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On the CostEffectiveness of PRAMs
, 1991
"... We introduce a formalism which allows to treat computer architecture as a formal optimization problem. We apply this to the design of shared memory parallel machines. Present computers of this type support the programming model of a shared memory. But simultaneous access to the shared memory by seve ..."
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Cited by 33 (12 self)
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We introduce a formalism which allows to treat computer architecture as a formal optimization problem. We apply this to the design of shared memory parallel machines. Present computers of this type support the programming model of a shared memory. But simultaneous access to the shared memory by several processors is in many situations processed sequentially. Asymptotically good solutions for this problem are offered by theoretical computer science. We modify these constructions under engineering aspects and improve the price/performance ratio by roughly a factor of 6. The resulting machine has surprisingly good price/performance ratio even if compared with distributed memory machines. For almost all access patterns of all processors into the shared memory, access is as fast as the access of only a single processor. 1 Introduction Commercially available parallel machines can be classified as distributed memory machines or shared memory machines. Exchange of data between different proce...
Smallest compact formulation for the permutahedron, working paper
, 2010
"... We consider the permutahedron, the convex hull of all permutations of {1, 2 · · · , n}. We show how to obtain an extended formulation for the permutahedron from any sorting network. By using the optimal AjtaiKomlósSzemerédi (AKS) sorting network, this extended formulation has Θ(n log n) variable ..."
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Cited by 28 (0 self)
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We consider the permutahedron, the convex hull of all permutations of {1, 2 · · · , n}. We show how to obtain an extended formulation for the permutahedron from any sorting network. By using the optimal AjtaiKomlósSzemerédi (AKS) sorting network, this extended formulation has Θ(n log n) variables and constraints. Furthermore, from basic polyhedral arguments, we show that any extended formulation has at least Ω(n log n) constraints. For any integer n, the permutahedron Pn is defined as the convex hull of all permutations of the set of numbers [n]: = {1, · · · , n}. In terms of a system of linear inequalities, it can be described by: Pn = {x ∈ Rn: x([n]) = g(n) x(S) ≤ g(S) ∀S: ∅ ̸ = S ⊂ [n]}, where g(k) = n∑ j=n+1−k n + 1 j = −
On the CostEffectiveness and Realization of the Theoretical PRAM Model
 SONDERFORSCHUNGSBEREICH 124 VLSI ENTWURFSMETHODEN UND PARALLELITAT, UNIVERSITAT SAARBRUCKEN
, 1991
"... Todays parallel computers provide good support for problems that can be easily embedded on the machines' topologies with regular and sparse communication patterns. But they show poor performance on problems that do not satisfy these conditions. A general purpose parallel computer should guarant ..."
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Cited by 17 (0 self)
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Todays parallel computers provide good support for problems that can be easily embedded on the machines' topologies with regular and sparse communication patterns. But they show poor performance on problems that do not satisfy these conditions. A general purpose parallel computer should guarantee good performance on most parallelizable problems and should allow users to program without special knowledge about the underlying architecture. Access to memory cells should be fast for local and non local cells and should not depend on the access pattern. A theoretical model that reaches this goal is the PRAM. But it was thought to be very expensive in terms of constant factors. Our goal is to show that the PRAM is a realistic approach for a general purpose architecture for any class of algorithms. To do that we sketch a measure of costeffectiveness that allows to determine constant factors in costs and speed of machines. This measure is based on the price/performance ratio and can be compu...
How to Sort N items using a sorting network of fixed I/O size
, 1999
"... Sorting networks of a fixed I/O size p have been used, thus far, for sorting a set of p elements. Somewhat surprisingly, the important problem of using such a sorting network for sorting arbitrarily large data sets has not been addressed in the literature. Our main contribution is to propose a si ..."
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Cited by 8 (1 self)
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Sorting networks of a fixed I/O size p have been used, thus far, for sorting a set of p elements. Somewhat surprisingly, the important problem of using such a sorting network for sorting arbitrarily large data sets has not been addressed in the literature. Our main contribution is to propose a simple sorting architecture whose main feature is the pipelined use of a sorting network of fixed I/O size p to sort an arbitrarily large data set of N elements. A noteworthy feature of our design is that no extra data memory space is required, other than what is used for storing the input. As it turns out, our architecture is feasible for VLSI implementation and its time performance is virtually independent of the cost and depth of the underlying sorting network. Specifically, we show that by using our design N elements can be sorted in ) time without memory access conflicts. Finally, we show how to use an AT optimal sorting network of fixed I/O size p to construct a similar architecture that sorts N elements in Key Words: computer architecture, sorting, parallel processing, pipelined processing, sorting networks.
RealTime Emulations of BoundedDegree Networks
 Information Processing Letters
, 1998
"... this paper, we survey the state of the art in realtime network emulations. In particular, we consider emulation schemes whereby a host network of one type can mimic, in a stepbystep fashion, any computation that can be performed by a guest network of another type. An emulation is called realtime ..."
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Cited by 5 (3 self)
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this paper, we survey the state of the art in realtime network emulations. In particular, we consider emulation schemes whereby a host network of one type can mimic, in a stepbystep fashion, any computation that can be performed by a guest network of another type. An emulation is called realtime if sizes of the guest and the host are equal, to within a constant factor, and the time required by the host and the time used by the guest are also equal, to within a constant factor. We restrict our attention in this paper to boundeddegree
Matching nuts and bolts in O(n log n) time
 SODA: 7th ACMSIAM Symposium on Discrete Algorithms
, 1996
"... ..."
Sorting Omega Networks Simulated with P Systems: Optimal Data Layouts
"... The paper introduces some sorting networks and their simulation with P systems, in which each processor/membrane can hold more than one piece of data, and perform operations on them internally. Several data layouts are discussed in this context, and an optimal one is proposed, together with its impl ..."
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Cited by 2 (2 self)
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The paper introduces some sorting networks and their simulation with P systems, in which each processor/membrane can hold more than one piece of data, and perform operations on them internally. Several data layouts are discussed in this context, and an optimal one is proposed, together with its implementation as a P system with dynamic communication graphs.
An size faulttolerant sorting network
 In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing
, 1996
"... Abstract This thesis studies sorting circuits, networks, and PRAM algorithms that are tolerant to faults. We consider both worstcase and random fault models, although we mainly focus on the more challenging problem of random faults. In the random fault model, the circuit, network, or algorithm is r ..."
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Abstract This thesis studies sorting circuits, networks, and PRAM algorithms that are tolerant to faults. We consider both worstcase and random fault models, although we mainly focus on the more challenging problem of random faults. In the random fault model, the circuit, network, or algorithm is required to sort all ninput permutations with probability at least 1 \Gamma 1n even if the result of each comparison is independently faulty with probability upper bounded by a fixed constant. In particular, ffl we construct a passivefaulttolerant sorting circuit with O(n log n log log n) comparators, thereby answering an open question posed by Yao and Yao in 1985, ffl we construct a reversalfaulttolerant sorting network with O(n loglog2 3 n) comparators, thereby answering an open question posed by Assaf and Upfal in 1990, ffl we design an optimal O(log n)step O(n)processor deterministic EREW PRAM faulttolerant sorting algorithm, thereby answering an open question posed by Feige, Peleg, Raghavan, and Upfal in 1990, and ffl we prove a tight lower bound of \Omega (n log2 n) on the number of comparators needed for any destructivefaulttolerant sorting or merging network, thereby answering an open question posed by Assaf and Upfal in 1990.
Valiant’s PolynomialSize Monotone Formula for Majority
, 2011
"... Summary: This text provides an exposition of Valiant’s proof of the existence of polynomialsize monotone formula for Majority. The exposition follows the main principles of Valiant’s proof, but deviates from it in some details. While it is easy to construct quasipolynomialsize monotone formulae f ..."
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Summary: This text provides an exposition of Valiant’s proof of the existence of polynomialsize monotone formula for Majority. The exposition follows the main principles of Valiant’s proof, but deviates from it in some details. While it is easy to construct quasipolynomialsize monotone formulae for majority (by relying on divideandconquer approaches) 1, it is less obvious how to construct polynomialsize formulae (let alone monotone ones; cf. [4] and the references therein). Notation. Suppose, for simplicity that n is odd, and consider the majority function MAJ: {0,1} n → {0,1} defined as MAJ(x) = 1 if wt(x)> n/2 and MAJ(x) = 0 otherwise, where wt(x) = {i ∈ [n] : xi = 1} denotes the Hamming weight of x = x1 · · · xn. Theorem 1 There exist polynomialsize monotone formulae for computing majority. The existence of polynomialsize (monotone) formulae is known to be equivalent to the existence of logarithmicdepth (monotone) circuits of bounded fanin. 2 Anyhow, we shall prove the existence of logarithmicdepth monotone formulae (of bounded fanin) for majority. Actually, two radically different proofs are known: The first proof uses a rather complicated construction of sorting networks of logarithmic depth [1, 2]. 3 The second proof, presented below, uses the probabilistic method.
Spiking Neural P Systems  A Natural Model for Sorting Networks
"... This paper proposes two simulations of sorting networks with spiking neural P systems. A comparison between different models is also made. ..."
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This paper proposes two simulations of sorting networks with spiking neural P systems. A comparison between different models is also made.