Optimal Parallel 5-Colouring of Planar Graphs

Hagerup, Torben; Chrobák, Marek; Diks, Krzysztof

doi:10.1137/0218020

Cited by 22 publications

(18 citation statements)

References 13 publications

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“…This result gener~lizes scheduling techniques directly and indirectly stated in recent ootgorithms for linked list ranking [6], selection [4], expression tree evaluation [6,9,18], and 5-coloring [14].…”

Section: For O(n)-operation O(logn)-time Pram Algorithmsmentioning

confidence: 77%

“…The components algorithm for planar graphs by Leiserson and Phillips [17] matches our bounds but uses n processors. Previous algorithms for 5-coloring use O(tognlog* n) time on the CRCW PRAM [13,14] and O(n 2) space on the EREW PRAM [14]. Previous algorithms for 7-coloring, maximal independent set, and maximal matching use n processors and O(logn log* n) time [13,9].…”

Section: For O(n)-operation O(logn)-time Pram Algorithmsmentioning

confidence: 98%

“…In general, SS/ is the set of degree-6 vertices in Gi = G-(S1 US2 U---t2 S'/-1) where G1 = G. This continues until the residual graph is empty. Since we implement this algorithm in the essentially the sarne way as the MIS algorithms for the CRCW PRAM in [19,13,14] and for the Et/~EW PI~AM in [9], we have the lemma below. Proof.…”

Section: Optimal Graph Algorithmsmentioning

confidence: 98%

“…Recent parallel algorithms for finding a maximal independent set in a planar graph G have used the following approach [19,13,9,14]. Find the set of vertices of degree 6 or less and call this set V6.…”

Section: Optimal Graph Algorithmsmentioning

confidence: 99%

“…An algorithm with a optimal PT-product can then be produced by scheduling these operations on S(n)/T(n) processors without increasing the time complexity. This approaeh's usefulness has been limited because current scheduling techniques are problem dependent [18,6,4,1,14] or apply only to algorithms with very restrictive properties [8,3].…”

Section: Introductionmentioning

confidence: 99%

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Optimal on-line load balancing

Shannon

1989

Proceedings of the First Annual ACM Symposium on Parallel Algorithms and Architectures

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A b s t r a c tWe present a general technique for simulating a broad class of T(n)-time and P(n)-processor algorithms on P(n)/T(n) processors using only O(T(n)) time for problems of size n. Surprisingly, this technique is not work conserving; many processors might be idle for long periods of time during the simulation. This generafizes and extends recent work on designing algorithms with optimal processor-time products for the parallel RAM model. These techniques enable us to design optimal processor-time product PRAM algorithms for the problems on planar graphs of maximal independent set, 5-coloring, 7-coloring,connected components, and maximal matching. These algorithms use linear space and from O(log n) to O(log n log* n) time, depending on the model (CRCW or EREW) and the problem. These algorithms are currently the fastest, most processor efficient, and most space efficient for these problems on the PRAM models.

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Section: For O(n)-operation O(logn)-time Pram Algorithmsmentioning

confidence: 77%

Section: For O(n)-operation O(logn)-time Pram Algorithmsmentioning

confidence: 98%

Section: Optimal Graph Algorithmsmentioning

confidence: 98%

Section: Optimal Graph Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal on-line load balancing

Shannon

1989

Proceedings of the First Annual ACM Symposium on Parallel Algorithms and Architectures

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Parallel Algorithms for Gray-Scale Digitized Picture Component Labeling on a Mesh-Connected Computer

Hambrusch

Miller

1994

Journal of Parallel and Distributed Computing

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We consider the problem of labeling connected components in a gray-scale image so that every component is connected, the maximum difference in the gray-scale values of the pixels within any component does not exceed a given value, and no component can be merged with a neighboring component. We develop two asymptotically optimal 0(n) time algorithms for generating such labelings on a mesh-connected computer when the image is mapped onto the mesh with one pixel per processor. The first algorithm operates directly on the image and is based on a divide-and-conquer approach. Although it is simple, it has the potential drawback of possibly assigning two adjacent pixels with the same gray-scale value to different components. The second algorithm avoids this potential drawback. It works wit.h a graph representation of the image and it allows larger components to be formed from smaller ones by taking into account properties and characteristics of the smaller components.

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