Near-Optimal Solutions to a 2-Dimensional Placement Problem

Karp, Richard M.; McKellar, A. C.; Wong, Carmen

doi:10.1137/0204023

Cited by 35 publications

(23 citation statements)

References 10 publications

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“…2. The resulting shapes appear to approximate some "ideal" rounded shape, with better and better approximation for growing k. Karp et al [13] and Bender et al [4] study the exact nature of this shape, shown in Fig. 3.…”

Section: Related Algorithmic Workmentioning

confidence: 95%

Communication-Aware Processor Allocation for Supercomputers: Finding Point Sets of Small Average Distance

Bender

Bunde

Demaine³

et al. 2007

Algorithmica

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We give processor-allocation algorithms for grid architectures, where the objective is to select processors from a set of available processors to minimize the average number of communication hops.The associated clustering problem is as follows: Given n points in d , find a size-k subset with minimum average pairwise L 1 distance. We present a natural approximation algorithm and show that it is a 7 4 -approximation for two-dimensional grids; in d dimensions, the approximation guarantee is 2 − 1 2d , which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d, and we report on experimental results.

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Section: Related Algorithmic Workmentioning

confidence: 95%

Communication-Aware Processor Allocation for Supercomputers: Finding Point Sets of Small Average Distance

Bender

Bunde

Demaine³

et al. 2007

Algorithmica

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“…This is considered in [9] for the scenario of all processors being of the same size; even without existing modules and uniform routing cost, this turns out to be a tough problem, as noted in [8]. We hope to provide results on this scenario for modules of differing size and non-uniform routing cost in the near future.…”

Section: Resultsmentioning

confidence: 99%

Optimal Free-Space Management and Routing-Conscious Dynamic Placement for Reconfigurable Devices

Ahmadinia

Bobda

Fekete

et al. 2007

IEEE Trans. Comput.

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We describe algorithmic results on two crucial aspects of allocating resources on computational hardware devices with partial reconfigurability. By using methods from the field of computational geometry, we derive a method that allows correct maintainance of free and occupied space of a set of n rectangular modules in time O(n log n); previous approaches needed a time of O(n 2 ) for correct results and O(n) for heuristic results. We also show a matching lower bound of Ω(n log n), so our approach is optimal. We also show that finding an optimal feasible communication-conscious placement (which minimizes the total weighted Manhattan distance between the new module and existing demand points) can be computed with Θ(n log n). Both resulting algorithms are practically easy to implement and show convincing experimental behavior.

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“…For the case of a city, a particularly appropriate metric is the Manhattan distance, which is the sum of the east-west distance along streets and the north-south distance along avenues. It is interesting that while the optimal shape of a city is not circular, it is extremely close to circular [8,9].…”

Section: Introductionmentioning

confidence: 99%

“…This paper generalizes two earlier studies. Karp, McKellar, and Wong [8] consider the two special extreme cases p = 1 and p = ∞. For these cases, they determine a differential equation that describes the boundary of the optimal region that minimizes the average distance between all points in the region.…”

Section: Introductionmentioning

confidence: 99%

Optimal shape of a blob

Bender

2007

Journal of Mathematical Physics

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This paper presents the solution to the following optimization problem: What is the shape of the two-dimensional region that minimizes the average L p distance between all pairs of points if the area of this region is held fixed? [The L p distance between two points x = (x 1 , x 2 ) andVariational techniques are used to show that the boundary curve of the optimal region satisfies a nonlinear integral equation. The special case p = 2 is elementary and for this case the integral equation reduces to a differential equation whose solution is a circle. Two nontrivial special cases, p = 1 and p = ∞, have already been examined in the literature. For these two cases the integral equation reduces to nonlinear secondorder differential equations, one of which contains a quadratic nonlinearity and the other a cubic nonlinearity.

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Near-Optimal Solutions to a 2-Dimensional Placement Problem

Cited by 35 publications

References 10 publications

Communication-Aware Processor Allocation for Supercomputers: Finding Point Sets of Small Average Distance

Communication-Aware Processor Allocation for Supercomputers: Finding Point Sets of Small Average Distance

Optimal Free-Space Management and Routing-Conscious Dynamic Placement for Reconfigurable Devices

Optimal shape of a blob

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