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32
Analysis of Bernstein's Factorization Circuit
, 2002
"... In [1], Bernstein proposed a circuitbased implementation of the matrix step of the number field sieve factorization algorithm. These circuits o er an asymptotic cost reduction under the measure "construction cost × run time". We evaluate the cost of these circuits, in agreement ..."
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In [1], Bernstein proposed a circuitbased implementation of the matrix step of the number field sieve factorization algorithm. These circuits o er an asymptotic cost reduction under the measure "construction cost &times; run time". We evaluate the cost of these circuits, in agreement with [1], but argue that compared to previously known methods these circuits can factor integers that are 1.17 times larger, rather than 3.01 as claimed (and even this, only under the nonstandard cost measure).
Sorting on the OTISMesh
 Proc. 14 th Int’l Parallel and Distributed Processing Symp
, 2000
"... In this paper we present sorting algorithms on the recently introduced N 2 processor OTISMesh, a network with diameter 4 p N \Gamma 3 consisting of N connected meshes of size p N \Theta p N . We show that kk sorting can be done in 8 p N + O(N 1 3 ) steps for k = 1; 2; 3; 4 and in 2k ..."
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Cited by 13 (0 self)
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In this paper we present sorting algorithms on the recently introduced N 2 processor OTISMesh, a network with diameter 4 p N \Gamma 3 consisting of N connected meshes of size p N \Theta p N . We show that kk sorting can be done in 8 p N + O(N 1 3 ) steps for k = 1; 2; 3; 4 and in 2k p N + O(kN 1 3 ) steps for k ? 4 with constant buffersize for all k. We show how our algorithms can be modified to achieve 4 p N+O(N 1 3 ) steps for k = 1; 2; 3; 4 and k p N+O(kN 1 3 ) steps for k ? 4 in the average case. Finally we show a lower bound of maxf4 p N ; 1 p 2 k p Ng steps for kk sorting. 1. Introduction Several models for parallel machines were studied in the past and it has turned out that no model ideally fits for all applications. Especially well studied topologies is the mesh of processors or meshconnected array which is a simple architecture that fulfills the demands of VLSI technology quite well. The twodimensional mesh is ideally suited for seve...
Lattice Sensor Networks: Capacity Limits, Optimal Routing and Robustness to Failures ∗
"... We study network capacity limits and optimal routing algorithms for regular sensor networks, namely, square and torus grid sensor networks, in both, the static case (no node failures) and the dynamic case (node failures). For static networks, we derive upper bounds on the network capacity and then w ..."
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We study network capacity limits and optimal routing algorithms for regular sensor networks, namely, square and torus grid sensor networks, in both, the static case (no node failures) and the dynamic case (node failures). For static networks, we derive upper bounds on the network capacity and then we characterize and provide optimal routing algorithms whose rate per node is equal to this upper bound, thus, obtaining the exact analytical expression for the network capacity. For dynamic networks, the unreliability of the network is modeled in two ways: a Markovian node failure and an energy based node failure. Depending on the probability of node failure that is present in the network, we propose to use a particular combination of two routing algorithms, the first one being optimal when there are no node failures at all and the second one being appropriate when the probability of node failure is high. The combination of these two routing algorithms defines a family of randomized routing algorithms, each of them being suitable for a given probability of node failure.
Optimal LoadBalancing
 in Proceedings of IEEE Infocom
, 2005
"... This paper is about loadbalancing packets across multiple paths inside a switch, or across a network. It is motivated by the recent interest in loadbalanced switches. Loadbalanced switches provide an appealing alternative to crossbars with centralized schedulers. A loadbalanced switch has no sch ..."
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Cited by 5 (3 self)
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This paper is about loadbalancing packets across multiple paths inside a switch, or across a network. It is motivated by the recent interest in loadbalanced switches. Loadbalanced switches provide an appealing alternative to crossbars with centralized schedulers. A loadbalanced switch has no scheduler, is particularly amenable to optics, and  most relevant here  guarantees 100% throughput. A uniform mesh is used to loadbalance packets uniformly across all 2hop paths in the switch. In this paper we explore whether this particular method of loadbalancing is optimal in the sense that it achieves the highest throughput for a given capacity of interconnect. The method we use allows the loadbalanced switch to be compared with ring, torus and hypercube interconnects, too. We prove that for a given interconnect capacity, the loadbalancing mesh has the maximum throughput. Perhaps surprisingly, we find that the best mesh is slightly nonuniform, or biased, and has a throughput of N/(2N1), where N is the number of nodes.
How Helpers Hasten hRelations
 IN EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2000
"... We study the problem of exchanging a set of messages among a group of processors using the model of simplex communication. Each processor has a unidirectional connection into a fast network. Messages may consist of different numbers of packets. Let h denote the maximum number of packets that a p ..."
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Cited by 5 (1 self)
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We study the problem of exchanging a set of messages among a group of processors using the model of simplex communication. Each processor has a unidirectional connection into a fast network. Messages may consist of different numbers of packets. Let h denote the maximum number of packets that a processor must send and receive. If all the packets need to be delivered directly, at least h communication steps are needed to solve the problem. We show that by allowing forwarding, only h +O(1) time steps are needed to exchange all the messages, and this is optimal. Our work was motivated by the importance of irregular message exchanges in distributedmemory parallel computers, but it can also be viewed as an answer to an open problem on scheduling file transfers posed by Coffmann, Garey, Johnsson, and LaPaugh in 1985.
Lattice networks: Capacity limits, optimal routing and queueing behavior
"... Abstract—Lattice networks are widely used in regular settings like grid computing, distributed control, satellite constellations, and sensor networks. Thus, limits on capacity, optimal routing policies, and performance with finite buffers are key issues and are addressed in this paper. In particular ..."
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Abstract—Lattice networks are widely used in regular settings like grid computing, distributed control, satellite constellations, and sensor networks. Thus, limits on capacity, optimal routing policies, and performance with finite buffers are key issues and are addressed in this paper. In particular, we study the routing algorithms that achieve the maximum rate per node for infinite and finite buffers in the nodes and different communication models, namely uniform communications, central data gathering and border data gathering. In the case of nodes with infinite buffers, we determine the capacity of the network and we characterize the set of optimal routing algorithms that achieve capacity. In the case of nodes with finite buffers, we approximate the queue network problem and obtain the distribution on the queue size at the nodes. This distribution allows us to study the effect of routing on the queue distribution and derive the algorithms that achieve the maximum rate. Index Terms—Border data gathering, data gathering, lattice networks, network capacity, queueing theory, routing, square grid, torus, uniform communication. I.
Algorithms for Data Migration
 ALGORITHMICA
, 2007
"... The data migration problem is the problem of computing a plan for moving data objects stored on devices in a network from one configuration to another. Load balancing or changing usage patterns might necessitate such a rearrangement ..."
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Cited by 2 (0 self)
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The data migration problem is the problem of computing a plan for moving data objects stored on devices in a network from one configuration to another. Load balancing or changing usage patterns might necessitate such a rearrangement
A Lower Bound for Oblivious Dimensional Routing.
, 2008
"... In this work we consider deterministic oblivious dimensional routing algorithms on ddimensional meshes. In oblivious dimensional routing algorithms the path of a packet depends only on the source and destination node of the packet. Furthermore packets use a shortest path with a minimal number of ..."
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In this work we consider deterministic oblivious dimensional routing algorithms on ddimensional meshes. In oblivious dimensional routing algorithms the path of a packet depends only on the source and destination node of the packet. Furthermore packets use a shortest path with a minimal number of bends. We present an Ω(kn(d+1)/2) step lower bound for oblivious dimensional kk routing algorithms on ddimensional meshes for odd d> 1 and show that the bound is tight. 1
Optimal Oblivious Routing on DDimensional Meshes
"... In this work we consider deterministic oblivious kk routing algorithms with buffer size O(k). Our main focus lie is the design of algorithms for d dimensional n n meshes, d > 1. For these networks we present asymptotically optimal O(k d ) step oblivious kk routing algorithms for all k ..."
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In this work we consider deterministic oblivious kk routing algorithms with buffer size O(k). Our main focus lie is the design of algorithms for d dimensional n n meshes, d > 1. For these networks we present asymptotically optimal O(k d ) step oblivious kk routing algorithms for all k and d > 1.