Logarithmic Diameter Research Articles

Large expanders in high genus unicellular maps

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.Mathematics Subject Classifications: 05C10, 05C48, 60C05, 60D05Keywords: Combinatorial maps, high genus, expander graphs

Logarithmic diameter bounds for some Cayley graphs

Abstract Let S ⊂ GL n ⁢ ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

Local routing in a tree metric 1-spanner

Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight \((1+\epsilon )\)-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after \(O(\log n)\) hops and requires \(O(\varDelta \log n)\) bits of storage per vertex where \(\varDelta \) is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.

An Efficient Exchanged Hyper Cube for Parallel and Distributed Network

Enormously parallel distribution memory designs are accepting and expanding regard to satisfy the expanding need on processing power. Numerous topologies have been projected for interconnecting the processors of distributed computing systems. The hypercube topology has attracted significant consideration because of a significant number of attractive properties. The engaging properties of the hypercube topology, for example, vertex and edge balance, recursive structure, logarithmic diameter, maximally fault-tolerance, simple routing and broadcasting, and the capacity to recreate other interconnection systems with least overhead have made it a brilliant possibility for some parallel processing applications. Numerous varieties of the hypercube topology have been accounted for the literature, mostly to add the computational power of the hypercube. One of the gorgeous versions of the hypercube was introduced for the improvement of the presented Exchanged hypercube. An Exchanged hypercube has the equivalent structural complexities of the hypercube. It protects the gorgeous properties of the hypercube and diameter the communication time by dropping the diameter by a factor of two. This paper presents the fundamental communication and some of the essential operations normally required in parallel computing on the Exchanged hypercube interconnection networks.

Cube-connected circulants: Bisection width, Wiener and forwarding indices

The family of cube-connected circulants has been recently introduced as an efficient model for communication networks. A cube-connected circulant has many favourable features including vertex-transitivity, logarithmic diameter, and recursive construction in general. However, our knowledge about many other characteristics of this family of graphs is limited. In this paper, we compute nearly matching bounds for the Wiener index, total distance, bisection width, vertex- and edge-forwarding indices of cube-connected circulants. Furthermore, we introduce the class of ‘partition-proportional graphs’. This gives a more general classification of graphs than the ‘orbit proportional’ classification in the literature. Then we develop results to obtain bounds for their edge-forwarding index. Using our results, we obtain tight bounds for the edge-forwarding index of cube-connected circulants and multiplicative circulants. We show that cube-connected circulants outperform other well-known network models in terms of maximum network throughput.

Topology-Hiding Computation on All Graphs

A distributed computation in which nodes are connected by a partial communication graph is called topology hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes (Moran–Orlov–Richelson, TCC’15; Hirt et al., Crypto’16) as well as for other graph families, such as cycles, trees, and low circumference graphs (Akavia–Moran, Eurocrypt’17), but the feasibility question for general graphs was open. In this work, we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under either the decisional Diffie–Hellman or quadratic residuosity assumption. Our techniques employ random or deterministic walks to generate paths covering the graph, upon which we apply the Akavia–Moran topology-hiding broadcast for chain graphs (paths). To prevent topology information revealed by the random walk, we design multiple graph-covering sequences that, together, are locally identical to receiving at each round a message from each neighbor and sending back a processed message from some neighbor (in a randomly permuted order).

The Impact of a Sparse Migration Topology on the Runtime of Island Models in Dynamic Optimization

Island models denote a distributed system of evolutionary algorithms which operate independently, but occasionally share their solutions with each other along the so-called migration topology. We investigate the impact of the migration topology by introducing a simplified island model with behavior similar to lambda islands optimizing the so-called Maze fitness function (Kötzing and Molter in Proceedings of parallel problem solving from nature (PPSN XII), Springer, Berlin, pp 113–122, 2012). Previous work has shown that when a complete migration topology is used, migration must not occur too frequently, nor too soon before the optimum changes, to track the optimum of the Maze function. We show that using a sparse migration topology alleviates these restrictions. More specifically, we prove that there exist choices of model parameters for which using a unidirectional ring of logarithmic diameter as the migration topology allows the model to track the oscillating optimum through nMaze-like phases with high probability, while using any graph of diameter less than cln n for some sufficiently small constant c>0 results in the island model losing track of the optimum with overwhelming probability. Experimentally, we show that very frequent migration on a ring topology is not an effective diversity mechanism, while a lower migration rate allows the ring topology to track the optimum for a wider range of oscillation patterns. When migration occurs only rarely, we prove that dense migration topologies of small diameter may be advantageous. Combined, our results show that the sparse migration topology is able to track the optimum through a wider range of oscillation patterns, and cope with a wider range of migration frequencies.

Microstructure Contributions to Vibrational Damping and Identification of Damage Mechanisms in Arundo Donax L: Reed Cane for Woodwind Instruments

ABSTRACTNatural cane reeds (Latin name Arundo Donax L and here termed ADL) have been used on woodwind instruments for centuries with little change. The reed acts as a mechanical valve controlling the energy input into the musical instrument and it is the musician’s first option for altering the instrument’s sound and response characteristics. Despite this, their consistency, variable performance, durability and sensitivity to ambient conditions make it difficult for the musician to find and maintain a reed that responds to their liking. Thus it is desirable to examine the material, microstructural and anatomical properties of the reed and their contributions to vibrational performance with input from mechanical engineers, materials scientists and musicians.The current study is part of an on-going research project, and this paper presents preliminary results. In the present work raw samples of ADL obtained from a manufacturer in pre-cut form are sectioned into longitudinal and transverse specimens for mechanical characterization. Prior to testing, samples are conditioned using an incubation system to 37 degrees Celsius and 90% relative humidity, mimicking in-use conditions of the reed. Initial microstructure analysis of each specimen is completed using optical microscopy to quantify fiber spatial arrangement, size and the existence of micro-cracks along the fiber-matrix interface. X-ray diffraction is also used to quantify the fraction of crystalline cellulose present in each sample. Specimens are then excited over a specific frequency range similar to that of in-use reeds using pressure waves in a non-contact setup. Values of internal friction are obtained as logarithmic decrement values for frequency-dependent decay. One set of specimens is then subjected to cyclic mechanical loading at low frequency (< 1Hz) and stresses up to 15MPa. The other set is maintained at the given environmental conditions using the incubator and aged through temperature and humidity cycling. Comparisons of post-testing microstructure damage and internal friction measurements are then completed to delineate specific degradation mechanisms due to mechanical/fatigue deterioration and moisture cycling. Internal friction is found to be dependent on both frequency, moisture and cyclic loading. Furthermore, the existence of microstructural cracks contributes to increasing decrement values at high frequencies in both fatigued and moisture cycled samples. Statistically significant correlations are discovered between logarithmic decrement and vascular bundle orientation at 700 Hz and logarithmic decrement and parenchyma cell diameter at 1000 Hz. Reductions in internal friction below 400 Hz indicate a decreasing loss modulus (E’’) with increased moisture cycles, although this trend will be tested against a larger sample set in further work.

Analytical performance modeling of de Bruijn inspired mesh-based network-on-chips

This paper proposes and evaluates a de Bruijn inspired topology as an efficient alternative to the popular mesh topology for Network-on-Chips (NoCs). The proposed topology (1) provides logarithmic diameter for NoC, (2) offers better performance under uniform, hotspot, and matrix transpose traffic patterns and, (3) consumes lower energy for packet delivery, however, the proposed topology imposes the cost nearly equal to the mesh topology. In addition to the proposed topology, an analytical performance model is proposed in the paper to accelerate the evaluation process of the proposed topology. The model considers a network of M/G/1 queues and accurately estimates the average message latency as a widely used representative for the network performance. Results obtained from the analytical model are in good agreement with those of simulations for a wide range of working conditions such as various network sizes, different message lengths, and different traffic patterns.

Exploiting generalized de-Bruijn/Kautz topologies for flexible iterative channel code decoder architectures

Modern iterative channel code decoder architectures have tight constrains on the throughput but require flexibility to support different modes and standards. Unfortunately, flexibility often comes at the expense of increasing the number of clock cycles required to complete the decoding of a data-frame, thus reducing the sustained throughput. The Network-on-Chip (NoC) paradigm is an interesting option to achieve flexibility, but several design choices, including the topology and the routing algorithm, can affect the decoder throughput. In this work logarithmic diameter topologies, in particular generalized de-Bruijn and Kautz topologies, are addressed as possible solutions to achieve both flexible and high throughput architectures for iterative channel code decoding. In particular, this work shows that the optimal shortest-path routing algorithm for these topologies, that is still available in the open literature, can be efficiently implemented resorting to a very simple circuit. Experimental results show that the proposed architecture features a reduction of about 14% and 10% for area and power consumption respectively, with respect to a previous shortest-path routing-table-based design.

Three Round Adaptive Diagnosis in Hierarchical Multiprocessor Systems

System level diagnosis is an important technique for fault detection and location in multiprocessor computing systems. Adaptive diagnosis, proposed by Nakajima, is a practical system level diagnostic scheme; the main design objective of an adaptive diagnostic scheme is to reduce the number of test rounds, as well as the total number of tests. The hierarchical crossed cube draws upon constructions used within both the hypercube and the crossed cube, giving it many desirable features such as symmetry and logarithmic diameter, making it suitable for massively parallel systems with thousands of processors. In this paper, we first show that the diagnosability of a hierarchical crossed cube, HCCk,n, is k+n, and then propose a scheme that completely diagnoses a HCCk,n within three test rounds, and at most N+⌈(k+n+2)/2⌉ × ⌊(k+n+2)/2⌋ tests, where N=2k+2n is the number of vertices of HCCk,n. Our diagnostic scheme has the optimal number of test rounds. Moreover, most of our proofs are applicable not just to hierarchical crossed cubes but also to hierarchical interconnection networks formed by replacing crossed cubes with other appropriate interconnection networks.

An Overlapping Structured P2P for REIK Overlay Network

Abstract REIK is based on a ring which embedded an inverse Kautz digraph, to enable multi-path P2P routing. It has the constant degree and the logarithmic diameter DHT scheme with constant congestion and Byzantine fault tolerance. However, REIK did not consider the interconnection of many independent smaller networks. In this paper, we propose a new approach to build overlay network, OLS-REIK which is an overlapping structured P2P for REIK overlay network. It is a more flexible interconnecting different REIK network. Peers can belong to several rings, allowing this interconnection. By connecting smaller structured overlay networks in an unstructured way, it provides a cost effective alternative to hierarchical structured P2P systems requiring costly merging. Routing of lookup messages is performed as in REIK within one ring, but a peer belonging to several rings forwards the request to the different rings it belongs to. Furthermore a small number of across point is enough to ensure a high exhaustiveness level.

Möbius–deBruijn: The product of Möbius cube and deBruijn digraph

Möbius cube and deBruijn digraph have been proved to be two of the most popular interconnection architectures, due to their desirable properties. Some of the attractive properties of one, however, are not found in the other. The Möbius–deBruijn architecture, proposed in this paper, is the product of Möbius cube and deBruijn digraph, which is a combination of the two architectures. It employs the Möbius cube as a unit cluster and connects many such clusters by means of given number of parallel deBruijn digraphs. Consequently, the Möbius–deBruijn provides some of the desirable properties of both the architectures, such as the flexibility in terms of embedding of parallel algorithms, the high level of fault-tolerant, and the efficient inter-cluster communication. The proposed architecture also possesses the logarithmic diameter, the optimal connectivity, and the simple routing mechanism amenable to network faults. The methodology to construct the Möbius–deBruijn can apply to the product of deBruijn digraph and other hypercube-like networks, and also applies to the product of Kautz digraph and hypercube-like networks.

Pegging Graphs Yields a Small Diameter

We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degree.

The Hyper-Kautz Network: A New Scalable Product Network

In this paper, we propose a new product network using a Cartesian product of hypercube and Kautz network, named as hyper-Kautz network. The proposed network possesses the properties of both hypercube and the Kautz network including logarithmic diameter, optimal connectivity, and simple routing algorithms.

Volume Ordering for Analysis and Modeling of Vascular Systems

Morphological characteristics of vascular systems are commonly presented in terms of Strahler order because the logarithms of quantities such as vessel diameter and length are often linearly related to Strahler order. However, the ability to interpret Strahler order geometrically or physiologically is compromised because the precision of the order number is limited to integer values. This limitation is overcome by the volume ordering scheme, in which volume order number is defined as the logarithm of the estimated perfused tissue volume for each vascular segment. While Strahler and volume order numbers are equivalent for completely symmetrical branching trees, they deviate in the presence of asymmetries. The physiology-based definition of volume ordering offers benefits in the analysis of vascular design, fractal characterization of vascular systems, and blood flow modeling. These benefits are illustrated based on arterial kidney data that show a linear relationship of logarithmic vessel diameter and conductance as a function of both Strahler order and volume order with differing proportionality constants, which are expected to depend on the branching characteristics of the particular organ investigated.

Asymptotically Optimal Solutions for Small World Graphs

We consider the problem of determining constructions with an asymptotically optimal oblivious diameter in small world graphs under the Kleinberg’s model. In particular, we give the first general lower bound holding for any monotone distance distribution, that is induced by a monotone generating function. Namely, we prove that the expected oblivious diameter is Ω(log 2n) even on a path of n nodes. We then focus on deterministic constructions and after showing that the problem of minimizing the oblivious diameter is generally intractable, we give asymptotically optimal solutions, that is with a logarithmic oblivious diameter, for paths, trees and Cartesian products of graphs, including d-dimensional grids for any fixed value of d.

Node-disjoint paths in hierarchical hypercube networks

The hierarchical hypercube network is suitable for massively parallel systems. One of its appealing properties is the low number of connections per processor, which can facilitate the VLSI design and fabrication. Other alluring features include symmetry and logarithmic diameter, which can derive easy and fast algorithms for communication. In this paper, a maximal number of node-disjoint paths are constructed between every two distinct nodes of the hierarchical hypercube network. Their maximal length is not greater than max { 2 m + 1 + 2 m + 1 , 2 m + 1 + m + 4 } , where 2 m + 1 is the diameter. The effectiveness of node-disjoint paths is further verified by experiments.

Finite groups of uniform logarithmic diameter

We demonstrate the existence of an infinite family of finite groups with 2 generators and logarithmic diameter with respect to any set of generators. This answers a question of A. Lubotzky. Moreover, in our groups, all minimal sets of generators have at most 3 elements.

The k‐degree Cayley graph and its topological properties

AbstractThis article introduces a new family of Cayley graphs, called k‐degree Cayley graphs, for building interconnection networks. The k‐degree Cayley graph possesses many valuable topological properties, such as regularity with degree k, logarithmic diameter, and maximal fault tolerance. We present an optimal shortest path routing algorithm for the k‐degree Cayley graph. Cycle‐embedding and clique‐embedding are also discussed. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 47(1), 26–36 2006