimizing integrated circuit (IC) designs, especially as the complexity and scale of ICs continue to grow. This paper introduces a via-aware parallel layer assignment algorithm to address the key challenges of routing and layer assignment in large-scale IC designs. The algorithm utilizes parallel processing techniques to divide the IC layout into distinct regions, improving load balancing and reducing routing congestion. Additionally, the via-aware strategy minimizes the number of vias used during layer assignment, significantly lowering manufacturing costs. Experimental data show that this method reduces runtime by approximately 22% and via count by 15% compared to traditional algorithms. Notably, when handling complex test cases, the algorithm demonstrates a substantial increase in efficiency, with a runtime speedup ratio of 2.16 and a via count reduction of 11.60% in most test cases. These results indicate that the proposed algorithm is highly effective in optimizing IC design for large-scale applications, with significant practical value.

In this paper, the problem of A-order of binary tree is studied with the PRAM(Parallel Random Access Machine) model of parallel computation and a parallel algorithm for A-order of binary tree is proposed. The process of the parallel algorithm is proceed with detailed description and verified analysis with an application instance. The parallel algorithm of A-order of binary tree provides using and reference for applying it to binary tree traverse sequence and solving the parallelism problem of application program.

We design an efficient sublinear time parallel construction of optimal binary search trees. The efficiency of the parallel algorithm corresponds to its total work (the product time × processors). Our algorithm works in O(n1−ɛ log n) time with the total work O(n2−2ɛ), for an arbitrarily small constant 0 < e ≤ 1/2. This is optimal within a factor n2ɛ with respect to the best known sequential algorithm given by Knuth, which needs only O(n2) time due to a monotonicity property of optimal binary search trees, see [6]). It is unknown how to explore this property in an efficient NC construction of binary search trees. Here we show that it can be effectively used in sublinear time parallel computation. Our improvement also relies on the use (in independently processed small subcomputations) of the parallelism present in Knuth's algorithm. The best known sublinear time algorithms for the construction of binary search trees (as an instance of a more general problem) have O(n3) work for time larger than n3/4, see [3] and [7]. For time √n these algorithms need n4 work, while our algorithm needs for this time only n3 work, thus improving the known algorithms by a linear factor. Also if time is O(n1−ɛ) and e is very small our improvement is close to O(n). Such improvement is similar to the one implied by the monotonicity property in sequential computations (from n3 sequential time for a more general dynamic programming problem to n2 time for the special case of optimal binary search trees).

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.

Phylogenetic trees are a central tool in many areas of life science and medicine. They demonstrate evolutionary patterns among species, genes, and patterns of ancestry among sets of individuals. The tree shapes and branch lengths of phylogenetic trees encode evolutionary and epidemiological information. To extract information from tree shapes and branch lengths, representation and comparison methods for phylogenetic trees are needed. Representing and comparing tree shapes and branch lengths of phylogenetic trees are challenging, for a tree shape is unlabeled and can be displayed in numerous different forms, and branch lengths of a tree shape are specific to edges whose positions vary with respect to the displayed forms of the tree shape. In this article, we introduce representation and comparison methods for rooted unlabeled phylogenetic trees based on a tree lattice that serves as a coordinate system for rooted binary trees with branch lengths and a graph polynomial that fully characterizes tree shapes. We show that the introduced tree representations and metrics provide distance-based likelihood-free methods for tree clustering, parameter estimation, and model selection and apply the methods to analyze phylogenies reconstructed from virus sequences. [Graph polynomial; likelihood-free inference; phylogenetics; tree lattice; tree metrics.].

Trees are a common structure to represent the intertask communication pattern of a parallel algorithm. In this paper, we consider the embedding of a complete binary tree in a star graph with the objective of minimizing congestion and dilation. We develop two embeddings: (i) a congestion-free, dilation-2, load-1 embedding of a level-p binary tree, and (ii) a congestion-free, dilation-2, load-2k embedding of a level-(p + k) binary tree, into an n-dimensional star graph, where and k is any positive integer. The first result offers a tree of size comparable or superior to existing results, but with less congestion and dilation. The second result provides more flexibility in the embeddable tree sizes compared to existing results. © 1999 John Wiley & Sons, Inc. Networks 33: 221–231, 1999

The event horizon is a very important concept that applies to both parallel and sequential discrete-event simulations. By exploiting the event horizon, parallel simulations can processes events optimistically in a risk-free manner (i.e., without requiring antimessages) using adaptable breathing time cycles with variable time widths. Additionally, exploiting the event horizon can significantly reduce the overhead of event list management that is common to virtually all discrete-event simulations. This paper is a continuation of work previously reported at PADS94. In that report, a complete mathematical formulation of the event horizon was derived under equilibrium conditions using the hold model. Various forms of the beta density function were consequently used to verify the predicted results of the analytic model. This second report describes how the concept of the event horizon can also be applied to event list management. By exploiting the event horizon, the performance of several priority queue data structures are improved including: linked lists, various binary trees, and heaps. A somewhat detailed description of these modified data structures along with other relevant background information is provided for completeness. Performance results for each of these priority queue data structure is provided.

Effect of selection on the topology of genealogical trees