Optimal Randomized Algorithms for Multipacket and Wormhole Routing on the Mesh

In this paper, we present a randomized algorithm for the multipacket (i.e., k - k) routing problem on an n x n mesh. The algorithm competes with high probability in at most kn + O(k log n) parallel communication steps, with a constant queue size of O(k). The previous best known algorithm [4] takes [5/4] kn + O([kn/f(n)]) steps with a queue size of O(k f(n)) (for any 1 ≤ f (n) ≤ n). We will also present a randomized algorithm for the wormhole model permutation routing problem for the mesh that completes in at the most kn + O(k log n) steps, with a constant queue size of O(k), where k is the number of flits that each packet is divided into. The previous best result [6] was also randomized and had a time bound of kn + O ([kn/f(n)]) with a queue size of O(k f(n)) for any 1 ≤ f(n). The two algorithms that we will present are optimal with respect to queue size. The time bounds are within a factor of two of the only known lower bound

Randomized Algorithms For Packet Routing on the Mesh

Packet routing is an important problem of parallel computing since a fast algorithm for packet routing will imply 1) fast inter-processor communication, and 2) fast algorithms for emulating ideal models like PRAMs on fixed connection machines.There are three different models of packet routing, namely 1) Store and forward, 2) Multipacket, and 3) Cut through. In this paper we provide a survey of the best known randomized algorithms for store and forward routing, k-k routing, and cut through routing on the Mesh Connected Computers

\u3cem\u3ek-k\u3c/em\u3e Routing, \u3cem\u3ek-k\u3c/em\u3e Sorting, and Cut Through Routing on the Mesh

In this paper we present randomized algorithms for k-k routing, k-k sorting, and cut through routing. The stated resource bounds hold with high probability. The algorithm for k-k routing runs in [k/2]n+o(kn) steps. We also show that k-k sorting can be accomplished within [k/2] n+n+o(kn) steps, and cut through routing can be done in [3/4]kn+[3/2]n+o(kn) steps. The best known time bounds (prior to this paper) for all these three problems were kn+o(kn). [kn/2] is a known lower bound for all the three problems (which is the bisection bound), and hence our algorithms are very nearly optimal. All the above mentioned algorithms have optimal queue length, namely k+o(k). These algorithms also extend to higher dimensional meshes

Mesh Connected Computers With Multiple Fixed Buses: Packet Routing, Sorting and Selection

Mesh connected computers have become attractive models of computing because of their varied special features. In this paper we consider two variations of the mesh model: 1) a mesh with fixed buses, and 2) a mesh with reconfigurable buses. Both these models have been the subject matter of extensive previous research. We solve numerous important problems related to packet routing, sorting, and selection on these models. In particular, we provide lower bounds and very nearly matching upper bounds for the following problems on both these models: 1) Routing on a linear array; and 2) k-k routing, k-k sorting, and cut through routing on a 2D mesh for any k ≥ 12. We provide an improved algorithm for 1-1 routing and a matching sorting algorithm. In addition we present greedy algorithms for 1-1 routing, k-k routing, cut through routing, and k-k sorting that are better on average and supply matching lower bounds. We also show that sorting can be performed in logarithmic time on a mesh with fixed buses. As a consequence we present an optimal randomized selection algorithm. In addition we provide a selection algorithm for the mesh with reconfigurable buses whose time bound is significantly better than the existing ones. Our algorithms have considerably better time bounds than many existing best known algorithms

Universal Wormhole Routing

In this paper, we examine the wormhole routing problem in terms of the “congestion” c and “dilation” d for a set of packet paths. We show, with mild restrictions, that there is a simple randomized algorithm for routing any set of P packets in O(cdη+cLηlogP) time with high probability, where L is the number of flits in a packet, and η=min{d,L}; only a constant number of flits are stored in each queue at any time. Using this result, we show that a fat-tree network of area Θ(A) can simulate wormhole routing on any network of comparable area with O(log^3 A) slowdown, when all worms have the same length. Variable-length worms are also considered. We run some simulations on the fat-tree which show that not only does wormhole routing tend to perform better than the more heavily studied store-and-forward routing in this context, but that performance superior to our provable bound is attainable in practice

Optimal Permutation Routing for Low-dimensional Hypercubes

We consider the offline problem of routing a permutation of tokens on the nodes of a d-dimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d-dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting side-result, that the number of possible communication steps in a d-dimensional hypercube is the same as the number of perfect matchings in a (d + 1)-dimensional hypercube, a combinatorial quantity for which there is no closed-form expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d. 2

\u3cem\u3eGRASP News\u3c/em\u3e, Volume 8, Number 1

A report of the General Robotics and Active Sensory Perception (GRASP) Laboratory. Edited by Thomas Lindsay

Efficient operation of coded packet networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. [109]-122).A fundamental problem faced in the design of almost all packet networks is that of efficient operation -- of reliably communicating given messages among nodes at minimum cost in resource usage. We present a solution to the efficient operation problem for coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. Such networks are in contrast to conventional, routed packet networks, where outgoing packets are restricted to being copies of received packets and where reliability is provided by the use of retransmissions. This thesis introduces four considerations to coded packet networks: 1. efficiency, 2. the lack of synchronization in packet networks, 3. the possibility of broadcast links, and 4. packet loss. We take these considerations and give a prescription for operation that is novel and general, yet simple, useful, and extensible. We separate the efficient operation problem into two smaller problems, which we call network coding -- the problem of deciding what coding operation each node should perform given the rates at which packets are injected on each link -- and subgraph selection -- the problem of deciding those rates.(cont.) Our main contribution for the network coding problem is to give a scheme that achieves the maximum rate of a multicast connection under the given injection rates. As a consequence, the separation of network coding and subgraph selection results in no loss of optimality provided that we are constrained to only coding packets within a single connection. Our main contribution for the subgraph selection problem is to give distributed algorithms that optimally solve the single-connection problem under certain assumptions. Since the scheme we propose for network coding can easily be implemented in a distributed manner, we obtain, by combining the solutions for each of the smaller problems, a distributed approach to the efficient operation problem. We assess the performance of our solution for three problems: minimum-transmission wireless unicast, minimum-weight wireline multicast, and minimum-energy wireless multicast. We find that our solution has the potential to offer significant efficiency improvements over existing techniques in routed packet networks, particularly for multi-hop wireless networks.by Desmond S. Lun.Ph.D

Energy Efficient and Cooperative Solutions for Next-Generation Wireless Networks

Energy efficiency is increasingly important for next-generation wireless systems due to the limited battery resources of mobile clients. While fourth generation cellular standards emphasize low client battery consumption, existing techniques do not explicitly focus on reducing power that is consumed when a client is actively communicating with the network. Based on high data rate demands of modern multimedia applications, active mode power consumption is expected to become a critical consideration for the development and deployment of future wireless technologies. Another reason for focusing more attention on energy efficient studies is given by the relatively slow progress in battery technology and the growing quality of service requirements of multimedia applications. The disproportion between demanded and available battery capacity is becoming especially significant for small-scale mobile client devices, where wireless power consumption dominates within the total device power budget. To compensate for this growing gap, aggressive improvements in all aspects of wireless system design are necessary. Recent work in this area indicates that joint link adaptation and resource allocation techniques optimizing energy efficient metrics can provide a considerable gain in client power consumption. Consequently, it is crucial to adapt state-of-the-art energy efficient approaches for practical use, as well as to illustrate the pros and cons associated with applying power-bandwidth optimization to improve client energy efficiency and develop insights for future research in this area. This constitutes the first objective of the present research. Together with energy efficiency, next-generation cellular technologies are emphasizing stronger support for heterogeneous multimedia applications. Since the integration of diverse services within a single radio platform is expected to result in higher operator profits and, at the same time, reduce network management expenses, intensive research efforts have been invested into design principles of such networks. However, as wireless resources are limited and shared by clients, service integration may become challenging. A key element in such systems is the packet scheduler, which typically helps ensure that the individual quality of service requirements of wireless clients are satisfied. In contrastingly different distributed wireless environments, random multiple access protocols are beginning to provide mechanisms for statistical quality of service assurance. However, there is currently a lack of comprehensive analytical frameworks which allow reliable control of the quality of service parameters for both cellular and local area networks. Providing such frameworks is therefore the second objective of this thesis. Additionally, the study addresses the simultaneous operation of a cellular and a local area network in spectrally intense metropolitan deployments and solves some related problems. Further improving the performance of battery-driven mobile clients, cooperative communications are sought as a promising and practical concept. In particular, they are capable of mitigating the negative effects of fading in a wireless channel and are thus expected to enhance next-generation cellular networks in terms of client spectral and energy efficiencies. At the cell edges or in areas missing any supportive relaying infrastructure, client-based cooperative techniques are becoming even more important. As such, a mobile client with poor channel quality may take advantage of neighboring clients which would relay data on its behalf. The key idea behind the concept of client relay is to provide flexible and distributed control over cooperative communications by the wireless clients themselves. By contrast to fully centralized control, this is expected to minimize overhead protocol signaling and hence ensure simpler implementation. Compared to infrastructure relay, client relay will also be cheaper to deploy. Developing the novel concept of client relay, proposing simple and feasible cooperation protocols, and analyzing the basic trade-offs behind client relay functionality become the third objective of this research. Envisioning the evolution of cellular technologies beyond their fourth generation, it appears important to study a wireless network capable of supporting machine-to-machine applications. Recent standardization documents cover a plethora of machine-to-machine use cases, as they also outline the respective technical requirements and features according to the application or network environment. As follows from this activity, a smart grid is one of the primary machine-to-machine use cases that involves meters autonomously reporting usage and alarm information to the grid infrastructure to help reduce operational cost, as well as regulate a customer's utility usage. The preliminary analysis of the reference smart grid scenario indicates weak system architecture components. For instance, the large population of machine-to-machine devices may connect nearly simultaneously to the wireless infrastructure and, consequently, suffer from excessive network entry delays. Another concern is the performance of cell-edge machine-to-machine devices with weak wireless links. Therefore, mitigating the above architecture vulnerabilities and improving the performance of future smart grid deployments is the fourth objective of this thesis. Summarizing, this thesis is generally aimed at the improvement of energy efficient properties of mobile devices in next-generation wireless networks. The related research also embraces a novel cooperation technique where clients may assist each other to increase per-client and network-wide performance. Applying the proposed solutions, the operation time of mobile clients without recharging may be increased dramatically. Our approach incorporates both analytical and simulation components to evaluate complex interactions between the studied objectives. It brings important conclusions about energy efficient and cooperative client behaviors, which is crucial for further development of wireless communications technologies